Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization

Leonid G. Fel ()
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Leonid G. Fel: Department of Civil Engineering, Technion—Israel Institute of Technology, Haifa 32000, Israel

Mathematics, 2025, vol. 13, issue 2, 1-43

Abstract: We consider a wide class of summatory functions F f ; N , p m = ∑ k ≤ N f p m k , 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 → ∞ G 1 ( N ) 1 + 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 ≥ 0 , − ∞ < b 1 , b 2 < ∞ , we calculate the renormalization function R f ; N , p m , defined as a ratio F f ; N , p m / F { f ; N , 1 } , and find its asymptotics R ∞ f ; p m when N → ∞ . We prove that a renormalization function is multiplicative, i.e., R ∞ f ; ∏ i = 1 n p i m i = ∏ i = 1 n R ∞ f ; p i m i with n distinct primes p i . We extend these results to the other summatory functions ∑ k ≤ N f ( p m k l ) , m , l , k ∈ Z + and ∑ k ≤ N ∏ i = 1 n f i k p m i , f i ≠ f j , m i ≠ m j . 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 σ n ( k ) and prime divisor β ( k ) functions, the Ramanujan sum C q ( n ) and Ramanujan τ Dirichlet series, and others.

Keywords: multiplicative number theory; summatory functions; asymptotic analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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