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Laplace convolutions of weighted averages of arithmetical functions

Marco Cantarini, Alessandro Gambini, Alessandro Zaccagnini

Abstract

Let $G(g;x):=\sum_{n\leq x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_{j}(n),\,j\in\left\{ 1,\dots,N\right\} $ as an integral involving the convolution (in the sense of Laplace) of $G_{j}(x),\,j\in\left\{ 1,\dots,N\right\} $. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.

Laplace convolutions of weighted averages of arithmetical functions

Abstract

Let be the summatory function of an arithmetical function . In this paper, we prove that we can write weighted averages of an arbitrary fixed number of arithmetical functions as an integral involving the convolution (in the sense of Laplace) of . Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.
Paper Structure (10 sections, 12 theorems, 119 equations)

This paper contains 10 sections, 12 theorems, 119 equations.

Table of Contents

  1. introduction
  2. Outline of the method
  3. Applications
  4. main identity
  5. the case g1g2 = Lambda
  6. the Cesàro weight case for the Goldbach numbers
  7. power series and Dirichlet series
  8. weighted averages of representations of a prime and an integer power
  9. a distributional approach to additive problems
  10. Acknowledgements

Key Result

Proposition 2

Let $f:\mathbb{R}\rightarrow\mathbb{C}$ and assume that: - $f$ has compact support on $\left[a,b\right]$, with $a<b$; - $f\in C^{1}\left(a,b\right)$; - $f^{\prime}$ is absolutely continuous on $\left(a,b\right)$; - Both $f\left(a^{+}\right)$ and $f^{\prime}\left(a^{+}\right)$ exist and are finite an

Theorems & Definitions (26)

  • Definition 1
  • Proposition 2
  • proof
  • Remark 3
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • Corollary 6
  • proof
  • ...and 16 more