Distribution of values of a real function: Means, moments, and symmetry.
Distributions of the values of some arithmetical functions
Divisors of the Euler and Carmichael functions
Equations and inequalities involving .
Étude d'une somme arithmétique multiple liée à la fonction de Möbius
Linear correlations amongst numbers represented by positive definite binary quadratic forms
Linear equations with the Euler totient function
Lower bounds for a certain class of error functions
Nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines
We show that the density functions of nearest neighbor spacing distributions for the zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which appears in value distribution of the logarithmic derivative of the Riemann zeta-function on vertical lines.
(Non)Automaticity of number theoretic functions
Denote by Liouville’s function concerning the parity of the number of prime divisors of . Using a theorem of Allouche, Mendès France, and Peyrière and many classical results from the theory of the distribution of prime numbers, we prove that is not –automatic for any . This yields that is transcendental over for any prime . Similar results are proven (or reproven) for many common number–theoretic functions, including , , , , , and others.
On a Brun-Titchmarsh inequality for multiplicative functions
On a multiplicative hybrid problem
On almost even arithmetical functions via orthonormal systems on Vilenkin groups
On consecutive integers divisible by the number of their divisors
We prove that there are no strings of three consecutive integers each divisible by the number of its divisors, and we give an estimate for the number of positive integers n ≤ x such that each of n and n + 1 is a multiple of the number of its divisors.
On fluctuations in the mean of a sum-of-divisors function
On fluctuations in the mean of a sum-of-divisors function, II
I give explicit values for the constant implied by an Omega-estimate due to Chen and Chen [CC] on the average of the sum of the divisors of n which are relatively coprime to any given integer a.
On invariants related to non-unique factorizations in block monoids and rings of algebraic integers
On isolated, respectively consecutive large values of arithmetic functions
On rough and smooth neighbors.
We study the behavior of the arithmetic functions defined byF(n) = P+(n) / P-(n+1) and G(n) = P+(n+1) / P-(n) (n ≥ 1)where P+(k) and P-(k) denote the largest and the smallest prime factors, respectively, of the positive integer k.
On sets characterizing number-theoretical functions (II) (The set of "prime plus one"'s is a set of quasi-uniqueness)