In mathematics, the Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter (zeta), is a function of a complex variable defined as for , and its analytic continuation elsewhere.
The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics.
Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's seminal 1859 article "On the Number of Primes Less Than a Given Magnitude" extended Euler's definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider the most important unsolved problem in pure mathematics.
The values of the Riemann zeta function at even positive integers were computed by Euler. The first of them, , provides a solution to the Basel problem. In 1979, Roger Apéry proved the irrationality of , and as a consequence, the number was named after him. The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms. Many generalizations of the Riemann zeta function, such as Dirichlet series, Dirichlet -functions and -functions, are known.
Definition
The Riemann zeta function is a function of a complex variable , where and are real numbers. (The notation , , and is used traditionally in the study of the zeta function, following Riemann.) When , the function can be written as a converging summation or as an integral:
where
is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for .
Leonhard Euler considered the above series in 1740 for positive integer values of , and later Chebyshev extended the definition to .
The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for such that and diverges for all other values of . Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values . For , the series is the harmonic series which diverges to , and
Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole at with residue .
Euler's product formula In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity
where, by definition, the left hand side is and the infinite product on the right hand side extends over all prime numbers (such expressions are called Euler products):
Both sides of the Euler product formula converge for . The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when , diverges, Euler's formula, which becomes , implies that there are infinitely many primes. Since the logarithm of is approximately , the formula can also be used to prove the stronger result that the sum of the reciprocals of the primes is infinite. On the other hand, combining that with the sieve of Eratosthenes shows that the density of the set of primes within the set of positive integers is zero.
The Euler product formula can be used to calculate the asymptotic probability that randomly selected integers within a bound are set-wise coprime. Intuitively, the probability that any single number is divisible by a prime (or any integer) is . Hence the probability that numbers are all divisible by this prime is , and the probability that at least one of them is not is . Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors and if and only if it is divisible by , an event which occurs with probability ). Thus the asymptotic probability that numbers are coprime is given by a product over all primes,
Riemann's functional equation This zeta function satisfies the functional equation
where is the gamma function. This is an equality of meromorphic functions valid on the whole complex plane. The equation relates values of the Riemann zeta function at the points and , in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that has a simple zero at each even negative integer , known as the trivial zeros of . When is an even positive integer, the product on the right is non-zero because has a simple pole, which cancels the simple zero of the sine factor. When is , the zero of the sine factor is cancelled by the simple pole of .
A proof of the functional equation proceeds as follows: We observe that if , then
As a result, if then
with the inversion of the limiting processes justified by absolute convergence (hence the stricter requirement on ).
For convenience, let
which is a special case of the theta function.
Because and are Fourier transform pairs, then, by the Poisson summation formula, we have
so that
Hence
The right side is equivalent to
or
So
which is convergent for all , because more quickly than any power of for , so the integral converges. As the RHS remains the same if is replaced by ,
which is the functional equation attributed to Bernhard Riemann.
The functional equation above can be obtained using both the reflection formula and the duplication formula.
First collect terms of :
Then multiply both sides by and use the reflection formula:
Use the duplication formula with
so that
The functional equation was established by Riemann in his 1859 paper "On the Number of Primes Less Than a Given Magnitude" and used to construct the analytic continuation in the first place.
In the adelic formulation developed in Tate's thesis, the so-called Gamma factor is interpreted as the local zeta factor at the Archimedean place.
Riemann's xi function
Riemann also found a symmetric version of the functional equation by setting
that satisfies:
The xi function is an entire function whose zeros are exactly the non-trivial zeros of the Riemann zeta function.
Returning to the functional equation's derivation in the previous section, we have
where
Using integration by parts,
Using integration by parts again with a factorization of ,
As ,
Remove a factor of to make the exponents in the remainder opposites.
Using the hyperbolic functions, namely , and letting gives
and by separating the integral and using the power series for ,
which led Riemann to his famous hypothesis.
Zeros, the critical line, and the Riemann hypothesis
The functional equation shows that the Riemann zeta function has zeros at . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from being in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields important results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip , which is called the critical strip. The set is called the critical line. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that all non-trivial zeros are on the critical line. In 1989, Conrey proved that more than 40% of the non-trivial zeros of the Riemann zeta function are on the critical line. This was improved to 41.7%, and then 67.2%.
For the Riemann zeta function on the critical line, see -function.
+ First few nontrivial zeros Zero
Number of zeros in the critical strip Let be the number of zeros of in the critical strip , whose imaginary parts are in the interval . Timothy Trudgian proved that, if , then
Hardy–Littlewood conjectures In 1914, G. H. Hardy proved that has infinitely many real zeros.
Hardy and J. E. Littlewood formulated two conjectures on the density and distance between the zeros of on intervals of large positive real numbers. In the following, is the total number of real zeros and the total number of zeros of odd order of the function lying in the interval .
These two conjectures opened up new directions in the investigation of the Riemann zeta function.
Zero-free region The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the line . It is also known that zeros do not exist in certain regions slightly to the left of the line , known as zero-free regions. For instance, Korobov and Vinogradov independently showed via the Vinogradov's mean-value theorem that for sufficiently large , for
for any and a number depending on . Asymptotically, this is the largest known zero-free region for the zeta function.
Explicit zero-free regions are also known. Platt and Trudgian verified computationally that if and . Mossinghoff, Trudgian and Yang proved that zeta has no zeros in the region
for , which is the largest known zero-free region in the critical strip for (for previous results see). Yang showed that if and which is the largest known zero-free region for . Bellotti proved (building on the work of Ford) the zero-free region and This is the largest currently known zero-free region for fixed . Bellotti also showed that for sufficiently large , the following better result is known: for
In October 2026, OpenAI claimed a proof for the zero-free region
The strongest result of this kind one can hope for is the truth of the Riemann hypothesis, which would have many profound consequences in the theory of numbers.
Other results It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence () contains the imaginary parts of all zeros in the upper half-plane in ascending order, then
The critical line theorem asserts that a positive proportion of the nontrivial zeros lies on the critical line. (The Riemann hypothesis would imply that this proportion is .)
In the critical strip, the zero with smallest non-negative imaginary part is (). The fact that, for all complex ,
implies that the zeros of the Riemann zeta function are symmetric about the real axis. Combining this symmetry with the functional equation, furthermore, one sees that the non-trivial zeros are symmetric about the critical line .
It is also known that no zeros lie on the line with real part .
A large class of modified zeta functions exists that share the same non-trivial zeros as the Riemann zeta function, where modification means replacing the prime numbers in the Euler product by real numbers, which was shown in a result by Grosswald and Schnitzer.
Specific values
For any positive even integer ,
where is the th Bernoulli number. The demonstration of the particular value
is known as the Basel problem. The reciprocal of this sum answers the question: 'What is the probability that two numbers selected from a uniform distribution from to ] are coprime as ?'
For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic -theory of the integers; see Special values of -functions. The value
is Apéry's constant.
For nonpositive integers, the series does not converge, but via analytic continuation one can show that
for (using the convention that ). In particular, vanishes at the negative even integers because for all odd other than . These are the so-called "trivial zeros" of the zeta function. Another particular value is