On distribution of primes in an arithmetical progression
The authors prove the following theorem: Let \((a,q)=1\), where \(1\leq q\leq \log^3x\) and \(x\geq \exp \exp (11.5)\). Then \[ \left| \sum_{_{\substack{\ell =1,\ldots,q\\ (\ell,q)=1}}} e\bigl(\frac{a\ell}{q}\bigr)\psi (x;q,\ell)-\frac{\mu (q)x}{\phi (q)}+\frac{\tilde E \tilde\chi(a)\tau(\tilde\chi)x^{\tilde \beta}}{\tilde \beta\phi (q)}\right| \leq 0.13xq^{0.5}(\log x)^{-10.35}. \] Here, apart from the usual notation, \(\tau(\tilde \chi)=\sum^q_{h=1}\tilde\chi(h)e(h/q)\) and \(\tilde E\) is 1 or 0 depending on whether there is, or is not, a real character \(\tilde\chi\pmod q\) such that \(L(s,\tilde \chi)\) has a zero \(\tilde\beta\geq 1-0.1077/\log q\). The theorem here is relevant to the determination of an explicit bound for the validity of the ternary Goldbach-Vinogradov theorem, in which the authors proved elsewhere that every odd \(N>10^{43001}\) is a sum of three odd primes.
scientific article; zbMATH DE number 4175055