§25.10 Zeros

§25.10(i) Distribution

The product representation (25.2.11) implies $\zeta\left(s\right)\neq 0$ for $\Re s>1$. Also, $\zeta\left(s\right)\neq 0$ for $\Re s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3). The functional equation (25.4.1) implies $\zeta\left(-2n\right)=0$ for $n=1,2,3,\dots$. These are called the trivial zeros. Except for the trivial zeros, $\zeta\left(s\right)\neq 0$ for $\Re s\leq 0$. In the region $0<\Re s<1$, called the critical strip, $\zeta\left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\Re s=\frac{1}{2}$. The Riemann hypothesis states that all nontrivial zeros lie on this line.

Calculations relating to the zeros on the critical line make use of the real-valued function

 25.10.1 $Z(t)\equiv\exp\left(i\vartheta(t)\right)\zeta\left(\tfrac{1}{2}+it\right),$ ⓘ Symbols: $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp\NVar{z}$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta(t)$: function Keywords: definition Source: Titchmarsh (1986b, (4.17.2), p. 89) Permalink: http://dlmf.nist.gov/25.10.E1 Encodings: TeX, pMML, png See also: Annotations for §25.10(i), §25.10 and Ch.25

where

 25.10.2 $\vartheta(t)\equiv\operatorname{ph}\Gamma\left(\tfrac{1}{4}+\tfrac{1}{2}it% \right)-\tfrac{1}{2}t\ln\pi$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase, $z$: complex variable, $\vartheta(t)$: function and $\chi(s)$: function Keywords: definition Source: Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta(t)\equiv-\frac{1}{2}\operatorname{ph}\chi(\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\Gamma\left(\overline{z}\right)=\overline{\Gamma\left(z\right)}$, and (4.8.10). Permalink: http://dlmf.nist.gov/25.10.E2 Encodings: TeX, pMML, png See also: Annotations for §25.10(i), §25.10 and Ch.25

is chosen to make $Z(t)$ real, and $\operatorname{ph}\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)$ assumes its principal value. Because $|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|$, $Z(t)$ vanishes at the zeros of $\zeta\left(\frac{1}{2}+it\right)$, which can be separated by observing sign changes of $Z(t)$. Because $Z(t)$ changes sign infinitely often, $\zeta\left(\frac{1}{2}+it\right)$ has infinitely many zeros with $t$ real.

§25.10(ii) Riemann–Siegel Formula

Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta\left(s\right)$ in that portion of the critical strip with $0. By comparing $N(T)$ with the number of sign changes of $Z(t)$ we can decide whether $\zeta\left(s\right)$ has any zeros off the line in this region. Sign changes of $Z(t)$ are determined by multiplying (25.9.3) by $\exp\left(i\vartheta(t)\right)$ to obtain the Riemann–Siegel formula:

 25.10.3 $Z(t)=2\sum_{n=1}^{m}\frac{\cos\left(\vartheta(t)-t\ln n\right)}{n^{1/2}}+R(t),$ $m=\left\lfloor\sqrt{t/(2\pi)}\right\rfloor$, ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\left\lfloor\NVar{x}\right\rfloor$: floor of $x$, $\ln\NVar{z}$: principal branch of logarithm function, $m$: nonnegative integer, $n$: nonnegative integer, $Z(t)$: zeros function and $\vartheta(t)$: function Keywords: Riemann–Siegel formula Source: Titchmarsh (1986b, (4.17.4), p. 89) Referenced by: §25.18(i), Erratum (V1.1.4) for Equation (25.10.3) Permalink: http://dlmf.nist.gov/25.10.E3 Encodings: TeX, pMML, png Clarification (effective with 1.1.4): The constraint $m=\left\lfloor\sqrt{t/(2\pi)}\right\rfloor$ was added. Suggested 2021-08-23 by Gergő Nemes See also: Annotations for §25.10(ii), §25.10 and Ch.25

where $R(t)=O\left(t^{-1/4}\right)$ as $t\to\infty$.

The error term $R(t)$ can be expressed as an asymptotic series that begins

 25.10.4 $R(t)=(-1)^{m-1}\left(\frac{2\pi}{t}\right)^{1/4}\frac{\cos\left(t-(2m+1)\sqrt{% 2\pi t}-\frac{1}{8}\pi\right)}{\cos\left(\sqrt{2\pi t}\right)}+O\left(t^{-3/4}% \right).$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function and $m$: nonnegative integer Keywords: asymptotic approximation Source: Titchmarsh (1986b, (4.17.5), p. 89) Referenced by: §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii) Permalink: http://dlmf.nist.gov/25.10.E4 Encodings: TeX, pMML, png See also: Annotations for §25.10(ii), §25.10 and Ch.25

Riemann also developed a technique for determining further terms. Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of $\zeta\left(s\right)$ in the critical strip are on the critical line (van de Lune et al. (1986)). More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)).

For further information on the Riemann–Siegel expansion see Berry (1995).