Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality (n) := P d|n d 1. As consequence we obtain that RH holds true i every natural number divisible by a… Expand

We show that the Extended Riemann Hypothesis, the prime k-tuples conjecture and a conjecture of Ihara about the distribution of Euler-Kronecker constants cannot be all true.Expand

Let g∈Z\{−1, 0, 1} and let h be the largest integer such that g is an hth power. Let p be a prime. Put wg(p)=2ϕ(p−1)/(p−1) if (g/p)=−1 (Legendre symbol) and (p−1, h)=1 and wg(p)=0 otherwise, with ϕ… Expand

Abstract Let Ψ n ( x ) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψ n ( x ) = ( x n − 1 ) / Φ n ( x ) , with Φ n ( x ) the nth… Expand

Abstract:A constant of the form , where the product ranges over all sufficiently large primes p and h is rational, is an example of a singular series. We show that this type of singular series can be… Expand

Abstract It follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is… Expand

Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p ≡ a(mod f)… Expand