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25 Zeta and Related FunctionsRiemann Zeta Function

§25.10 Zeros

  1. §25.10(i) Distribution
  2. §25.10(ii) Riemann–Siegel Formula

§25.10(i) Distribution

The product representation (25.2.11) implies ζ(s)0 for s>1. Also, ζ(s)0 for 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 ζ(2n)=0 for n=1,2,3,. These are called the trivial zeros. Except for the trivial zeros, ζ(s)0 for s0. In the region 0<s<1, called the critical strip, ζ(s) has infinitely many zeros, distributed symmetrically about the real axis and about the critical line s=12. 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)exp(iϑ(t))ζ(12+it),


25.10.2 ϑ(t)phΓ(14+12it)12tlnπ

is chosen to make Z(t) real, and phΓ(14+12it) assumes its principal value. Because |Z(t)|=|ζ(12+it)|, Z(t) vanishes at the zeros of ζ(12+it), which can be separated by observing sign changes of Z(t). Because Z(t) changes sign infinitely often, ζ(12+it) 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 ζ(s) in that portion of the critical strip with 0<t<T. By comparing N(T) with the number of sign changes of Z(t) we can decide whether ζ(s) has any zeros off the line in this region. Sign changes of Z(t) are determined by multiplying (25.9.3) by exp(iϑ(t)) to obtain the Riemann–Siegel formula:

25.10.3 Z(t)=2n=1mcos(ϑ(t)tlnn)n1/2+R(t),

where R(t)=O(t1/4) as t.

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

25.10.4 R(t)=(1)m1(2πt)1/4cos(t(2m+1)2πt18π)cos(2πt)+O(t3/4).

Riemann also developed a technique for determining further terms. Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of ζ(s) 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).