O jisté větě číselné
Odd perfect numbers
Odd perfect numbers are divisible by at least seven distinct primes
Odd perfect numbers of a special form
We show that there is an effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent.
On -tuples of twin practical numbers
Un intero positivo si dice pratico se ogni intero con può essere espresso come una somma di divisori distinti positivi di . In questo articolo è affrontato il problema dell'esistenza di infinite cinquine di numeri pratici della forma .
On a certain class of arithmetic functions
A homothetic arithmetic function of ratio is a function such that for every . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of in terms of the period and the ratio of .
On a class of arithmetic convolutions involving arbitrary sets of integers.
On a class of arithmetical functions connected with multiplicative functions.
On a class of multiplicative arithmetic functions.
On a class of -convolutions characterized by the identical equation
The identical equation for multiplicative functions established by R. Vaidyanathaswamy in the case of Dirichlet convolution in 1931 has been generalized to multiplicativity preserving -convolutions satisfying certain conditions (cf. [7]) which can be called as Lehmer-Narkiewicz convolutions for some reasons. In this paper we prove the converse.
On a Conjecture of A. Ivić and W. Schwarz
On A Conjecture of Emma Lehmer.
On a conjecture of Mąkowski and Schinzel
On a conjecture of Mąkowski and Schinzel concerning the composition of the arithmetic functions σ and ϕ
For any positive integer n let ϕ(n) and σ(n) be the Euler function of n and the sum of divisors of n, respectively. In [5], Mąkowski and Schinzel conjectured that the inequality σ(ϕ(n)) ≥ n/2 holds for all positive integers n. We show that the lower density of the set of positive integers satisfying the above inequality is at least 0.74.
On a conjecture of Narkiewicz about functions with non-decreasing normal order
On a divisibility problem
On a divisibility problem
Let be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if , then which improves a previous result of the second author.
On a function related to Ramanujan's tau function.
On a kind of generalized Lehmer problem
For , let be fixed numbers of the set , and let
On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions
The paper deals with lower bounds for the remainder term in asymptotics for a certain class of arithmetic functions. Typically, these are generated by a Dirichlet series of the form ζ 2(s)ζ(2s−1)ζ M(2s)H(s), where M is an arbitrary integer and H(s) has an Euler product which converges absolutely for R s > σ0, with some fixed σ0 < 1/2.