Laplace convolutions of weighted averages of arithmetical functions

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mi>G</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>g</m:mi><m:mo>;</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo>:=</m:mo><m:mrow><m:msub><m:mo largeop="true" symmetric="true">∑</m:mo><m:mrow><m:mi>n</m:mi><m:mo>≤</m:mo><m:mi>x</m:mi></m:mrow></m:msub><m:mrow><m:mi>g</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>n</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:mrow></m:mrow></m:math> {G(g;x):=∑n≤ xg(n)} be the summatory function of an arithmetical function <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>g</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>n</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {g(n)} . In this paper, we prove that we can write weighted averages of an arbitrary fixed number N of arithmetical functions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mrow><m:mrow><m:msub><m:mi>g</m:mi><m:mi>j</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>n</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow><m:mo rspace="4.2pt">,</m:mo><m:mi>j</m:mi></m:mrow><m:mo>∈</m:mo><m:mrow><m:mo stretchy="false">{</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:mi>N</m:mi><m:mo stretchy="false">}</m:mo></m:mrow></m:mrow></m:math> {gⱼ(n),\,j∈\{1,,N\}} as an integral involving the convolution (in the sense of Laplace) of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:msub><m:mi>G</m:mi><m:mi>j</m:mi></m:msub><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi>x</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {Gⱼ(x)} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>j</m:mi><m:mo>∈</m:mo><m:mrow><m:mo stretchy="false">{</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mi mathvariant="normal">…</m:mi><m:mo>,</m:mo><m:mi>N</m:mi><m:mo stretchy="false">}</m:mo></m:mrow></m:mrow></m:math> {j∈\{1,,N\}} . 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.