A Bombieri-Vinogradov type exponential sum result with applications
In this paper the exponential sums \[ S_d=\sum\limits_{n\sim x\atop{n\equiv c\pmod d}}\Lambda (n)e(\alpha n) \] are considered. Here \(\Lambda (n)\) is von Mangoldt function, \(x\) is a large positive number, \(c\neq 0\) is a integer, and \(\alpha\) is a real number with a rational approximation \(a/q\) satisfying \[ \bigg | \alpha -\frac{a}{q} \bigg |<\frac{1}{q^2},\;\;(a,q)=1,\;q\geq 1\,. \] For well-factorable function \(\lambda\) of level \(D\) the author prove that \[ \sum\limits_{d\sim D\atop{(d,c)=1}}\lambda_d \sum\limits_{k\sim K}c_k \sum\limits_{n\sim x\atop{n\equiv c\pmod d}}\Lambda (n)e(\alpha nk) \ll\tag{1} \] \[ K(\log x)^Cx^{3/4+\eta} \bigg( \frac{x}{q}+\frac{q}{K}+D^2+x^{7/9+4\eta}+ \min\bigg\{ d^{4+20\eta}, \frac{x}{D} \bigg\} \bigg)^{1/4-\eta}\,, \] where \(\eta>0\) and \(c_k\ll \tau (k)^C\) for some \(C>0\). This estimate improves the previous result of \textit{H. Mikawa} [Tsukuba J. Math. 24, No. 2, 351--360 (2000; Zbl 1002.11066)]. As a corollary a Bombieri-Vinogradov type result for sums \(S_d\) is obtained. Using (1) the author shows that for any irrational \(\alpha\), real \(\beta\) and some \(\theta>0\) there are infinitely many primes \(p\) such that \(p+2\) has at most two prime factors and \(\|\alpha p+\beta\|<p^{-\theta}\).
scientific article; zbMATH DE number 5575285
scientific article