how to pronounce riemann hypothesis

July 1, 2024

The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved

Number theorists have been trying to prove a conjecture about the distribution of prime numbers for more than 160 years

By Manon Bischoff

Weiquan Lin/Getty Images

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

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The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values greater than 1 that are only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.*

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

A Periodic Table for Numbers

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 B.C.E. that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line.

The exact number of prime numbers may differ from the estimate given by the theorem, however. For example: According to the prime number theorem, there are approximately 100 / ln(100) ≈ 22 prime numbers in the interval between 1 and 100. But in reality there are 25. There is therefore a deviation of 3. This is where the Riemann hypothesis comes in. This hypothesis gives mathematicians a way to estimate the deviation. More specifically, it states that this deviation cannot become arbitrarily large but instead must scale at most with the square root of n , the length of the interval under consideration.

The Riemann hypothesis therefore does not predict exactly where prime numbers are located but posits that their appearance on the number line follows certain rules. According to the Riemann hypothesis, the density of primes decreases according to the prime number theorem, and the primes are evenly distributed according to this density. This means that there are no large areas in which there are no prime numbers at all, while others are full of them.

You can also imagine this idea by thinking about the distribution of molecules in the air of a room: the overall density on the floor is somewhat higher than on the ceiling, but the particles—following this density distribution—are nonetheless evenly scattered, and there is no vacuum anywhere.

A Strange Connection

Riemann formulated the conjecture named after him in 1859, in a slim, six-page publication (his only contribution to the field of number theory). At first glance, however, his work has little to do with prime numbers.

He dealt with a specific function, the so-called zeta function ζ( s ), an infinitely long sum that adds the reciprocal values of natural numbers that are raised to the power of s :

Even before Riemann’s work, experts knew that such zeta functions are related to prime numbers. Thus, the zeta function can also be expressed as a function of all prime numbers p as follows:

Riemann recognized the full significance of this connection with prime numbers when he used not only real values for s but also complex numbers. These numbers contain both a real part and roots from negative numbers, the so-called imaginary part.

You can imagine complex numbers as a two-dimensional construct. Rather than mark a point on the number line, they instead lie on the plane. The x coordinate corresponds to the real part and the y coordinate to the imaginary part:

Никита Воробьев/Wikimedia

The complex zeta function that Riemann investigated can be visualized as a landscape above the plane. As it turns out, there are certain points amid the mountains and valleys that play an important role in relation to prime numbers. These are the points at which the zeta function becomes zero (so-called zeros), where the landscape sinks to sea level, so to speak.

The colors represent the values of the complex zeta function, with the white dots indicating its zeros.

Jan Homann/Wikimedia

Riemann quickly found that the zeta function has no zeros if the real part is greater than 1. This means that the area of the landscape to the right of the straight line x = 1 never sinks to sea level. The zeros of the zeta function are also known for negative values of the real part. They lie on the real axis at x = –2, –4, –6, and so on. But what really interested Riemann—and all mathematicians since—were the zeros of the zeta function in the “critical strip” between 0 ≤ x ≤ 1.

In the critical strip (dark blue), the Riemann zeta function can have “nontrivial” zeros. The Riemann conjecture states that these are located exclusively on the line x = 1/2 (dashed line).

LoStrangolatore/Wikimedia ( CC BY-SA 3.0 )

Riemann knew that the zeta function has an infinite number of zeros within the critical strip. But interestingly, all appear to lie on the straight line x = 1 / 2 . Thus Riemann hypothesized that all zeros of the zeta function within the critical strip have a real part of x = 1 / 2 . That statement is actually at the crux of understanding the distribution of prime numbers. If correct, then the placement of prime numbers along the number line never deviates too much from the prime number set.

On the Hunt for Zeros

To date, billions and billions of zeta function zeros have now been examined— more than 10 13 of them —and all lie on the straight line x = 1 / 2 .

But that alone is not a valid proof. You would only have to find a single zero that deviates from this scheme to disprove the Riemann hypothesis. Therefore we are looking for a proof that clearly demonstrates that there are no zeros outside x = 1 / 2 in the critical strip.

Thus far, such a proof has been out of reach, so researchers took a different approach. They tried to show that there is, at most, a certain number N of zeros outside this straight line x = 1 / 2 . The hope is to reduce N until N = 0 at some point, thereby proving the Riemann conjecture. Unfortunately, this path also turns out to be extremely difficult. In 1940 mathematician Albert Ingham was able to show that between 0.75 ≤ x ≤ 1 there are at most y 3/5+ c zeros with an imaginary part of at most y , where c is a constant between 0 and 9.

In the following 80 years, this estimation barely improved. The last notable progress came from mathematician Martin Huxley in 1972 . “This has limited us from doing many things in analytic number theory,” Tao wrote in his social media post . For example, if you wanted to apply the prime number theorem to short intervals of the type [ x , x + x θ ], you were limited by Ingham’s estimate to θ > 1 / 6 .

Yet if Riemann’s conjecture is true, then the prime number theorem applies to any interval (or θ = 0), no matter how small (because [ x , x + x θ ] = [ x , x + 1] applies to θ = 0).

Now Maynard, who was awarded the prestigious Fields Medal in 2022 , and Guth have succeeded in significantly improving Ingham’s estimate for the first time. According to their work, the zeta function in the range 0.75 ≤ x ≤ 1 has at most y (13/25)+ c zeros with an imaginary part of at most y . What does that mean exactly? Blomer explains: “The authors show in a quantitative sense that zeros of the Riemann zeta function become rarer the further away they are from the critical straight line. In other words, the worse the possible violations of the Riemann conjecture are, the more rarely they would occur.”

“This propagates to many corresponding improvements in analytic number theory,” Tao wrote . It makes it possible to reduce the size of the intervals for which the prime number theorem applies. The theorem is valid for [ x , x + x 2/15 ], so θ > 1 / 6 = 0.166... becomes θ > 2 ⁄ 15 = 0.133...

For this advance, Maynard and Guth initially used well-known methods from Fourier analysis for their result. These are similar techniques to what is used to break down a sound into its overtones. “The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them,” Tao explained . From there, however, Maynard and Guth “do a number of clever and unexpected maneuvers,” Tao wrote.

Blomer agrees. “The work provides a whole new set of ideas that—as the authors rightly say—can probably be applied to other problems. From a research point of view, that’s the most decisive contribution of the work,” he says.

So even if Maynard and Guth have not solved Riemann’s conjecture, they have at least provided new food for thought to tackle the 160-year-old puzzle. And who knows—perhaps their efforts hold the key to finally cracking the conjecture.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

*Editor’s Note (7/9/24): This sentence was edited after posting to better clarify that prime numbers exclude 1.

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The Riemann hypothesis, a Step Closer to Being Solved

• 09/07/2024 10/07/2024
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The Biggest Problem in Mathematics Is Finally a Step Closer to Being Solved Scientific American Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed … link

The Riemann hypothesis is the most important open question in number theory—if not all of mathematics. It has occupied experts for more than 160 years. And the problem appeared both in mathematician David Hilbert’s groundbreaking speech from 1900 and among the “Millennium Problems” formulated a century later. The person who solves it will win a million-dollar prize.

But the Riemann hypothesis is a tough nut to crack. Despite decades of effort, the interest of many experts and the cash reward, there has been little progress. Now mathematicians Larry Guth of the Massachusetts Institute of Technology and James Maynard of the University of Oxford have posted a sensational new finding on the preprint server arXiv.org. In the paper, “the authors improve a result that seemed insurmountable for more than 50 years,” says number theorist Valentin Blomer of the University of Bonn in Germany.

Other experts agree. The work is “a remarkable breakthrough,” mathematician and Fields Medalist Terence Tao wrote on Mastodon , “though still very far from fully resolving this conjecture.”

The Riemann hypothesis concerns the basic building blocks of natural numbers: prime numbers, values only divisible by 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, and so on.

Every other number, such as 15, can be clearly broken down into a product of prime numbers: 15 = 3 x 5. The problem is that the prime numbers do not seem to follow a simple pattern and instead appear randomly among the natural numbers. Nineteenth-century German mathematician Bernhard Riemann proposed a way to deal with this peculiarity that explains how prime numbers are distributed on the number line—at least from a statistical point of view.

Proving this conjecture would provide mathematicians with nothing less than a kind of “periodic table of numbers.” Just as the basic building blocks of matter (such as quarks, electrons and photons) help us to understand the universe and our world, prime numbers also play an important role, not just in number theory but in almost all areas of mathematics.

There are now numerous theorems based on the Riemann conjecture. Proof of this conjecture would prove many other theorems as well—yet another incentive to tackle this stubborn problem.

Interest in prime numbers goes back thousands of years. Euclid proved as early as 300 BCE that there are an infinite number of prime numbers. And although interest in prime numbers persisted, it was not until the 18th century that any further significant findings were made about these basic building blocks.

As a 15-year-old, physicist Carl Friedrich Gauss realized that the number of prime numbers decreases along the number line. His so-called prime number theorem (not proven until 100 years later) states that approximately n / ln( n ) prime numbers appear in the interval from 0 to n . In other words, the prime number theorem offers mathematicians a way of estimating the typical distribution of primes along a chunk of the number line….

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The Riemann Hypothesis, explained

It’s been called the most difficult problem in mathematics. What is the Riemann Hypothesis?

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You remember prime numbers, right? Those numbers you can’t divide into other numbers, except when you divide them by themselves or 1?Right. Here is a 3000 year old question:

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,  p . What is  p ? 31. What is the next  p ? It’s 37. The  p  after that? 41. And then? 43. How, but… …how do you know what comes next?

Present an argument or formula which (even barely) predicts what the next prime number will be (in any given sequence of numbers), and your name will be forever linked to one of the greatest achievements of the human mind, akin to Newton, Einstein and Gödel. Figure out why the primes act as they do, and you will never have to do anything else, ever again.

Introduction

The properties of the prime numbers have been studied by many of history’s mathematical giants. From the first proof of the infinity of the primes by Euclid, to Euler’s product formula which connected the prime numbers to the zetafunction. From Gauss and Legendre’s formulation of the prime number theoremto its proof by Hadamard and de la Vallée Poussin. Bernhard Riemann still reigns as the mathematician who made the single biggest breakthrough in prime number theory. His work, all contained in an 8 page paper published in 1859made new and previously unknown discoveries about the distribution of the primes and is to this day considered to be one of the most important papers in number theory.

Since its publication, Riemann’s paper has been the main focus of prime number theory and was indeed the main reason for the proof of something called the  prime number theorem  in 1896. Since then several new proofs have been found, including elementary proofs by Selberg and Erdós. Riemann’s hypothesis about the roots of the zeta function however, remains a mystery.

How many primes are there?

Let’s start off easy. We all know that a number is either  prime  or  composite . All composite numbers are made up of, and can be broken down (factorized) into a product (a x b) of prime numbers. Prime numbers are in this way the “building blocks” or “fundamental elements” of numbers. They were proven to be infinite in number by Euclid, 300 years BCE. His elegant proof goes as follows:

Euclid’s theorem Assume that the set of prime numbers is not infinite. Make a list of all the primes. Next, let P be the product of all the primes in the list (multiply all the primes in the list). Add 1 to the resulting number, Q = P +1. As with all numbers, this number Q has to be either prime or composite: - If Q is prime, you’ve found a prime that was not in your “list of all the primes”. - If Q is not prime, it is composite, i.e made up of prime numbers, one of which, p, would divide Q (since all composite numbers are products of prime numbers). Every prime p that makes up P obviously divides P. If p divides both P and Q, then it would have to also divide the difference between the two, which is 1. No prime number divides 1, and so the number p cannot be on your list, another contradiction that your list contains all prime numbers.

There will always be another prime p not on the list which divides Q. Therefore there must be infinitely many primes numbers.

Why are primes so hard to understand?

The mere fact that any novice understands the problem I laid out above, speaks volumes about how difficult it is. Even the arithmetic properties of primes, while heavily studied, are still poorly understood. The scientific community is so confident in our lacking ability to understand how prime numbers behave that the factorization of large numbers (figuring out which two primes multiply together to make a number) is one of the the very foundations of encryption theory. Here’s one way of looking at it:

We understand composite numbers well. Those are all the non-primes. They are made up of primes, but you can easily write a formula that predicts and/or generates composites. Such a “composite filter” is called a  sieve . The most famous example is named the “Sieve of Eratosthenes” from c. 200 BCE. What it does, is simply mark the multiples of each prime up to a set limit. So, take the prime 2, and mark 4,6,8,10 and so on. Next, take 3, and mark 6,9,12,15 and so on. What you’ll be left with is only primes. Although very simple to understand, the sieve of Erathosthenes is as you can imagine, not very efficient.

One function simplifying your work significantly is 6n +/- 1. This simple function spits out all primes except 2 and 3, and removes all multiples of 3 and all even numbers. Put in for n = 1,2,3,4,5,6,7 and behold the result: 5,7,11,13,17,19,23,25,29,31,35,37,41,43. The only non-prime numbers generated by the function are 25 and 35, which can be factorized into 5 x 5 and 5 x 7, respectively. The next non-primes are, as you can imagine, 49 = 7 x 7, 55 = 5 x 11and so on. Simple right?

Illustrating this visually, I’ve used something that I’m calling “compositeladders”, a simple way to see how the composite numbers generated by the function are laid out for each prime, and combined. In the first three columns of the image below, you neatly see the prime numbers 5, 7 and 11 with each respective composite ladder up to and including 91. The chaos of the fourth column, showing how the sieve has removed all but the prime numbers, is a fair illustration of why prime numbers are so hard to understand.

Fundamental Resources

So what does this all have to do with this thing you may have heard of called the “Riemann hypothesis”? Well, said simply, in order to understand more about primes, mathematicians in the 1800s stopped trying to predict with absolute certainty where a prime number was, and instead started looking at the phenomenon of prime numbers as a whole. This  analytic  approach is what Riemann was a master of, and where his famous hypothesis was made. Before I can explain it however, it is necessary to get familiar with some fundamental resources.

The Harmonic Series

The harmonic series is an infinite series of numbers first studied by Nicholas Oresme in the 14th century. Its name relates to the concept of harmonics in music, overtones higher than the fundamental frequency of a tone. The series is as follows:

This sum was proven to be divergent by Oresme (not having a finite limit, not approaching/tending towards any particular number, but running off into infinity).

Zeta Functions

The harmonic series is a special case of a more general type of function called a  zeta function  ζ(s). The real valued zeta function is given for  r  and  n,  two real numbers:

If you put in for n = 1, you get the harmonic series, which diverges. For all values of n > 1 however, the series  converges , meaning the sum  tends towards  some number as the value of r increases, i.e it does not run off into infinity.

The Euler Product Formula

The first connection between zeta functions and prime numbers was made by Euler when he showed that for  n  and  p , two natural numbers (integers larger than zero) where  p  is prime:

This expression first appeared in a paper in 1737 entitled  Variae observationes circa series infinitas . The expression states that the  sum of the zeta function  is equal to the  product of the reciprocal of one minus the reciprocal of primes to the power s.  This astonishing connection laid the foundation for modern prime number theory, which from this point on used the zeta function ζ(s) as a way of studying primes.

The proof of the formula is one of my favorites, and so I will include it even though it is not strictly necessary for our purposes (it’s just so lovely!):

Proof of Euler’s product formula

Euler begins with the general zeta function

First, he multiplies both sides by the second term:

He then subtracts the resulting expression from the zeta function:

He repeats this process, next multiplying both sides by the third term

And then subtracting the resulting expression from the zeta function

Repeating this process to infinity, one would in the end be left with the expression:

If this process is familiar to you, it is because what Euler constructed was in fact, a sieve, much like the Sieve of Eratosthenes. He is filtering out non-prime numbers from the zeta function.

Next, divide the expression by all of the prime reciprocal terms, and obtain:

Shortened, we have shown that:

Wasn’t that beautifully done? Put in for s = 1, and find the infinite harmonic series, re-proving the infinity of the primes.

The Möbius Function

August Ferdinand Möbius later rewrote the Euler product formula to create a new sum. In addition to containing reciprocals of primes, Möbius’ function also contains every natural number that is the product of odd and even numbers of prime factors. The numbers left out of his series are those that divide by some prime squared. His sum, denoted by  μ(n)  is as follows:

The sum contains reciprocals of:

• Every prime;
• Every natural number which is a product of an odd number of different primes, prefixed by a minus sign; and
• Every natural number that is the product of an even number of different primes, prefixed by a plus sign;

Below are the first terms:

The sum does not contain reciprocals of numbers which divide by some prime squared, e.g 4,8,9 and so on.

The Möbius function  μ(n)  only takes on three possible values which either prefix (1 or -1) or remove (0) terms from the sum:

Although first formally defined by Möbius, this quirky sum was remarkably considered by Gauss in a sidenote more than 30 years earlier, when he wrote:

“The sum of all primitive roots (of a prime number p) is either  ≡  0 (when p-1 is divisible by a square), or  ≡  ±1 (mod p) (when p-1 is the product of unequal prime numbers); if the number of these is even the sign is positive, but if the number is odd, the sign is negative.”

The Prime Counting Function

Back to primes. To understand how primes are distributed as you go higher up the number line, without knowing where they are, it is useful to instead count how many there are up to a certain number.

The prime counting function π(x), introduced by Gauss, does just that, gives the number of primes less than or equal to a given real number. Given that there is no known formula for finding primes, the prime counting formula is known to us only as a plot, or  step function  increasing by 1 whenever  x  is prime. The plot belowshows the function up to x = 200.

The Prime Number Theorem

The prime number theorem, also formulated by Gauss (and Legendre, independently) states:

In English, it states: “As x goes to infinity, the prime counting function π(x) will approximate the function x/ln(x)” .  In other words, if you count high enough, and plot the number of primes up to a very large number  x,  then plot  x  divided by the natural logarithm of  x , the ratio between the two will approach 1. The two functions are plotted below for x = 1000:

In terms of probability, the prime number theorem states that if you pick a natural number x at random, the probability P(x) that that number will be a prime number is about 1 / ln(x). This means that the average gap between consecutive prime numbers among the first  x  integers is approximately ln(x).

The Logarithmic Integral Function

The function Li(x) is defined for all positive real numbers except   x = 1. It is defined by an integral from  2  to  x :

Plotting this function alongside the prime counting function and the formula from the prime number theorem, we see that Li(x) is actually a better approximation than x/ln(x):

How much better of an approximation it is, can be seen if we make a table with large values of x, the number of primes up to x and the error of the old (prime number theorem) and new (logarithmic integral) functions:

As can be easily seen here, the logarithmic integral function is far better of an approximation than the function from the prime number theorem, only “overshooting” by 314,890 primes for x = 10 to the power of 14. However, both functions do converge towards the prime counting function π(x). Li(x)   does so much faster, but as  x  goes to infinity, the ratio between the prime counting function and both functions Li(x) and x/ln(x) goes towards 1. Visualized:

The Gamma Function

The gamma function Γ(z) has been an important object of study since the problem of extending the factorial function to non-integer arguments was studied by Daniel Bernoulli and Christian Goldbach in the 1720s. It is an extension of the factorial function  n ! (1 x 2 x 3 x 4 x 5 x ….  n ), shifted down by 1:

Its plot is very curious:

The gamma function Γ(z)   is defined for all  complex  values of  z  larger than zero.Complex numbers, as you probably know, are a class of numbers with an  imaginary part,  written as Re( z ) + Im( z ), where Re( z ) is the real part (ordinary real number) and Im( z ) is the imaginary part, denoted by the letter  i . A complex number is typically written in the form  z = σ + it  where sigma  σ  is the real part and  i t is the imaginary part. Complex numbers are useful because they allow mathematicians and engineers to evaluate and work on problems where ordinary real numbers will not allow it. Visualized, complex numbers extend the traditional one-dimensional “number line” into a two-dimensional “number plane”, called the  complex plane,  in which the real part of a complex number is plotted on the x-axis and the imaginary part is plotted on the y-axis.

In order to be able to use the gamma function Γ(z), it is typically rewritten to the form

Using this identity, one can obtain values for z below zero. It does not however give values for negative integers, as they are not defined (technically they are singularities, or simple poles).

Zeta and Gamma

The link between the zeta function and the gamma function is given by the following integral:

Bernhard Riemann

Now that we’ve covered the necessary fundamental resources, we can finally begin making the connection between prime numbers and the Riemann Hypothesis.

German mathematician Bernhard Riemann was born in Breselenz in 1826. Student of Gauss, Riemann published work in the fields of analysis and geometry. His biggest contribution was likely in the field of differential geometry, where he laid the groundwork for the geometric language later used in Einstein’s General Theory of Relativity.

His sole effort in number theory, the 1859 paper  Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse , “On prime numbers less than a given magnitude” is considered the most important paper in the field. In four short pages he outlined:

• A definition of the Riemann zeta function ζ(s), a complex-valued zeta function;
• The  analytic continuation  of the zeta function to all complex numbers s≠1;
• A definition for the Riemann xi function ξ(s), an entire function related to the Riemann zeta function through the gamma function;
• Two proofs of the functional equation of the Riemann zeta function;
• A definition of the Riemann prime counting function J(x) by using the prime counting function and the Möbius function;
• An explicit formula for the number of primes less than a given number by using the Riemann prime counting function, defined using the non-trivial zeros of the Riemann zeta function.

An incredible feat of engineering and creativity, the likes of which probably hasn’t been seen since. Absolutely astounding.

The Riemann Zeta Function

We’ve seen the intimate relationship between prime numbers and the zeta function shown by Euler in his product formula. Beyond this connection however, not much was known about the relationship and it would take the invention of complex numbers to show explicitly just how interconnected the two are.

Riemann was the first to consider the zeta function ζ(s) for a complex variable  s, where  s = σ + i t .

Dubbed the Riemann zeta function ζ(s), it is an infinite series which is analytic (has definable values) for all complex numbers with real part larger than 1 (Re(s) > 1). In this area, it converges  absolutely .

In order to analyze the function in areas beyond the regular area of convergence (when the real part of the complex variable  s  is larger than 1), the function needs to be redefined. Riemann successfully does this by  analytic continuation  to an absolutely convergent function in the half plane Re(s) > 0.

This new definition for the zeta function is analytic everywhere in the half plane Re(s) > 0, except at s = 1 where there is a singularity/simple pole. This is called a  meromorphic function  in this domain, because it is  holomorphic  (complex differentiable in a neighborhood of every point in its domain) except for at the simple pole s = 1. It is also a great example of something called a  Dirichlet L-function .

In his paper, Riemann does not stop there. He goes on to analytically continue his zeta function ζ(s) to the  entire  complex plane, using the gamma function Γ(z) . In the interest of keeping this article simple, I will not show this calculation here, but I strongly urge you to read it for yourself as it demonstrates Riemann’s remarkable intuition and technique supremely well (edit 03.13.20: the calculation is available in  Veisdal (2013) pp. 28 ).

His method makes use of the integral representation of gamma Γ(z) for complex variables and something called the Jacobi theta function ϑ(x), which together can be rewritten so that the zeta function appears. Solving for zeta,

In this form, one can see that the term ψ(s)   decreases more rapidly than any power of x, and so the integral converges for all values of s.

Going even further, Riemann notices that the first term in the braces (-1 / s(1 - s) ) is invariant (does not change) if one substitutes s by 1 - s. Doing so, Riemann further extends the usefulness of the equation by removing the two poles at s=0 and s=1, and defining the Riemann xi function ξ(s) with no singularities:

Zeros of the Riemann Zeta Function

The roots/zeros of the zeta function, when ζ(s)=0, can be divided into two types which have been dubbed the “trivial” and the “non-trivial” zeros of the Riemann zeta function.

Existence of zeros with real part Re(s) < 0

The trivial zeros are the zeros which are easy to find and explain. They are most easily noticable in the following functional form of the zeta function:

This product becomes zero when the sine term becomes zero. It does so at kπ .  So, e.g for a negative even integer  s = -2n , the zeta function becomes zero. For positive even integers  s = 2n  however, the zeros are cancelled out by the poles of the gamma function Γ(z). This is easier to see in the original functional form, where if you put in for s = 2n, the first part of the term becomes undefined.

So, the Riemann zeta function has zeros at every negative even integer s = -2n. These are the trivial zeros, and they can be seen in the plot of the function below:

Existence of zeros with real part Re(s) > 1

From Euler’s product formulation of zeta, we can immediately see that zeta ζ(s) cannot be zero in the area with real part of  s  larger than 1 because a convergent infinite product can only be zero if one of its factors is zero. The proof of the infinity of the primes denies this.

Existence of zeros with real part 0 ≤ Re(s) ≤ 1

We’ve now found the trivial zeros of zeta in the negative half plane when Re(s) < 0 and shown that there cannot be any zeros in the area Re(s) > 1.

The area between these two areas however, called the critical strip, is where much of the focus of analytic number theory has taken place for the last few hundred years.

In the plot above I have graphed the real parts of zeta ζ(s) in red and the imaginary parts in blue. We see the first two trivial zeros in the lower left when the real part of  s  is -2 and -4. In between 0 and 1, I have highlighted the critical strip and marked off where the real and imaginary parts of zeta ζ(s) intersect. These are the non-trivial zeros of the Riemann zeta function. Going to higher values we see more zeros, and two seemingly random functions which appear to be getting denser as the imaginary part of  s  gets larger.

The Riemann Xi Function

We’ve defined the Riemann Xi function ξ(s) (the version of the functional equation which has removed the singularities, and so is defined for all values of s) as:

This function satisfies the relationship

Which means that the function is symmetric about the vertical line Re( s ) = 1/2 so that ξ(1) = ξ(0), ξ(2) = ξ(-1) and so on. This functional relationship (the symmetry of  s  and 1- s ) combined with the Euler product formula shows that the Riemann xi function ξ(s) can only have zeros in the range 0 ≤ Re( s ) ≤ 1. The zeros of the Riemann xi function in other words correspond to the non-trivial zeros of the Riemann Zeta function. In a sense, the critical line R(s) = 1/2 for the Riemann Zeta function ζ( s ) corresponds to the real line (Im( s ) = 0) for the Riemann xi function ξ( s ).

Looking at the two charts above, one should immediately take note of the fact that all the non-trivial zeros of the Riemann Zeta function ζ( s ) (the zeros of the Riemann xi function) have real part Re(s) equal to 1/2. Riemann briefly remarked on this phenomenon in his paper, a fleeting comment which would end up as one of his greatest legacies.

The Riemann Hypothesis

The non-trivial zeros of the Riemann zeta function ζ(s) have real part Re(s) = 1/2.

This is the modern formulation of the unproven conjecture made by Riemann in his famous paper. In words, it states that the points at which zeta is zero, ζ(s) = 0, in the critical strip 0 ≤ Re(s) ≤ 1, all have real part Re(s) = 1/2. If true, all non-trivial zeros of Zeta will be of the form ζ(1/2 +  i t).

An equivalent statement (Riemann’s actual statement) is that all the roots of the Riemann xi function ξ(s) are real.

In the plot below, the line Re(s) = 1/2 is the horizontal axis. The real part Re( s ) of zeta ζ( s ) is the red graph and the imaginary part Im( s ) is the blue graph. The non-trivial zeros are the intersections between the red and blue graph on the horizontal line.

If the Riemann hypothesis turns out to be true, all the non-trivial zeros of the function will appear on this line as intersections between the two graphs.

Reasons to Believe the Hypothesis

There are many reasons to believe the truth of Riemann’s hypothesis about the zeros of the zeta function. Perhaps the most compelling reason for mathematicians is the consequences it would have for the distribution of prime numbers. The numerical verification of the hypothesis to very high values suggests its truth. In fact, the numerical evidence for the hypothesis is far strong enough to be regarded as experimentally verified in other fields such as physics and chemistry. However, the history of mathematics contains several conjectures that had been shown numerically to very high values and still were proven false. Derbyshire (2004) tells the story of the Skewes number, a very very large number that gave an upper bound, proving the falsity of one of Gauss’ conjectures that the logarithmic integral Li( x ) is always greater than the prime counting function. It was disproven by Littlewood without an example, and then shown to must fail above Skewes’ very, very large number ten to the power of ten, to the power of ten, to the power of 34, showing that even though Gauss’ idea had been proven to be wrong, an example of exactly  where  is far beyond the reach of numerical calculation even today. This could also be the case for Riemann’s hypothesis, which has “only” been verified up to ten to the power of twelve non-trivial zeros.

The Riemann Zeta Function and Prime Numbers

Using the truth of the Riemann hypothesis as a starting point, Riemann began studying its consequences. In his paper he writes;  “…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after some fleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of my investigation.”  His next objective was relating the zeros of the zeta function to the prime numbers.

Recall the prime counting function π( x )   which denotes the number of primes up to and including a real number  x . Riemann used π( x ) to define his own prime counting function, the Riemann prime counting function J( x ). It is defined as:

The first thing to notice about this function is that it is not infinite. At some term, the counting function will be zero because there are no primes for  x  < 2 .  So, taking J(100) as an example, the function will be made up of seven terms because the eight term will include an eight root of 100, which is approximately equal to 1.778279.., so this prime counting term becomes zero and the sum becomes J(100) = 28.5333….

Like the prime counting function, the Riemann prime counting function J( x ) is a step function which increases in value when:

To relate the value of J( x ) to how many primes there are up to and including  x , we recover the prime counting function π( x )   by a process called Möbius inversion (which I will not show here). The resulting expression is

Remembering that the possible values of the Möbius function are

This means that we can now write the prime counting function as a function of the Riemann prime counting function, giving us

This new expression is still a finite sum because J( x ) is zero when  x  < 2 because there are no primes less than 2.

If we now look at our example of J(100), we get the sum

Which we know to be the number of primes below 100.

Translating the Euler Product Formula

Riemann next uses the Euler product formula as a starting point and derives a method for analytically evaluating the prime numbers in the infinitesimal language of calculus. Starting with Euler:

By first taking the logarithm of both sides, then rewriting the denominators in the parenthesis, he derives the relationship

Next, using the well known Maclaurin Taylor series, he expands each log term on the right hand side, creating an infinite sum of infinite sums, one for each term in the prime number series.

Looking at one such term, e.g:

This term, and every other term in the calculation, represents part of the area under the J( x ) function. Written as an integral:

In other words, using the Euler product formula, Riemann showed that it is possible to represent the discrete prime counting step function as a continuous sum of integrals. Below our example term is shown as part of the area under the Riemann prime counting function graph.

So, each expression in the finite sum that makes up the prime reciprocal series of Euler product formula can be expressed as integrals, making an infinite sum of integrals that correspond to the area under the Riemann prime counting function. For the prime number 3, this infinite product of integrals is:

Collecting all of these infinite sums together into one integral, the integral under the Riemann prime counting function J( x ) can be written as simply:

Or, the more popular form

Riemann had with this connected his zeta function ζ( s ) with his Riemann prime counting function J( x ) in an identity statement equivalent to the Euler product formula, in the language of calculus.

The Error Term

Having obtained his analytic version of the Euler product formula, Riemann next went on to formulate his own prime number theorem. The explicit form he gave was:

This is Riemann’s explicit formula. It is an improvement on the prime number theorem, a more accurate estimate of how many primes exist up to and including a number  x . The formula has four terms:

• The first term, or “principle term” is the logarithmic integral Li( x ), which is the better estimation of the prime counting function π( x ) from the prime number theorem. It is by far the largest term, and like we have seen earlier, an overestimate on how many primes there are up to a given value  x .
• The second term, or “periodic term” is the sum of the logarithmic integral of x to the power  ρ , summed over  ρ , which are the non-trivial zeros of the Riemann zeta function. It is the term that adjusts the overestimate of the principle term.
• The third is the constant -log(2) = -0.6993147…
• The fourth and final term is an integral which is zero for  x  < 2 because there are no primes smaller than 2. It has its maximum value at 2, when its integral equals approximately 0.1400101….

The two latter terms are infinitesimal in their contributions to the function’s value as  x  gets large. The main “contributers” for large numbers are the logarithmic integral function and the periodic sum. See their effects in the chart below:

In the chart above, I have approximated the prime counting function π( x ) by using the explicit formula for the Riemann prime counting function J( x ), and summed over the first 35 non-trivial zeros of the Riemann zeta function ζ(s). We see that the periodic term causes the function to “resonate” and begin to approach the shape of the prime counting function π( x ).

Below you can see the same chart, using more non-trivial zeros.

Using Riemann’s explicit function, one can approximate the number of primes up to and including a given number  x  to a very high accuracy. In fact, Von Koch proved in 1901 that using the non-trivial zeros of the Riemann zeta function to error-correct the logarithmic integral function is equivalent to the “best possible” bound for the error term in the prime number theorem.

“..These zeros act like telephone poles, and the special nature of Riemann’s zeta function dictates precisely how the wire — its graph — must be strung between them..” — Dan Rockmore

Since the death of Riemann in 1866 at the modest age of 39, his groundbreaking paper has remained a landmark in the field of prime- and analytic number theory. To this day Riemann’s hypothesis about the non-trivial zeros of the Riemann zeta function remains unsolved, despite extensive research by numerous great mathematicians for hundreds of years. Numerous new results and conjectures associated with the hypothesis are published each year, in the hope that one day a proof will be tangible.

This essay is an abridged version of my undergraduate thesis “Prime Numbers and the Riemann Zeta Function”. The thesis itself is  available here . For those interested in further exploring the topic, I especially recommend John Derbyshire’s book ‘ Prime Obsession ’ (Amazon Affiliate Link).

Is ‘Philosophy of Mathematics’ somehow useful?

An amazing formula, perfect numbers.

Has one of math’s greatest mysteries, the Riemann hypothesis, finally been solved?

Professor of Mathematics, University of Richmond

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Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis .

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002 .

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

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Riemann Hypothesis

A more general statement known as the generalized Riemann hypothesis conjectures that neither the Riemann zeta function nor any Dirichlet L-series has a zero with real part larger than 1/2.

Legend holds that the copy of Riemann's collected works found in Hurwitz's library after his death would automatically fall open to the page on which the Riemann hypothesis was stated (Edwards 2001, p. ix).

Proof of the Riemann hypothesis is number 8 of Hilbert's problems and number 1 of Smale's problems .

In 2000, the Clay Mathematics Institute ( http://www.claymath.org/ ) offered a $1 million prize ( http://www.claymath.org/millennium/Rules_etc/ ) for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line ), does not earn the $1 million award.

The Riemann hypothesis is equivalent to the statement that all the zeros of the Dirichlet eta function (a.k.a. the alternating zeta function)

By modifying a criterion of Robin (1984), Lagarias (2000) showed that the Riemann hypothesis is equivalent to the statement that

There is also a finite analog of the Riemann hypothesis concerning the location of zeros for function fields defined by equations such as

According to Fields medalist Enrico Bombieri, "The failure of the Riemann hypothesis would create havoc in the distribution of prime numbers" (Havil 2003, p. 205).

In Ron Howard's 2001 film A Beautiful Mind , John Nash (played by Russell Crowe) is hindered in his attempts to solve the Riemann hypothesis by the medication he is taking to treat his schizophrenia.

In the Season 1 episode " Prime Suspect " (2005) of the television crime drama NUMB3RS , math genius Charlie Eppes realizes that character Ethan's daughter has been kidnapped because he is close to solving the Riemann hypothesis, which allegedly would allow the perpetrators to break essentially all internet security.

In the novel Life After Genius (Jacoby 2008), the main character Theodore "Mead" Fegley (who is only 18 and a college senior) tries to prove the Riemann Hypothesis for his senior year research project. He also uses a Cray Supercomputer to calculate several billion zeroes of the Riemann zeta function. In several dream sequences within the book, Mead has conversations with Bernhard Riemann about the problem and mathematics in general.

Portions of this entry contributed by Len Goodman

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Riemann hypothesis

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Riemann hypothesis , in number theory , hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function , which is connected to the prime number theorem and has important implications for the distribution of prime numbers . Riemann included the hypothesis in a paper, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (“On the Number of Prime Numbers Less Than a Given Quantity”), published in the November 1859 edition of Monatsberichte der Berliner Akademie (“Monthly Review of the Berlin Academy”).

Riemann extended the study of the zeta function to include the complex numbers x + i y , where i = Square root of √ −1 , except for the line x = 1 in the complex plane. Riemann knew that the zeta function equals zero for all negative even integers −2, −4, −6,… (so-called trivial zeros) and that it has an infinite number of zeros in the critical strip of complex numbers that fall strictly between the lines x = 0 and x = 1. He also knew that all nontrivial zeros are symmetric with respect to the critical line x = 1 / 2 . Riemann conjectured that all of the nontrivial zeros are on the critical line, a conjecture that subsequently became known as the Riemann hypothesis.

In 1914 English mathematician Godfrey Harold Hardy proved that an infinite number of solutions of ζ( s ) = 0 exist on the critical line x = 1 / 2 . Subsequently it was shown by various mathematicians that a large proportion of the solutions must lie on the critical line, though the frequent “proofs” that all the nontrivial solutions are on it have been flawed. Computers have also been used to test solutions, with the first 10 trillion nontrivial solutions shown to lie on the critical line.

A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography .

The Riemann hypothesis has long been considered the greatest unsolved problem in mathematics . It was one of 10 unsolved mathematical problems (23 in the printed address) presented as a challenge for 20th-century mathematicians by German mathematician David Hilbert at the Second International Congress of Mathematics in Paris on Aug. 8, 1900. In 2000 American mathematician Stephen Smale updated Hilbert’s idea with a list of important problems for the 21st century; the Riemann hypothesis was number one. In 2000 it was designated a Millennium Problem , one of seven mathematical problems selected by the Clay Mathematics Institute of Cambridge, Mass., U.S., for a special award. The solution for each Millennium Problem is worth $1 million. In 2008 the U.S. Defense Advanced Research Projects Agency ( DARPA ) listed it as one of the DARPA Mathematical Challenges, 23 mathematical problems for which it was soliciting research proposals for funding—“Mathematical Challenge Nineteen: Settle the Riemann Hypothesis. The Holy Grail of number theory.”

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How to properly pronounce "riemann"?

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• The word 'Riemann' is derived from the surname of Bernhard Riemann, a German mathematician.

What is the definition of 'Riemann'?

• Riemann is a proper noun that refers to the surname of a German mathematician or the mathematical concepts associated with Bernhard Riemann.

Who is Bernhard Riemann?

• Bernhard Riemann was a prominent German mathematician who made significant contributions to various areas of mathematics, particularly in the field of analysis, number theory, and differential geometry.

What are the mathematical concepts associated with Riemann?

• The mathematical concepts associated with Riemann include Riemann surfaces, Riemannian geometry, Riemann zeta function, Riemann hypothesis, and Riemann-Christoffel curvature tensor.

What is Riemann's contribution to mathematics?

• Bernhard Riemann made several important contributions to mathematics, including his work on Riemannian geometry, Riemann surfaces, and the Riemann zeta function. His work on the foundations of geometry and his formulation of the Riemann hypothesis also had a profound impact on the field of number theory.

What is Riemannian geometry?

• Riemannian geometry is a branch of differential geometry that studies curved spaces using the tools of calculus and analysis. It was developed by Bernhard Riemann and is used in various fields, including general relativity and mathematical physics.

What are Riemann surfaces?

• Riemann surfaces are complex two-dimensional manifolds that generalize the concept of a curve in the complex plane. They were introduced by Bernhard Riemann and have applications in complex analysis and algebraic geometry.

What is the Riemann zeta function?

• The Riemann zeta function is a complex-valued function that plays a crucial role in number theory. It is defined for complex numbers with real part greater than 1 and is used to study the distribution of prime numbers.

What is the Riemann hypothesis?

• The Riemann hypothesis is a conjecture formulated by Bernhard Riemann in 1859. It states that all non-trivial zeros of the Riemann zeta function have a real part of 1/2. The hypothesis is one of the most important unsolved problems in mathematics and has deep connections with prime numbers.

What is the Riemann-Christoffel curvature tensor?

• The Riemann-Christoffel curvature tensor, also known as the Riemann tensor, is a mathematical object that describes the intrinsic curvature of a Riemannian manifold. It is used in differential geometry and general relativity to study the curvature of space-time.

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How related is the distribution of primes to the Riemann Hypothesis?

I do not grasp all concepts of the Riemann Hypothesis (better yet: as a layman I barely grasp anything). However, I understand that there is a certain link between the Riemann Hypothesis and prime numbers and their distribution.

My question is:

Would a "formula" or other system that enables you to calculate the distribution of prime numbers enable mathematicians to solve the Riemann Hypothesis? Are the directly linked, or does solving prime number distribution not automatically solve the Riemann Hypothesis?

• number-theory
• prime-numbers
• riemann-hypothesis

• 5 $\begingroup$ I can't remember if there's actually an answer to your question in it, but I thought you might be interested in this in-progress book by Barry Mazur and William Stein. $\endgroup$ –  Dylan Moreland Commented Oct 3, 2011 at 21:58
• $\begingroup$ @DylanMoreland: thanks for that link! $\endgroup$ –  user17095 Commented Oct 3, 2011 at 21:59
• $\begingroup$ Just perusing that book-in-progress, I noticed this (p. 15): <<<<The number $p = 2^{43,112,609} - 1$ is the largest prime we know, where by "know" we mean that we know it so explicitly that we can compute things about it.>>>> However, the first million digits of the $10^{10^{10^{10}}}$th prime are readily computed -- but I don't think that qualifies it as a "known" prime. $\endgroup$ –  r.e.s. Commented Oct 16, 2011 at 17:06

3 Answers 3

would a 'formula' or other system that enables you to calculate the distribution of prime numbers enable mathematicians to solve the Riemann Hypothesis?

There is an exact formula, known as "the explicit formula " of Riemann, for the prime number counting function $\pi(n)$ in terms of the zeros of $\zeta(s)$ . (Really it uses a minor modification of $\pi(n)$ , extended to positive real values of $n$ , but the idea is the same.)

The explicit formula displays an equivalence between asymptotics of the prime number distribution and location of zeros of $\zeta(s)$ . Knowledge of the real part of the location of the zeta zeros translates into knowledge of the distribution of primes. The closer the zeros are to the line with real part $1/2$ , the better the control over the distribution of primes.

This is all in Riemann's paper approximately 150 years ago, that introduced the Riemann hypothesis. The prime number theorem is equivalent to a demonstration that no zeros have real part equal to $1$ , which was done at the end of the 19th century. The infinitude of primes is equivalent to the pole of $\zeta(s)$ at $s=1$ , as was shown by Euler.

The difficulty in finding all the zeros is not the lack of a formula, but that the explicit formula relates two complicated sets without proving anything about either set individually. To restrict the location of zeta zeros through a formula for prime numbers, the prime formula would have to be strong enough to estimate $\pi(n)$ with an error of order $n^{1-\epsilon}$ for a positive $\epsilon$ , which would be considered an incredible breakthrough. Using Riemann's explicit formula it would be possible to take any argument about the prime distribution and translate it relatively easily into an argument about the zeta function, so it's not the case that formulations in terms of primes are likely to be any more amenable to proof than talking about the zeta zeros. In fact it is usually easier to start from the zeta function.

• $\begingroup$ Sorry, I do not understand this - but that is okay. Just to confirm: the answer would be "No, getting this formula does not help because it already exists. Calculating the prime number distribution with a formula does not solve the Riemann hypothesis.". Is that in simple words what you say? $\endgroup$ –  user17095 Commented Oct 12, 2011 at 15:44
• 1 $\begingroup$ The explicit formula and the theory of the zeta function provide a tight relationship between the distribution of primes and the location of zeros of $\zeta(s)$. A "formula for the prime number distribution already exists" in the sense that one can use this to compute the exact number of primes less than N without finding them one by one. But to prove RH you would need some additional facts about the primes (not in relation to zeta), or about the zeta function (not in relation to primes), or another relationship between primes and the zeta function in addition to the explicit formula. $\endgroup$ –  zyx Commented Oct 12, 2011 at 19:47
• $\begingroup$ For example, a proof that Li(x) is a good approximate formula for $\pi(x)$ would solve RH, but right now it is only known to be a consequence of RH: if RH is true then this approximate formula for the prime number distribution has the expected level of accuracy. $\endgroup$ –  zyx Commented Oct 12, 2011 at 19:55
• $\begingroup$ You wrote "The closer the zeros are to the line with real part 1/2, the better the control over the distribution of primes" ? What is meant by control? $\endgroup$ –  Anthony Commented Aug 5, 2021 at 20:12

As lhf writes, there is a strong link between the error estimate in the prime number theorem and the Riemann hypothesis. Indeed, RH is equivalent to a certain bound on this error estimate.

More precisely, the prime number theorem states that $\pi(x)$ (the number of primes $\leq x$) is asymptotic to $\mathop{\mathrm {Li}}(x)$, and the Riemann hypothesis is equivalent to the statement that the error in this approximation is bounded (for large $x$) by $x^{1/2 + \epsilon}$ for any $\epsilon > 0$.

The asymptotic prime number distribution has been known for over a century now. The Riemann Hypothesis is about the error term in that asymptotic equation. In this sense, they are very closely linked.

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I thInk I just disproved the Riemann Hypothesis. Where do I go from here?

hey. I'm an undergrad who doesn't know shit about shit. I'm transferring from a community college to an out of town university this fall, so I don't really have any professors to go to right now, but I think I disproved the Riemann hypothesis. How do I go about getting this checked out by the community to decide if I actually did something cool or if I just missed something?

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