An effective Bombieri-Vinogradov theorem and its applications. (English) Zbl 1399.11160

The standard proof of the famous Bombieri-Vinogradov Theorem on averaging the error term in results on primes in arithmetic progression is ineffective as it depends on an ineffective theorem of Siegel. The aim of this paper is to use instead a result of A. Page [Proc. London Math. Soc. (2), 39, 116–141 (1935; JFM 61.1070.01)] which is applied here in the form that at most one of the functions \(L(s,\chi)\) for non-principal characters \(\chi (\mod q)\) for \(q\leq Y=Y(x)\) has a zero \(\beta +i\gamma\) with \(\beta >1-\frac{0.05}{\log Y}\), \(\left| \gamma \right|\leq T=T(x)\) with \(Y,T\) defined in term of \(L=\log x\). Let \(\psi (y;q,a)=\sum \limits_{ n\leq y,\; n\equiv a\mod q}\Lambda (n)\) where \(\Lambda (n)\) is the von Mangoldt function. Then using Page’s result but otherwise a standard argument the author establishes that there exists a computable constant \(B>0\) such that \[ \sum_{q\leq x^{1/2}L^{-B}}\max_{y\leq x}\max_{(a,q)=1}\left| \psi (y;q,a)-\frac{y}{\varphi (q)}\right| = O(xL^{-1}(\log L)^{9}) \] with \(L=\log x\) and the constant implied by the \(O\) symbol is computable. The above result can be used to obtain an effective version of several well known theorems in the literature, such as those concerning the Titchmarsh divisor problem and similar ones, and C. Hooley’s paper [Acta Math. 97, 189–210 (1957; Zbl 0078.03504)] on the representation of large integers as the sum of two squares and a prime.


11N13 Primes in congruence classes
11M20 Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\)
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