On Berndt's Method in Arithmetical Functions and Contour Integration

Arithmetical Functions I: Multiplicative Functions

Truth be told, this definition is a bit embarrassing. It would mean that taking any function from calculus whose domain contains [1,+∞) and restricting it to positive integer values, we get an arithmetical function. For instance, e −3x cos2 x+(17 log(x+1)) is an arithmetical function according to this definition, although it is, at best, dubious whether this function holds any significance in n...

On the binomial convolution of arithmetical functions

Let n = ∏ p p νp(n) denote the canonical factorization of n ∈ N. The binomial convolution of arithmetical functions f and g is defined as (f ◦g)(n) = ∑ d|n (∏ p (νp(n) νp(d) )) f(d)g(n/d), where ( a b ) is the binomial coefficient. We provide properties of the binomial convolution. We study the Calgebra (A,+, ◦,C), characterizations of completely multiplicative functions, Selberg multiplicative...

On the Symmetry of Arithmetical Functions

We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums. 1. Introduction and statement of the results. We pursue the study of the symmetry in (almost all) short intervals of arithmetical functions f (see [C1]), where this time we give (non-trivial) results for a new class of such (real) f ; the key-property they have is a non-negative exponential sum ...

On Some Rings of Arithmetical Functions

In this paper we consider several constructions which from a given B-product ∗B lead to another one ∗̃B . We shall be interested in finding what algebraic properties of the ring RB = 〈CN, +, ∗B 〉 are shared also by the ring RB̃ = 〈C N, +, ∗B 〉. In particular, for some constructions the rings RB and RB̃ will be isomorphic and therefore have the same algebraic properties. §

A Numerical Integration Method by Using Generalized Series of Functions