Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

We consider a wide class of summatory functions Ff;N,pm=∑k≤Nfpmk, m∈Z+∪{0} associated with the multiplicative arithmetic functions f of a scaled variable k∈Z+, where p is a prime number. Assuming an asymptotic behavior of the summatory function, F{f;N,1}=N→∞G1(N)1+OG2(N), where G1(N)=Na1logNb1, G2(N)=N−a2logN−b2 and a1,a2≥0, −∞<b1,b2<∞, we calculate the renormalization function Rf;N,pm, defined as a ratio Ff;N,pm/F{f;N,1}, and find its asymptotics R∞f;pm when N→∞. We prove that a renormalization function is multiplicative, i.e., R∞f;∏i=1npimi=∏i=1nR∞f;pimi with n distinct primes pi. We extend these results to the other summatory functions ∑k≤Nf(pmkl), m,l,k∈Z+ and ∑k≤N∏i=1nfikpmi, fi≠fj, mi≠mj. We apply the derived formulas to a large number of basic summatory functions including the Euler ϕ(k) and Dedekind ψ(k) totient functions, divisor σn(k) and prime divisor β(k) functions, the Ramanujan sum Cq(n) and Ramanujan τ Dirichlet series, and others.