On squares in Piatetski-Shapiro sequences (Q2075024)

Studying sequences containing infinitely many primes is a long-standing problem in number theory. \textit{I. I. Piatetski-Shapiro} [Mat. Sb., Nov. Ser. 33(75), 559--566 (1953; Zbl 0053.02702)] supplied a sequence sparser than the sequence \((qn+a)\) by studying the distribution of prime numbers of the form \(\lfloor n^c \rfloor\), where \(1 < c < 2\) is a fixed real number. This sequence can be considered as the simplest generalization of polynomials with non-integer degrees. Subsequently, several authors studied Piatetski-Shapiro type theorems, which incidentally may be of some help to exhibit the progress on exponential sums estimates. In the paper under review, the author establishes an asymptotic formula for the sum \[Q_c(s,N) := \sum_{\substack{n \le N \\ \lfloor n^c \rfloor = s\square}} 1\] where \(c \in \left( 1,2 \right)\), \(s \in \mathbb{Z}_{\ge 1}\) and the symbol \(\square\) meaning an unspecified square. Using classical techniques involving exponential sums and exponent pairs, the author derives the formula \[Q_c(s,N) = \frac{N^{1-c/2}}{\sqrt{s} (2-c)} + O \left( s^{- \rho(k,\ell)} N^{\theta(c,k,\ell)+\varepsilon} \right) \] where \((k, \ell)\) is an exponent pair and \[\rho(k,\ell) := \frac{\ell}{2(k+1)} \quad \text{and} \quad \theta(c,k,\ell) := \frac{c \ell + 2k}{2(k+1)}.\] This improves on a previous result by \textit{K. Liu} et al. [Acta Arith. 181, No. 3, 239--252 (2017; Zbl 1428.11051)]. The author also studies the mean \[\mathfrak{Q}_c(S,N) := \sum_{s \le S} \mu(s)^2 Q_c(s,N)\] with \(S \le N^c\). The difference \[\left| \mathfrak{Q}_c(S,N) - \frac{2 \sqrt{S}N^{1-c/2}}{\zeta(2)(2-c)} \right|\] which was also investigated in the above-mentioned paper of Liu et al., is here enhanced by the author, who first removes an error term, and secondly using the best exponent pair to date given in [J. Am. Math. Soc. 30, No. 1, 205--224 (2017; Zbl 1352.11065)], he proves that \(\mathfrak{Q}_c(S,N) \ll N^{1-\varepsilon}\) as soon as \(S \le N^{c - \varepsilon}\) and \(1 < c < \frac{249}{152} \approx 1.638\).