On primitive divisors of Mersenne numbers
Let denote the error term in the Dirichlet divisor problem, and let E(T) denote the error term in the asymptotic formula for the mean square of |ζ(1/2+it)|. If E*(t) := E(t) - 2πΔ*(t/(2π)) with Δ*(x) = -Δ(x) + 2Δ(2x) - 1/2Δ(4x) and , then we obtain a number of results involving the moments of |ζ(1/2+it)| in short intervals, by connecting them to the moments of E*(T) and R(T) in short intervals. Upper bounds and asymptotic formulae for integrals of the form ∫T2T(∫t-Ht+H |ζ(1/2+iu|2 duk dtare...
Let σ(n) denote the sum of positive divisors of the integer n, and let ϕ denote Euler's function, that is, ϕ(n) is the number of integers in the interval [1,n] that are relatively prime to n. It has been conjectured by Mąkowski and Schinzel that σ(ϕ(n))/n ≥ 1/2 for all n. We show that σ(ϕ(n))/n → ∞ on a set of numbers n of asymptotic density 1. In addition, we study the average order of σ(ϕ(n))/n as well as its range. We use similar methods to prove a conjecture of Erdős that ϕ(n-ϕ(n)) < ϕ(n)...
The arithmetic function counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of leads in a natural way to a certain error term which is known to be in mean-square. In this article it is proved that as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently...