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On Robin's criterion for the Riemann Hypothesis
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 aExpand
Values of the Euler Φ-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields
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
On primes in arithmetic progression having a prescribed primitive root
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 ϕExpand
Inverse cyclotomic polynomials
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 nthExpand
Artin's Primitive Root Conjecture – A Survey
  • P. Moree
  • Mathematics, Computer Science
  • Integers
  • 13 December 2004
We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). Expand
Approximation of singular series and automata
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 beExpand
Character sums for primitive root densities
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 isExpand
On primes in arithmetic progression having a prescribed primitive root. II
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
Cyclotomic Numerical Semigroups
Given a numerical semigroup $S$, we let $\mathrm P_S(x)=(1-x)\sum_{s\in S}x^s$ be its semigroup polynomial. Expand