Still another proof of Parseval's equation for almost-even arithmetical functions

Abstract

Still another proof is given for Parseval's well-known equation

$$\sum\limits_{1 \leqslant r< \infty } {|a_r |^2 \cdot \varphi (r) = \parallel f\parallel \begin{array}{*{20}c} 2 \\ 2 \\ \end{array} } $$

forB ²-almost-even arithmetical functionsf with Ramanujan coefficients$a_r = \frac{1}{{\varphi (r)}} \cdot M(f \cdot c_r )$. An explicit “best approximation” off by even functions, constructed from characteristic functions of subsets {n; g.c.d.(n, r) =k} ⊂$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{\mathbb{N}}$ is used.