Mohammad Zaki

ARITHMETICAL FUNCTIONS IN SEVERAL VARIABLES

In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R).

An algebraic independence result for Euler products of finite degree

We investigate the algebraic independence of some derivatives of certain multiplicative arithmetical functions over the field C of complex numbers.

Natural boundaries of a family of Dirichlet series

We study the natural boundary of the Dirichlet series Fq,b,H,α(s) = ∑ m,k≥1 mk≡b (mod q) H(α log(m + k)) Λ(m)Λ(k) (m + k)s , where Λ is the classical Von Mangoldt function, H is a smooth periodic function with period 1, α > 0 is a real number and b, q > 0 are integers with (b, q) = 1.

On the singularities of multiple L-functions

We investigate the singularities of a class of multiple L-functions considered by Akiyama and Ishikawa [2].

ARITHMETIC FUNCTIONS ON GAUSSIAN INTEGERS

In this paper we present a short study of the ring of complex-valued arithmetic functions defined on the ring of Gaussian integers. We investigate an absolute value and the corresponding metric structure on this ring.