Localized oscillations over the divisors (Q2890306)

For a multiplicative function \(f\) define NEWLINE\[NEWLINE\Delta(n,f)=\sup_{u\in\mathbb{R},\,0\leq v\leq 1}\left|\sum_{d|n,\,e^u<d\leq e^{u+v}}f(d)\right|.NEWLINE\]NEWLINE When \(f\equiv 1\) this is just Hooley's \(\Delta\)-function. In this case the maximum over \(v\) is unnecessary, since \(f\) is real and positive, but the present paper is concerned primarily with the situation in which \(f=\chi\), a non-principal quadratic character. The primary object of study is the weighted mean-value NEWLINE\[NEWLINES(x,\chi,t,g):=\sum_{n\leq x}g(n)|\Delta(n,\chi)|^{2t}NEWLINE\]NEWLINE where \(t\geq 1\) is a fixed real number and \(g\) is a non-negative multiplicative function. The paper imposes simple conditions on \(g\), including the assumption that NEWLINE\[NEWLINE\sum_{p\leq x}g(p)=y\,\text{li}(x)+O(x\exp\{-c(\log x)^{\eta}\})NEWLINE\]NEWLINE for suitable positive constants \(c\) and \(\eta\), and concludes that NEWLINE\[NEWLINES(x,\chi,g,t)\ll_{\chi,g,t} x (\log x)^{y-1+(2^{2t-1}y-y-t)^++o(1)}.NEWLINE\]NEWLINE In particular one has NEWLINE\[NEWLINE\sum_{n\leq x}|\Delta(n,\chi)|^{2}\ll_{\chi}x(\log x)^{o(1)}.NEWLINE\]NEWLINENEWLINENEWLINEIn the general case one can improve the exponent of \(\log x\) for integral \(t\), to \(2^{2t-1}y-2t-1\) for suitable \(y\). In particular, it suffices to have \(y\geq 4/7\) for \(t\geq 2\).NEWLINENEWLINEThe study of \(S(x,\chi,g,t)\), with \(\chi\) being the non-trivial character modulo 4, forms the key step in the authors' forthcoming treatment of Manin's conjecture for Chatelêt surfaces of the type \(F(x)=y^2+z^2\), where \(F\) is an irreducible quartic polynomial. For this application it is important to be able to handle multiplicative functions \(g\) other than \(g\equiv 1\).NEWLINENEWLINEThe paper also examines the function \(f(n)=\mu(n)\), and gives an upper bound for \(S(x,\mu,g,t)\).NEWLINENEWLINELower bounds are also considered, and it is shown for example that NEWLINE\[NEWLINES(x,\mu,\mu^2,1)\gg x(\log\log x)^{c+o(1)}NEWLINE\]NEWLINE with NEWLINE\[NEWLINEc=\frac{\log 4}{|1-(\log 3)^{-1})|}=0.57\ldotsNEWLINE\]