Riemann zeta function

The Riemann zeta function or Euler-Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable defined as \[\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots\] for \(\mathrm{Re}(s) > 1\), 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 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler 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, ζ(2), provides a solution to the Basel problem. In 1979, Roger Apéry proved the irrationality of ζ(3), 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 L-functions and L-functions, are known.

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it, where σ and t are real numbers. (The notation s, σ, and t is used traditionally in the study of the zeta function, following Riemann.) When Re(s) = σ > 1, the function can be written as a converging summation or as an integral:

\(\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s} = \frac{1}{\Gamma(s)} \int_0^\infty \frac{x ^ {s-1}}{e ^ x - 1} \, \mathrm{d}x\,,\)

where

\(\Gamma(s) = \int_0^\infty x^{s-1}\,e^{-x} \, \mathrm{d}x\)

is the gamma function. The Riemann zeta function is defined for other complex values via analytic continuation of the function defined for σ > 1.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to Re(s) > 1.

The above series is a prototypical Dirichlet series that converges absolutely to an analytic function for s such that σ > 1 and diverges for all other values of s. Riemann showed that the function defined by the series on the half-plane of convergence can be continued analytically to all complex values s ≠ 1. For s = 1, the series is the harmonic series which diverges to +∞, and \[\lim_{s \to 1} (s - 1)\zeta(s) = 1.\] Thus the Riemann zeta function is a meromorphic function on the whole complex plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.

In 1737, the connection between the zeta function and prime numbers was discovered by Euler, who proved the identity

\(\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}},\)

where, by definition, the left hand side is ζ(s) and the infinite product on the right hand side extends over all prime numbers p (such expressions are called Euler products):

\(\prod_{p \text{ prime}} \frac{1}{1-p^{-s}} = \frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}} \cdots \frac{1}{1-p^{-s}} \cdots\)

Both sides of the Euler product formula converge for Re(s) > 1. 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 s = 1, diverges, Euler's formula (which becomes Π_p ⁠p/p − 1⁠) implies that there are infinitely many primes. Since the logarithm of p/(p − 1) is approximately 1/p, 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 s 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) p is 1/p. Hence the probability that s numbers are all divisible by this prime is 1/p, and the probability that at least one of them is not is 1 − 1/p. Now, for distinct primes, these divisibility events are mutually independent because the candidate divisors are coprime (a number is divisible by coprime divisors n and m if and only if it is divisible by nm, an event which occurs with probability 1/(nm)). Thus the asymptotic probability that s numbers are coprime is given by a product over all primes,

\(\prod_{p \text{ prime}} \left(1-\frac{1}{p^s}\right) = \left( \prod_{p \text{ prime}} \frac{1}{1-p^{-s}} \right)^{-1} = \frac{1}{\zeta(s)}.\)

This zeta function satisfies the functional equation \[\zeta(s) = 2^s \pi^{s-1}\ \sin\left( \frac{\pi s}{2} \right)\ \Gamma(1-s)\ \zeta(1-s)\ ,\] where Γ(s) 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 s and 1 − s, in particular relating even positive integers with odd negative integers. Owing to the zeros of the sine function, the functional equation implies that ζ(s) has a simple zero at each even negative integer s = −2n, known as the trivial zeros of ζ(s). When s is an even positive integer, the product \(\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\) on the right is non-zero because Γ(1 − s) has a simple pole, which cancels the simple zero of the sine factor. When s is 0, the zero of the sine factor is cancelled by the simple pole of ζ(1).

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 \(\pi^{-s/2}\Gamma(s/2)\) is interpreted as the local zeta factor at the Archimedean place.

Riemann also found a symmetric version of the functional equation by setting \[\xi(s) =\frac{s(s-1)}{2}\pi^{-\frac{s}{2}}\Gamma\left( \frac{s}{2} \right)\zeta(s) = (s-1)\pi^{-\frac{s}{2}}\Gamma\left(\frac{s}{2}+1\right)\zeta(s)\] that satisfies: \[\xi(s) = \xi(1 - s) ~.\] 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 \[\xi(s) =\frac12 + \frac{s(s-1)}{2} \int_1^\infty \left(x^{-\frac{s}{2}-\frac{1}{2}} + x^{\frac{s}{2}-1}\right)\psi(x) dx,\] where \[\psi(x)=\sum_{n=1}^{\infty}e^{-\pi n^{2}x}.\]

Using integration by parts, \[\xi(s) =\frac12 - \left[\left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi(x)\right]_1^\infty + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx\] \[\xi(s) =\frac12 + \psi(1) + \int_1^\infty \left(sx^{\frac{1-s}{2}} + (1-s)x^{\frac{s}{2}}\right)\psi'(x) dx\]

Using integration by parts again with a factorization of x, \[\xi(s) =\frac12 + \psi(1) - 2\left[x^{\frac32}\psi'(x)\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\right]_1^\infty + 2\int_1^\infty \left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right)\frac{d}{dx}\left[x^{\frac32}\psi'(x)\right] dx\] \[\xi(s) =\frac12 +\psi(1) + 4\psi'(1) + 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx\]

As \(\frac12 +\psi(1) + 4\psi'(1)=0\), \[\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]\left(x^{\frac{s-1}{2}} + x^{-\frac{s}{2}}\right) dx\]

Remove a factor of x to make the exponents in the remainder opposites. \[\xi(s) = 2\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\left(x^{\frac{s-1/2}{2}} + x^{\frac{1/2-s}{2}}\right) dx\]

Using the hyperbolic functions, namely cos(x) = cosh(ix), and letting s = 1/2 + it gives \[\xi(s) = 4\int_1^\infty \frac{d}{dx}\left[x^{\frac32}\psi'(x)\right]x^{-\frac14}\cos\left(\frac{t}2\log x\right) dx\] and by separating the integral and using the power series for cos, \[\xi(s) = \sum_{n=0}^\infty a_{2n}t^{2n}\] which led Riemann to his famous hypothesis.

The functional equation shows that the Riemann zeta function has zeros at −2, −4, .... These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 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 {s ∈ \(\C\) | 0 < Re(s) < 1}, which is called the critical strip. The set {s ∈ \(\C\) | Re(s) = 1/2} 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 has since been improved to 41.7%.

For the Riemann zeta function on the critical line, see Z-function.

Let N(T) be the number of zeros of ζ(s) in the critical strip 0 < Re(s) < 1, whose imaginary parts are in the interval 0 < Im(s) < T. Timothy Trudgian proved that, if T > e, then

\(\left|N(T) - \frac{T}{2\pi} \log{\frac{T}{2\pi e}}\right| \leq 0.112 \log T + 0.278 \log\log T + 3.385 + \frac{0.2}{T}\).

In 1914, G. H. Hardy proved that ζ(⁠1/2⁠ + it) has infinitely many real zeros.

Hardy and J. E. Littlewood formulated two conjectures on the density and distance between the zeros of ζ(1/2 + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N₀(T) the total number of zeros of odd order of the function ζ(1/2 + it) lying in the interval (0, T].

1. For any ε > 0, there exists a T₀(ε) > 0 such that when

   \(T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac14+\varepsilon},\)

   the interval (T, T + H] contains a zero of odd order.
2. For any ε > 0, there exists a T₀(ε) > 0 and c_ε > 0 such that the inequality

   \(N_0(T+H)-N_0(T) \geq c_\varepsilon H\)

   holds when

   \(T \geq T_0(\varepsilon) \quad\text{ and }\quad H=T^{\frac12+\varepsilon}.\)

These two conjectures opened up new directions in the investigation of the Riemann zeta function.

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 Re(s) = 1. It is also known that zeros do not exist in certain regions slightly to the left of the line Re(s) = 1, known as zero-free regions. For instance, Korobov and Vinogradov independently showed via the Vinogradov's mean-value theorem that for sufficiently large |t|, ζ(σ + it) ≠ 0 for

\(\sigma \geq 1 - \frac{c}{(\log|t|)^{2/3 + \varepsilon}}\)

for any ε > 0 and a number c > 0 depending on ε. 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 ζ(σ + it) ≠ 0 if σ ≠ 1/2 and |t| ≤ 3⋅10. Mossinghoff, Trudgian and Yang proved that zeta has no zeros in the region

\(\sigma\ge 1 - \frac{1}{5.558691\log|t|}\)

for |t| ≥ 2, which is the largest known zero-free region in the critical strip for 3⋅10 < |t| < exp(64.1) ≈ 7⋅10 (for previous results see). Yang showed that ζ(σ + it) ≠ 0 if

\(\sigma \geq 1 - \frac{\log\log|t|}{21.233\log|t|}\) and \(|t| \geq 3\)

which is the largest known zero-free region for exp(170.2) < |t| < exp(4.8⋅10). Bellotti proved (building on the work of Ford) the zero-free region

\(\sigma \ge 1 - \frac{1}{53.989(\log|t|)^{2/3}(\log\log|t|)^{1/3}}\) and \(|t| \ge 3\).

This is the largest known zero-free region for fixed |t| ≥ exp(4.8⋅10). Bellotti also showed that for sufficiently large |t|, the following better result is known: ζ(σ + it) ≠ 0 for

\(\sigma \geq 1 - \frac{1}{48.0718(\log|t|)^{2/3}(\log\log|t|)^{1/3}}.\)

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.

It is known that there are infinitely many zeros on the critical line. Littlewood showed that if the sequence (γ_n) contains the imaginary parts of all zeros in the upper half-plane in ascending order, then

\(\lim_{n\rightarrow\infty}\left(\gamma_{n+1}-\gamma_n\right)=0.\)

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 1.)

In the critical strip, the zero with smallest non-negative imaginary part is 1/2 + 14.13472514... i (OEIS: A058303). The fact that, for all complex s ≠ 1,

\(\zeta(s)=\overline{\zeta(\overline{s})}\)

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 Re(s) = 1/2.

It is also known that no zeros lie on the line with real part 1.

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.

For any positive even integer 2n, \[\zeta(2n) = \frac{|{B_{2n}}|(2\pi)^{2n}}{2(2n)!},\] where B_2n is the (2n)th Bernoulli number. The demonstration of the particular value \[\zeta(2) = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \cdots = \frac{\pi^2}{6}\] 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 1 to n] are coprime as n → ∞?'

For odd positive integers, no such simple expression is known, although these values are thought to be related to the algebraic K-theory of the integers; see Special values of L-functions. The value \[\zeta(3) = 1 + \frac{1}{2^3} + \frac{1}{3^3} + \cdots = 1.202056903159594285399...\] is Apéry's constant.

For nonpositive integers, the series does not converge, but via analytic continuation one can show that \[\zeta(-n)= -\frac{B_{n+1}}{n+1}\] for n ≥ 0 (using the convention that B₁ = 1/2). In particular, ζ vanishes at the negative even integers because B_m = 0 for all odd m other than 1. These are the so-called "trivial zeros" of the zeta function. Another particular value is \[\zeta(-1) = -\tfrac{1}{12}\] This gives a pretext for assigning a finite value to the divergent series 1 + 2 + 3 + 4 + ⋯, which has been used in certain contexts (Ramanujan summation) such as string theory. Analogously, \[\zeta(0) = -\tfrac{1}{2}\] can be viewed as assigning a finite result to the divergent series 1 + 1 + 1 + 1 + ⋯.

The value \[\zeta\bigl(\tfrac12\bigr) = -1.46035450880958681288\ldots\] is employed in calculating kinetic boundary layer problems of linear kinetic equations.

Although \[\zeta(1) = 1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots\] diverges, its Cauchy principal value \[\lim_{\varepsilon \to 0} \frac{\zeta(1+\varepsilon)+\zeta(1-\varepsilon)}{2}\] exists and is equal to the Euler-Mascheroni constant γ = 0.5772....

Taking the limit s → +∞ through the real numbers, one obtains ζ(+∞) = 1. But at complex infinity on the Riemann sphere the zeta function has an essential singularity.

The reciprocal of the zeta function may be expressed as a Dirichlet series over the Möbius function μ(n):

\(\frac{1}{\zeta(s)} = \sum_{n=1}^\infty \frac{\mu(n)}{n^s}\)

for every complex number s with real part greater than 1. There are a number of similar relations involving various well-known multiplicative functions; these are given in the article on the Dirichlet series.

The Riemann hypothesis is equivalent to the claim that this expression is valid when the real part of s is greater than 1/2.

The critical strip of the Riemann zeta function has the remarkable property of universality. This zeta function universality states, roughly, that vertical translates of \(\zeta(s)\) can uniformly approximate any nonvanishing holomorphic function on suitable compact subsets of the strip \(1/2<\operatorname{Re}(s)<1\). Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975. More recent work has included effective versions of Voronin's theorem and extending it to Dirichlet L-functions.

Let the functions F(T; H) and G(s₀; Δ) be defined by the equalities

\(F(T;H) = \max_{|t-T|\le H}\left|\zeta\left(\tfrac{1}{2}+it\right)\right|,\qquad G(s_{0};\Delta) = \max_{|s-s_{0}|\le\Delta}|\zeta(s)|.\)

Here T is a sufficiently large positive number, 0 < H ≪ log log T, s₀ = σ₀ + iT, 1/2 ≤ σ₀ ≤ 1, 0 < Δ < 1/3. Estimating the values F and G from below shows, how large (in modulus) values ζ(s) can take on short intervals of the critical line or in small neighborhoods of points lying in the critical strip 0 ≤ Re(s) ≤ 1.

The case H ≫ log log T was studied by Kanakanahalli Ramachandra; the case Δ > c, where c is a sufficiently large constant, is trivial.

Anatolii Karatsuba proved, in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates

\(F(T;H) \ge T^{- c_1},\qquad G(s_0; \Delta) \ge T^{-c_2},\)

hold, where c₁ and c₂ are certain absolute constants.