Riemann zeta function

Let ${\displaystyle N(T)}$ be the number of zeros of ${\displaystyle \zeta (s)}$ in the critical strip ${\displaystyle 0<\operatorname {Re} (s)<1}$, whose imaginary parts are in the interval ${\displaystyle 0<\operatorname {Im} (s)<T}$. Trudgian proved that, if ${\displaystyle T>e}$, then[12]

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

In 1914, Godfrey Harold Hardy proved that ζ (1/2 + it) has infinitely many real zeros.[13]

Hardy and John Edensor 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].

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 Re(s) = 1 line.[14] A better result[15] that follows from an effective form of Vinogradov's mean-value theorem is that ζ (σ + it) ≠ 0 whenever ${\displaystyle \sigma \geq 1-{\frac {1}{57.54(\log {|t|})^{\frac {2}{3}}(\log {\log {|t|}})^{\frac {1}{3}}}}}$ and |t| ≥ 3.

In 2015, Mossinghoff and Trudgian proved[16] that zeta has no zeros in the region

${\displaystyle \sigma \geq 1-{\frac {1}{5.573412\log |t|}}}$

for |t| ≥ 2. This is the largest known zero-free region in the critical strip for ${\displaystyle 3.06\cdot 10^{10}<|t|<\exp(10151.5)\approx 5.5\cdot 10^{4408}}$.

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

${\displaystyle \lim _{n\rightarrow \infty }\left(\gamma _{n+1}-\gamma _{n}\right)=0.}$

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

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

for all complex s ≠ 1 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 a line with real part 1.

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

${\displaystyle {\frac {1}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}}$

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 that there exists some location on the critical strip that approximates any holomorphic function arbitrarily well. Since holomorphic functions are very general, this property is quite remarkable. The first proof of universality was provided by Sergei Mikhailovitch Voronin in 1975.[22] More recent work has included effective versions of Voronin's theorem[23] and extending it to Dirichlet L-functions.[24][25]

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

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

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,[26][27] in particular, that if the values H and Δ exceed certain sufficiently small constants, then the estimates

${\displaystyle F(T;H)\geq T^{-c_{1}},\qquad G(s_{0};\Delta )\geq T^{-c_{2}},}$

hold, where c₁ and c₂ are certain absolute constants.

The function

${\displaystyle S(t)={\frac {1}{\pi }}\arg {\zeta \left({\tfrac {1}{2}}+it\right)}}$

is called the argument of the Riemann zeta function. Here arg ζ(1/2 + it) is the increment of an arbitrary continuous branch of arg ζ(s) along the broken line joining the points 2, 2 + it and 1/2 + it.

There are some theorems on properties of the function S(t). Among those results[28][29] are the mean value theorems for S(t) and its first integral

${\displaystyle S_{1}(t)=\int _{0}^{t}S(u)\,\mathrm {d} u}$

on intervals of the real line, and also the theorem claiming that every interval (T, T + H] for

${\displaystyle H\geq T^{{\frac {27}{82}}+\varepsilon }}$

contains at least

${\displaystyle H{\sqrt[{3}]{\ln T}}e^{-c{\sqrt {\ln \ln T}}}}$

points where the function S(t) changes sign. Earlier similar results were obtained by Atle Selberg for the case

${\displaystyle H\geq T^{{\frac {1}{2}}+\varepsilon }.}$

An extension of the area of convergence can be obtained by rearranging the original series.[30] The series

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }\left({\frac {n}{(n+1)^{s}}}-{\frac {n-s}{n^{s}}}\right)}$

converges for Re(s) > 0, while

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=1}^{\infty }{\frac {n(n+1)}{2}}\left({\frac {2n+3+s}{(n+1)^{s+2}}}-{\frac {2n-1-s}{n^{s+2}}}\right)}$

converges even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > −k for any negative integer −k.

The Mellin transform of a function f(x) is defined as[31]

${\displaystyle \int _{0}^{\infty }f(x)x^{s}\,{\frac {\mathrm {d} x}{x}}}$

in the region where the integral is defined. There are various expressions for the zeta function as Mellin transform-like integrals. If the real part of s is greater than one, we have

${\displaystyle \Gamma (s)\zeta (s)=\int _{0}^{\infty }{\frac {x^{s-1}}{e^{x}-1}}\,\mathrm {d} x\quad }$ and ${\displaystyle \quad \Gamma (s)\zeta (s)={\frac {1}{2s}}\int _{0}^{\infty }{\frac {x^{s}}{\cosh(x)-1}}\,\mathrm {d} x}$,

where Γ denotes the gamma function. By modifying the contour, Riemann showed that

${\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}}{e^{x}-1}}\,\mathrm {d} x}$

for all s (where H denotes the Hankel contour).

We can also find expressions which relate to prime numbers and the prime number theorem. If π(x) is the prime-counting function, then

${\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }{\frac {\pi (x)}{x(x^{s}-1)}}\,\mathrm {d} x,}$

for values with Re(s) > 1.

A similar Mellin transform involves the Riemann function J(x), which counts prime powers pⁿ with a weight of 1/n, so that

${\displaystyle J(x)=\sum {\frac {\pi \left(x^{\frac {1}{n}}\right)}{n}}.}$

Now

${\displaystyle \ln \zeta (s)=s\int _{0}^{\infty }J(x)x^{-s-1}\,\mathrm {d} x.}$

These expressions can be used to prove the prime number theorem by means of the inverse Mellin transform. Riemann's prime-counting function is easier to work with, and π(x) can be recovered from it by Möbius inversion.

The Riemann zeta function can be given by a Mellin transform[32]

${\displaystyle 2\pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)=\int _{0}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t,}$

in terms of Jacobi's theta function

${\displaystyle \theta (\tau )=\sum _{n=-\infty }^{\infty }e^{\pi in^{2}\tau }.}$

However, this integral only converges if the real part of s is greater than 1, but it can be regularized. This gives the following expression for the zeta function, which is well defined for all s except 0 and 1:

${\displaystyle \pi ^{-{\frac {s}{2}}}\Gamma \left({\frac {s}{2}}\right)\zeta (s)={\frac {1}{s-1}}-{\frac {1}{s}}+{\frac {1}{2}}\int _{0}^{1}\left(\theta (it)-t^{-{\frac {1}{2}}}\right)t^{{\frac {s}{2}}-1}\,\mathrm {d} t+{\frac {1}{2}}\int _{1}^{\infty }{\bigl (}\theta (it)-1{\bigr )}t^{{\frac {s}{2}}-1}\,\mathrm {d} t.}$

The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development is then[33]

${\displaystyle \zeta (s)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {\gamma _{n}}{n!}}(1-s)^{n}.}$

The constants γ_n here are called the Stieltjes constants and can be defined by the limit

${\displaystyle \gamma _{n}=\lim _{m\rightarrow \infty }{\left(\left(\sum _{k=1}^{m}{\frac {(\ln k)^{n}}{k}}\right)-{\frac {(\ln m)^{n+1}}{n+1}}\right)}.}$

The constant term γ₀ is the Euler–Mascheroni constant.

For all s ∈ C, s ≠ 1, the integral relation (cf. Abel–Plana formula)

${\displaystyle \zeta (s)={\frac {1}{s-1}}+{\frac {1}{2}}+2\int _{0}^{\infty }{\frac {\sin(s\arctan t)}{\left(1+t^{2}\right)^{s/2}\left(e^{2\pi t}-1\right)}}\,\mathrm {d} t}$

holds true, which may be used for a numerical evaluation of the zeta function.

Another series development using the rising factorial valid for the entire complex plane is

${\displaystyle \zeta (s)={\frac {s}{s-1}}-\sum _{n=1}^{\infty }{\bigl (}\zeta (s+n)-1{\bigr )}{\frac {s(s+1)\cdots (s+n-1)}{(n+1)!}}.}$

This can be used recursively to extend the Dirichlet series definition to all complex numbers.

The Riemann zeta function also appears in a form similar to the Mellin transform in an integral over the Gauss–Kuzmin–Wirsing operator acting on x^{s − 1}; that context gives rise to a series expansion in terms of the falling factorial.[34]

On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product expansion

${\displaystyle \zeta (s)={\frac {e^{\left(\log(2\pi )-1-{\frac {\gamma }{2}}\right)s}}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)e^{\frac {s}{\rho }},}$

where the product is over the non-trivial zeros ρ of ζ and the letter γ again denotes the Euler–Mascheroni constant. A simpler infinite product expansion is

${\displaystyle \zeta (s)=\pi ^{\frac {s}{2}}{\frac {\prod _{\rho }\left(1-{\frac {s}{\rho }}\right)}{2(s-1)\Gamma \left(1+{\frac {s}{2}}\right)}}.}$

This form clearly displays the simple pole at s = 1, the trivial zeros at −2, −4, ... due to the gamma function term in the denominator, and the non-trivial zeros at s = ρ. (To ensure convergence in the latter formula, the product should be taken over "matching pairs" of zeros, i.e. the factors for a pair of zeros of the form ρ and 1 − ρ should be combined.)

A globally convergent series for the zeta function, valid for all complex numbers s except s = 1 + 2πi/ln 2n for some integer n, was conjectured by Konrad Knopp in 1926 [35] and proven by Helmut Hasse in 1930[36] (cf. Euler summation):

${\displaystyle \zeta (s)={\frac {1}{1-2^{1-s}}}\sum _{n=0}^{\infty }{\frac {1}{2^{n+1}}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s}}}.}$

The series appeared in an appendix to Hasse's paper, and was published for the second time by Jonathan Sondow in 1994.[37]

Hasse also proved the globally converging series

${\displaystyle \zeta (s)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{(k+1)^{s-1}}}}$

in the same publication.[36] Research by Iaroslav Blagouchine[38][35] has found that a similar, equivalent series was published by Joseph Ser in 1926.[39]

Peter Borwein developed an algorithm that applies Chebyshev polynomials to the Dirichlet eta function to produce a very rapidly convergent series suitable for high precision numerical calculations.[40]

${\displaystyle \zeta (k)={\frac {2^{k}}{2^{k}-1}}+\sum _{r=2}^{\infty }{\frac {(p_{r-1}\#)^{k}}{J_{k}(p_{r}\#)}}\qquad k=2,3,\ldots .}$

Here p_n# is the primorial sequence and J_k is Jordan's totient function.[41]

The function ζ can be represented, for Re(s) > 1, by the infinite series

${\displaystyle \zeta (s)=\sum _{n=0}^{\infty }B_{n,\geq 2}^{(s)}{\frac {(W_{k}(-1))^{n}}{n!}},}$

where k ∈ {−1, 0}, W_k is the kth branch of the Lambert W-function, and B^(μ)
_{n, ≥2} is an incomplete poly-Bernoulli number.[42]

The function ${\displaystyle g(x)=x\left(1+\left\lfloor x^{-1}\right\rfloor \right)-1}$ is iterated to find the coefficients appearing in Engel expansions.[43]

The Mellin transform of the map ${\displaystyle g(x)}$ is related to the Riemann zeta function by the formula

${\displaystyle {\begin{aligned}\int _{0}^{1}g(x)x^{s-1}\,dx&=\sum _{n=1}^{\infty }\int _{\frac {1}{n+1}}^{\frac {1}{n}}(x(n+1)-1)x^{s-1}\,dx\\[6pt]&=\sum _{n=1}^{\infty }{\frac {n^{-s}(s-1)+(n+1)^{-s-1}(n^{2}+2n+1)+n^{-s-1}s-n^{1-s}}{(s+1)s(n+1)}}\\[6pt]&={\frac {\zeta (s+1)}{s+1}}-{\frac {1}{s(s+1)}}\end{aligned}}}$

Certain linear combinations of Dirichlet series whose coefficients are terms of the Thue-Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 [44]). For instance:

${\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {5t_{n-1}+3t_{n}}{n^{2}}}&=4\zeta (2)={\frac {2\pi ^{2}}{3}},\\\sum _{n\geq 1}{\frac {9t_{n-1}+7t_{n}}{n^{3}}}&=8\zeta (3),\end{aligned}}}$

where ${\displaystyle (t_{n})_{n\geq 0}}$ is the ${\displaystyle n^{\rm {th}}}$ term of the Thue-Morse sequence. In fact, for all ${\displaystyle s}$ with real part greater than ${\displaystyle 1}$, we have

${\displaystyle (2^{s}+1)\sum _{n\geq 1}{\frac {t_{n-1}}{n^{s}}}+(2^{s}-1)\sum _{n\geq 1}{\frac {t_{n}}{n^{s}}}=2^{s}\zeta (s).}$

In the theory of musical tunings, the zeta function can be used to find equal divisions of the octave (EDOs) that closely approximate the intervals of the harmonic series. For increasing values of ${\displaystyle t\in \mathbb {R} }$, the value of

${\displaystyle \left\vert \zeta \left({\frac {1}{2}}+{\frac {2\pi {i}}{\ln {(2)}}}t\right)\right\vert }$

peaks near integers that correspond to such EDOs.[47] Examples include popular choices such as 12, 19, and 53.[48]

The zeta function evaluated at equidistant positive integers appears in infinite series representations of a number of constants.[49]

• ${\displaystyle \sum _{n=2}^{\infty }{\bigl (}\zeta (n)-1{\bigr )}=1}$

In fact the even and odd terms give the two sums

• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n)-1{\bigr )}={\frac {3}{4}}}$

and

• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (2n+1)-1{\bigr )}={\frac {1}{4}}}$

Parametrized versions of the above sums are given by

• ${\displaystyle \sum _{n=1}^{\infty }(\zeta (2n)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}+{\frac {1}{2}}\left(1-\pi t\cot(t\pi )\right)}$

and

• ${\displaystyle \sum _{n=1}^{\infty }(\zeta (2n+1)-1)\,t^{2n}={\frac {t^{2}}{t^{2}-1}}-{\frac {1}{2}}\left(\psi ^{0}(t)+\psi ^{0}(-t)\right)-\gamma }$

with ${\displaystyle |t|<2}$ and where ${\displaystyle \psi }$ and ${\displaystyle \gamma }$ are the polygamma function and Euler's constant, respectively, as well as

• ${\displaystyle \sum _{n=1}^{\infty }{\frac {\zeta (2n)-1}{n}}\,t^{2n}=\log \left({\dfrac {1-t^{2}}{\operatorname {sinc} (\pi \,t)}}\right)}$

all of which are continuous at ${\displaystyle t=1}$. Other sums include

• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}=1-\gamma }$
• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\left(\left({\tfrac {3}{2}}\right)^{n-1}-1\right)={\frac {1}{3}}\ln \pi }$
• ${\displaystyle \sum _{n=1}^{\infty }{\bigl (}\zeta (4n)-1{\bigr )}={\frac {7}{8}}-{\frac {\pi }{4}}\left({\frac {e^{2\pi }+1}{e^{2\pi }-1}}\right).}$
• ${\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\operatorname {Im} {\bigl (}(1+i)^{n}-(1+i)^{n}{\bigr )}={\frac {\pi }{4}}}$

where Im denotes the imaginary part of a complex number.

There are yet more formulas in the article Harmonic number.