On Vandiver’s arithmetical function – I

We study certain properties of Vandiver’s arithmetic function V(n) = \prod_{d|n} (d+1).

Sums of Averages of GCD-Sum Functions II

Let $$ \gcd (k,j) $$ gcd ( k , j ) denote the greatest common divisor of the integers k and j, and let r be any fixed positive integer. Define $$\begin{aligned} M_r(x; f) := \sum _{k\le x}\frac{1}{k^{r+1}}\sum _{j=1}^{k}j^{r}f(\gcd (j,k)) \end{aligned}$$ M r ( x ; f ) : = ∑ k ≤ x 1 k r + 1 ∑ j = 1 k j r f ( gcd ( j , k ) ) for any large real number $$x\ge 5$$ x ≥ 5 , where f is any arithmetical function. Let $$\phi $$ ϕ , and $$\psi $$ ψ denote the Euler totient and the Dedekind function, respectively. In this paper, we refine asymptotic expansions of $$M_r(x; \mathrm{id})$$ M r ( x ; id ) , $$M_r(x;{\phi })$$ M r ( x ; ϕ ) and $$M_r(x;{\psi })$$ M r ( x ; ψ ) . Furthermore, under the Riemann Hypothesis and the simplicity of zeros of the Riemann zeta-function, we establish the asymptotic formula of $$M_r(x;\mathrm{id})$$ M r ( x ; id ) for any large positive number $$x>5$$ x > 5 satisfying $$x=[x]+\frac{1}{2}$$ x = [ x ] + 1 2 .

Distribution and non-vanishing of special values of L-series attached to Erdős functions

In a written correspondence with A. Livingston, Erdős conjectured that for any arithmetical function [Formula: see text], periodic with period [Formula: see text], taking values in [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text], the series [Formula: see text] does not vanish. This conjecture is still open in the case [Formula: see text] or when [Formula: see text]. In this paper, we obtain the characteristic function of the limiting distribution of [Formula: see text] for any positive integer [Formula: see text] and Erdős function [Formula: see text] with the same parity as [Formula: see text]. Moreover, we show that the Erdős conjecture is true with “probability” one.

Sums of weighted averages of gcd-sum functions

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]

Counting Functions to Generate The Primes in the RSA Algorithm and Diffie-Hellman Key Exchange

The Rivest–Shamir–Adleman (RSA) and the Diffie-Hellman (DH) key exchange are famous methods for encryption. These methods depended on selecting the primes p and q in order to be secure enough . This paper shows that the named methods used the primes which are found by some arithmetical function .In the other sense, no need to think about getting primes p and q and how they are secure enough, since the arithmetical function enable to build the primes in such complicated way to be secure. Moreover, this article gives new construction of the RSA algorithm and DH key exchange using the primes p,qfrom areal number x.

On the Riesz means of δ k (n)

International audience Let k ≥ 1 be an integer. Let δ k (n) denote the maximum divisor of n which is co-prime to k. We study the error term of the general m-th Riesz mean of the arithmetical function δ k (n) for any positive integer m ≥ 1, namely the error term E m,k (x) where 1 m! n≤x δ k (n) 1 − n x m = M m,k (x) + E m,k (x). We establish a non-trivial upper bound for E m,k (x) , for any integer m ≥ 1.