Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

Abstract

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number. Assuming an asymptotic behavior of summatory function, F{f;N,1}\stackrel{N\to \infty}{=}G_1(N) [1+ {\cal O}(G_2(N))], where G_1(N)=N^{a_1}(log N)^{b_1}, G_2(N)=N^{-a_2}(log N)^{-b_2} and a_1, a_2\geq 0, -\infty < b_1, b_2< \infty, we calculate a renormalization function defined as a ratio, R(f;N,p^m)=F{f;N,p^m}/F{f;N,1}, and find its asymptotics R_{\infty}(f;p^m) when N\to \infty. We prove that the renormalization function is multiplicative, i.e., R_{\infty}(f;\prod_{i=1}^n p_i^{m_i})= \prod_{i=1}^n R_{\infty}(f;p_i^{m_i}) with n distinct primes p_i. We extend these results on the others summatory functions \sum_{k\leq N}f(p^m k^l), m,l,k\in \mathbb Z}_+ and \sum_{k\leq N}\prod_{i=1}^n f_i(k p^{m_i}), f_i\neq f_j, m_i\neq m_j. We apply the derived formulas to a large number of basic summatory functions including the Euler \phi(k) and Dedekind \psi(k) totient functions, divisor \sigma_n(k) and prime divisor \beta(k) functions, the Ramanujan sum C_q(n) and Ramanujan \tau(k) Dirichlet series, and others.