Harmonic analysis on the positive rationals. I: Basic results (Q2825292)

By using powerful methods of harmonic analysis on the positive rationals, the authors prove essentially the following result: Let \(\alpha\), \(\gamma\), \(x\) be real numbers and \(D\) an integer satisfying \(0<\alpha<1\), \(0<\gamma<1\), \(1\leq D\leq x\). Let \(g\) be a multiplicative function with values in the complex unit disc. Then there are nonprincipal Dirichlet characters \(\chi_j \pmod D\), their number bounded in terms of \(\alpha\) alone, such that NEWLINE\[NEWLINE\begin{multlined} \sum_{\substack{ n\leq y,\\ n\equiv a\pmod D}} g(n)= {1\over\varphi(D)}\sum_{\substack{ n\leq y,\\ (n,D)=1}} g(n)+ \sum_j{\chi_j(a)\over \varphi(D)} \sum_{n\leq y} g(n)\chi_j(n)\\ +O\Biggl({y\over \varphi(D)\log y} \prod_{\substack{ p\leq D,\\ (p,D)= 1}}\,\Biggl(1+ {|g(p)|\over p}\Biggr)\Biggl({\log y\over\log D}\Biggr)^\alpha\Biggr),\end{multlined}NEWLINE\]NEWLINE uniformly for \((a,D)= 1\), \(x^\gamma\leq y\leq x\). In particular, the error term is \(\ll D^{-1} y(\log D/\log y)^{1-\alpha}\).