Distribution of The Zeros of The Riemann Zeta Function: 1 Generalities
Distribution of The Zeros of The Riemann Zeta Function: 1 Generalities
Distribution of The Zeros of The Riemann Zeta Function: 1 Generalities
1 Generalities
Let us recall (see section on the analytic continuation of ζ(s) in The Riemann
Zeta-function : generalities) that zeta vanishes at negative odd integers. These
zeros are called the trivial zeros of ζ(s). The functional equation
πs
ζ(s) = χ(s)ζ(1 − s), χ(s) = 2s π s−1 sin Γ(1 − s). (1)
2
entails that other zeros of ζ(s) (they are called the non trivial zeros) are symetric
with respect to the critical line <(s) = 1/2 : for each non trivial zero s = σ + it,
the value s0 = 1 − σ + it is also a zero of ζ(s).
valid for all complex numbers s with <(s) > 1, shows that ζ(s) does not vanish
for <(s) > 1 (a convergent infinite product can not converge to zero because
its logarithm is a convergent series). Thus it suffices now to prove that ζ(s)
1 This pages are from //numbers.computation.free.fr/Constants/constants.html
Numbers, constants and computation 2
does not vanish on the line <(s) = 1. This property is in fact the key in the
proof of the prime number theorem, and Hadamard and De La Vallée Poussin
obtained this result independantly in 1896 by different mean (this problem is
in fact a first step in a determination of a zero-free region, important to obtain
good error terms in the prime number theorem). We present here the argument
of De La Vallée Poussin which is simpler to expose and more elegant, in a form
close to the presentation of [3].
which in particular, entails that the average value of S(T ) is zero. It is also
known that S(T ) is not too small, and more precisely, a result from Selberg
states that there exists a constant A > 0 for which the inequality
|S(T )| > A(log T )1/3 (log log T )−7/3
Numbers, constants and computation 4
Titchmarsh observed that the dominant term in the sum above is obtained with
n = 1 and is cos θ(t) and thus one should expect that on average, the sign of
Z(t) would be the sign of cos θ(t). For that reason he defined tν as the solution
of θ(tν ) = νπ (we have tν ∼ 2πν/ log ν) and showed that on average, the value
of Z(tν ) is 2(−1)ν . More precisely, he proved that
X X
Z(t2ν ) ∼ 2N, Z(t2ν+1 ) ∼ −2N (9)
ν≤N ν≤N
by showing that in the ν summation, the terms cos(θ(t2ν ) − t2ν log n)n−1/2
and cos(θ(t2ν+1 ) − t2ν+1 log n)n−1/2 of (8) for n ≥ 2 cancellate and give a
contribution of inferior order. It is now easy to prove that Z(t) has an infinity
of zeros since if not, it would keep the same sign for t large enough, thus one of
the two estimates in (9) would not be satisfied.
N0 (T ) > AT
holds for all values of T (see [3] for a proof). Selberg in 1932 improved consid-
erably this result by showing the existence of a constant A > 0 for which
N0 (T ) > AT log T
line. The proof is quite complicated and is given in [3]. The most significant
result on N0 (T ) has been obtained by Conrey in 1989 (see [1]) and states that
for T large enough, one as
N0 (T ) ≥ αN (T ), α = 0.40219.
Thus more than one third of the zeros lie on the critical line.
References
[1] J. B. Conrey, “More than two fifths of the zeros of the Riemann zeta function
are on the critical line”, J. reine angew. Math., 399 (1989), pp. 1-26.
[2] H. M. Edwards, Riemann’s Zeta Function, Academic Press, 1974.
[3] E. C. Titchmarsh, The theory of the Riemann Zeta-function, Oxford Science
publications, second edition, revised by D. R. Heath-Brown (1986).