Gram's Law and the Argument of the Riemann Zeta Function
Higher Mahler measure of an n-variable family
We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables. Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity.
Higher Mahler measures and zeta functions
Horizontal monotonicity of the modulus of the zeta function, L-functions, and related functions
As usual, let s = σ + it. For any fixed value of t with |t| ≥ 8 and for σ < 0, we show that |ζ(s)| is strictly decreasing in σ, with the same result also holding for the related functions ξ of Riemann and η of Euler. The following inequality related to the monotonicity of all three functions is proved: ℜ (η'(s)/η(s)) < ℜ (ζ'(s)/ζ(s)) < ℜ (ξ'(s)/ξ(s)). It is also shown that extending the above monotonicity result for |ζ(s)|, |ξ(s)|, or |η(s)| from σ <...
Hyperbolic manifolds and special values of Dedekind zeta-functions.
Hypergeometric series and the Riemann zeta function
Hypergeometric series associated with the Hurwitz-Lerch zeta function.
Hypothèse de Riemann et formes automorphes de Maass
Inequalities for zeros of ζ(s)
Inequalities involving a logarithmically convex function and their applications to special functions.
Infinite products over visible lattice points.
Integrals of logarithmic and hypergeometric functions
Integrals of logarithmic and hypergeometric functions are intrinsically connected with Euler sums. In this paper we explore many relations and explicitly derive closed form representations of integrals of logarithmic, hypergeometric functions and the Lerch phi transcendent in terms of zeta functions and sums of alternating harmonic numbers.
Irrationalité de valeurs de zêta
Les valeurs aux entiers pairs (strictement positifs) de la fonction de Riemann sont transcendantes, car ce sont des multiples rationnels de puissances de . En revanche, on sait très peu de choses sur la nature arithmétique des , pour entier. Apéry a démontré en 1978 que est irrationnel. Rivoal a prouvé en 2000 qu’une infinité de sont irrationnels, mais sans pouvoir en exhiber aucun autre que . Il existe plusieurs points de vue sur la preuve d’Apéry ; celui des séries hypergéométriques...
Irrationalité de selon Apéry
Joint value approximation and joint universality for several types of zeta functions
Kloosterman Sums and Fourier Coefficients of Cusp Forms.
Large deviations of Montgomery type and its application to the theory of zeta-functions
Large deviations of sums of independent random variables
Large gaps between consecutive zeros of the Riemann zeta-function. II
Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.
Large values of certain number-theoretic error terms