§25.10 Zeros

The product representation (25.2.11) implies $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s>1$. Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s=1$, 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 $n=1,2,3,\mathrm{\dots }$. These are called the trivial zeros. Except for the trivial zeros, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re }s\le 0$. In the region , called the critical strip, $\zeta \left(s\right)$ has infinitely many zeros, distributed symmetrically about the real axis and about the critical line $\mathrm{\Re }s=\frac{1}{2}$. 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

where

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

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 . 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 $\exp \left(\mathrm{i}\vartheta (t)\right)$ to obtain the Riemann–Siegel formula:

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

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

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).