Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization - Leonid G. Fel (Mathematics)
We consider a wide class of summatory functions <semantics>F<mfenced separators="" open="{" close="}">f;N,pm</mfenced>=∑k≤Nf<mfenced separators="" open="(" close=")">pmk</mfenced></semantics>, <semantics>m∈Z+∪{0}</semantics> associated with the multiplicative arithmetic functions f of a scaled variable <semantics>k∈Z+</semantics>, where p is a prime number. Assuming an asymptotic behavior of the summatory function, <semantics>F{f;N,1}<mover>=N→∞</mover>G1(N)<mfenced separators="" open="[" close="]">1+O<mfenced separators="" open="(" close=")">G2(N)</mfenced></mfenced></semantics>, where <semantics>G1(N)=Na1<mfenced separators="" open="(" close=")">logN</mfenced>b1</semantics>, <semantics>G2(N)=N−a2<mfenced separators="" open="(" close=")">logN</mfenced>−b2</semantics> and <semantics>a1,a2≥0</semantics>, <semantics>−∞<b1,b2<∞</semantics>, we calculate the renormalization function <semantics>R<mfenced separators="" open="(" close=")">f;N,pm</mfenced></semantics>, defined as a ratio <semantics>F<mfenced separators="" open="{" close="}">f;N,pm</mfenced>/F{f;N,1}</semantics>, and find its asymptotics <semantics>R∞<mfenced separators="" open="(" close=")">f;pm</mfenced></semantics> when <semantics>N→∞</semantics>. We prove that a renormalization function is multiplicative, i.e., <semantics>R∞<mfenced separators="" open="(" close=")">f;<msubsup>∏i=1n</msubsup><msubsup>pimi</msubsup></mfenced>=<msubsup>∏i=1n</msubsup>R∞<mfenced separators="" open="(" close=")">f;<msubsup>pimi</msubsup></mfenced></semantics> with n distinct primes <semantics>pi</semantics>. We extend these results to the other summatory functions <semantics>∑k≤Nf(pmkl)</semantics>, <semantics>m,l,k∈Z+</semantics> and <semantics>∑k≤N<msubsup>∏i=1n</msubsup>fi<mfenced separators="" open="(" close=")">kpmi</mfenced></semantics>, <semantics>fi≠fj</semantics>, <semantics>mi≠mj</semantics>. We apply the derived formulas to a large number of basic summatory functions including the Euler <semantics>ϕ(k)</semantics> and Dedekind <semantics>ψ(k)</semantics> totient functions, divisor <semantics>σn(k)</semantics> and prime divisor <semantics>β(k)</semantics> functions, the Ramanujan sum <semantics>Cq(n)</semantics> and Ramanujan <semantics>τ</semantics> Dirichlet series, and others.