Large values of the sum of divisors function in arithmetic progressions
Let \(\sigma(n)\) denote as usual the sum of divisors of \(n\), and for \(k\geq 2\), \((\ell,k)=1\), set \[ \sigma_{k,\ell}(n)=\prod_{{p^ m\| n,}\atop {p\equiv \ell \bmod k}}\sigma(p^ m). \] It is well-known that the function \(f(n)=\sigma(n)/(n \log \log n)\) satisfies \(\limsup_{n\to \infty}f(n)=e^{\gamma}\), and in an earlier paper [ibid. 63, 187--213 (1984; Zbl 0516.10036)] the author showed that the statement ``\(f(n)<e^{\gamma}\) for sufficiently large \(n\)'' is equivalent to the Riemann Hypothesis. In the paper under review he considers the analogous problem for the functions \(\sigma_{k,\ell}(n)\). He first shows that if \[ f_{k,\ell}(n)=\frac{\sigma_{k,\ell}(n)}{n \log (\phi (k) \log n)^{1/\phi (k)}}, \] then \(\alpha_{k,\ell}:=\limsup_{n\to \infty}f_{k,\ell}(n)\) is finite and non-zero. He then proves that, subject to some minor restrictions on \(k\) and \(\ell\), the statement ``\(f_{k,\ell}(n)<\alpha_{k,\ell}\) for sufficiently large \(n\)'' is equivalent to the truth of the Riemann Hypothesis for \(L\)-functions belonging to characters modulo \(k\).
scientific article; zbMATH DE number 4029693