Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes

  title={Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes},
  author={Gordon H. Hardy and John Edensor Littlewood},
  journal={Acta Mathematica},
On Selberg’s Central Limit Theorem for Dirichlet L-functions
L’accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (, implique l’accord avec les conditions générales d’utilisation (http://jtnb.Expand
The numerical evaluation of the Riesz function
The behaviour of the generalised Riesz function defined by Sm,p(x) = ∞ ∑ k=0 (−)x k!ζ(mk + p) (m ≥ 1, p ≥ 1) is considered for large positive values of x. A numerical scheme is given to compute thisExpand
A weighted central limit theorem for $\log|\zeta(1/2+it)|$
for any fixed V , as T goes to infinity. Analogous statements hold also in more generality, for example in the case of the imaginary part of log ζ(1/2 + it) or other L−functions (see e.g. [3]). InExpand
Extreme values for $S_n(\sigma,t)$ near the critical line
Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta function at the point $\sigma+it$ of the critical strip. For $n\geq 1$ and $t>0$ we define $$ S_{n}(\sigma,t) =Expand
Maximum of the Riemann zeta function on a short interval of the critical line
We prove the leading order of a conjecture by Fyodorov, Hiary and Keating, about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as $TExpand
The sixth moment of the Riemann zeta function and ternary additive divisor sums
Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second momentExpand
Twisted second moments and explicit formulae of the Riemann zeta-function
Several aspects connecting analytic number theory and the Riemann zeta-function are studied and expanded. These include: 1. explicit formulae relating the Mobius function to the non-trivial zeros ofExpand
Statistique des zéros non-triviaux de fonctions L de formes modulaires
Cette these se propose d’obtenir des resultats statistiques sur les zeros non-triviaux de fonctions L. Dans le cas des fonctions L de formes modulaires, on prouve qu’une proportion positive expliciteExpand
Non-vanishing of the symmetric square L-function at the central point
Using the mollifier method, we show that for a positive proportion of holomorphic Hecke eigenforms of level one and weight bounded by a large enough constant, the associated symmetric squareExpand
Möbius convolutions and the Riemann hypothesis
The well-known necessary and sufficient criteria for the Riemann hypothesis of M. Riesz and of Hardy and Littlewood are embedded in a general theorem for a class of entire functions, which in turn is seen to be a consequence of a rather transparent convolution criterion. Expand