On a generalization of the Selberg formula
Define the generalized von Mangoldt function \(\Lambda_k\) by \(\Lambda_k(n)= \sum_{d|n}\mu(d)(\log {n\over d})^k\), and for \((a, q)= 1\) let \(\psi_k(x;q, a)= \sum_{\substack{ n\leq x\\ n\equiv a\pmod q}} \Lambda_k(n)\). For the case \(k= 1\), the Siegel-Walfisz theorem yields the asymptotic formula \(\psi((x; q, a)\sim{x\over\varphi(q)}\), valid uniformly in the range \(q< (\log x)^A\) for any fixed \(A\). In [Recent progress in analytic number theory, Durham Symposium 1979. Vol. 1, 15--23 (1981; Zbl 0459.10029)], \textit{J. B. Friedlander} used a sieve method developed by E. Bombieri to prove that \(\psi_2(x;q, a)\sim{2x\over\psi(q)}\log x\) uniformly for \(\log q< \varepsilon(x)\log x\) for a fixed positive function \(\varepsilon(x)\to 0\) as \(x\to\infty\), and he showed that analytic methods yield this asymptotic formula in the smaller range of validity \(\log q< \varepsilon(x){\log x\over\log\log x}\). The present author generalizes Friedlander's lemmas 3.1 and 3.2, and then establishes an estimate for the sum \(\psi_k(x;\chi)= \sum_{n\leq x} \chi(n)\Lambda_k(n)\), where \(\chi\) denotes a character modulo \(q\). He deduces that for \(k\geq 2\) and \(\varepsilon(x)\) as above \[ \psi_k(x; q,a)\sim{kx\over\varphi(q)}(\log x)^{k- 1} \] uniformly for \(\log q<(\varepsilon(x))^{{1\over k-1}}{\log x\over\log\log x}\). He also derives some other related results.
scientific article; zbMATH DE number 5190746
scientific article