A Bombieri-Vinogradov-type result for exponential sums over Piatetski-Shapiro primes (Q2692192)

The prime numbers of the form \(p=[n^{1/\gamma}]\) are called the Piatetski-Shapiro primes of type \(\gamma\). In this paper, the author establishes a new Bombieri-Vinogradov-type result for exponential sums running over Piatetski-Shapiro primes \(p\) of type \(\gamma\in(865/886,1)\) belonging to the arithmetic progression \(p\equiv a\) modulo \(d\) with \(a\in\mathbb{Z}\) and \(a\neq 0\). More precisely, letting \(1<c<3\), \(c\neq 2\), and \(A>0\), then the author shows that for any \(|t|<X^{1/4-c}\) any sufficiently small \(\varepsilon>0\), \[ \sum_{\substack{d\leq X^\theta\\ \gcd(d,a)=1}}\ \Big|\sum_{\substack{p\leq X\\ p\equiv a\pmod{d}\\ p=[n^{1/\gamma}]}} p^{1-\gamma}e(tp^c)\log p-\frac{\gamma}{\varphi(d)}\int_2^X e(ty^c)\,dy\Big| \ll\frac{X}{(\log X)^A}, \] where \(\theta=\theta(\gamma)=443\gamma/55-173/22-\varepsilon\), and \(e(x)=e^{2\pi i x}\).