Multiplicative functions of polynomial values in short intervals
Multiplicative functions over short segments
Note on multiplicative functions satisfying congruence property. II.
On an arithmetic function considered by Pillai
For every positive integer let be the largest prime number . Given a positive integer , we study the positive integer such that if we define recursively for , then is a prime or . We obtain upper bounds for as well as an estimate for the set of whose takes on a fixed value .
On composite n for which φ(n) | n -1
On Euler-von Mangoldt's equation
On Gelfond’s conjecture about the sum of digits of prime numbers
The goal of this paper is to outline the proof of a conjecture of Gelfond [6] (1968) in a recent work in collaboration with Christian Mauduit [11] concerning the sum of digits of prime numbers, reflecting the lecture given in Edinburgh at the Journées Arithmétiques 2007.
On Jacobsthal's g(n)-Function.
On multiplicatively perfect numbers.
On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions ϕ and σ
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)...
On sums of Ramanujan sums
On the Average Order of an Arithmetical Function
On the congruences and
On the counting function for the generalized Niven numbers
Given an integer base and a completely -additive arithmetic function taking integer values, we deduce an asymptotic expression for the counting functionunder a mild restriction on the values of . When , the base sum of digits function, the integers counted by are the so-called base Niven numbers, and our result provides a generalization of the asymptotic known in that case.
On the distribution of perfect totients.
On the frequencies of large values of divisor functions
On the height of cyclotomic polynomials
On the largest -primitive subset of .
On the largest prime factor of
For an integer we denote by the largest prime factor of . We obtain several upper bounds on the number of solutions of congruences of the form and use these bounds to show that
On the largest prime factor of the partition function of n