Helmut Maier

Cyclotomic polynomials whose orders contain many prime factors

Let &PHgr; n (z) = ∑ ϕ (n) m=0 a (m, n) z m be the n th cyclonomic polynomial and set A(n) = max 0≤m≤ϕ(n) |a=(m, n)|. In previous papers the author has shown that for almost all integers A(n)≤ n &PHgr;(n) ; whenever lim n→∞ &PHgr;(n) = ∞ . In this paper we show that for most integers n with at least C log log n prime factors ( C > 2/log 2) this inequality is wrong.

The Ternary Goldbach Problem with a Prime and Two Isolated Primes

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

Asymptotics and Equidistribution of Cotangent Sums Associated with the Estermann and Riemann Zeta Functions

The Nyman–Beurling criterion is a well-known approach to the Riemann Hypothesis. Certain integrals over Dirichlet series appearing in this approach can be expressed in terms of cotangent sums. These cotangent sums are also associated with the Estermann zeta function. In this paper improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. The main result of this paper is the existence of a unique positive measure μ on \(\mathbb{R}\) with respect to which normalized versions of these cotangent sums are equidistributed. We also consider the moments of order 2k as a function of k.