Pretentiously detecting power cancellation (Q2842340)

Let \(f\), \(g\) be multiplicative functions taking values in the complex unit disc, and define NEWLINE\[NEWLINE\mathbb D(f,g)^2:= \sum_p{1-\text{Re}(f(p)\overline g(p))\over p}.NEWLINE\]NEWLINE When this quantity is finite, Granville and Soundararajan called the functions \(f(n)\) and \(g(n)\) pretentious to each other, and they have written a series of papers on this topic; see, for example, the report by \textit{A. Granville} [J. Théor. Nombres Bordx. 21, No. 1, 159--173 (2009; Zbl 1236.11086)]. The aim of the present paper is to extend the idea of pretentiousness in two different ways in order to preserve power cancellation. NEWLINENEWLINEFor any multiplicative functions \(f(n)\), \(g(n)\) define the multiplicative function \(h(n)\) by \(g(n)= (f* h)(n)\), where the right-side represents the Dirichlet convolution. For \(\beta>0\) let NEWLINE\[NEWLINE\begin{aligned} H_\beta(f,g) &:= \sum_p \sum^\infty_{k=1} {|h(p^k)|\over p^{k\beta}},\\ \widehat{\mathbb D}_\beta(f,g) &:= \sum_p \sum^\infty_{j=1} {|g(p^j)-f(p^j)|\over p^{j\beta}}.\end{aligned}NEWLINE\]NEWLINE The authors explore the interrelationship between the finiteness of \(H_\beta(f,g)\) and \(\widehat{\mathbb D}_\beta(f,g)\) for some or all \(\beta> 0\) under various assumptions. They use these functions to determine conditions under which \(S_f(x):=\sum_{n\leq x} f(n)\ll x^\alpha\) \((\alpha>0)\) implies that \(S_g(x)\ll x^{\max(\alpha,\beta)}\) or just \(S_g(x)\ll x^\alpha\). Altogether the authors establish six theorems, and their proofs involve some manipulations of multiple sums and other techniques from linear algebra and complex analysis.