On the distribution of \(p^\alpha\) modulo one
The authors give a new upper bound for the discrepancy \[ D(N):= \sup_{0\leq\gamma\leq 1} \Bigl|\sum_{\substack{ p\leq N\\ \{p^\alpha\}\leq \gamma}} 1-\pi(N)\gamma \Bigr| \] for the sequence \((p^\alpha)\) when \(5/3\leq \alpha\leq 3\) and \(\alpha\neq 2\). As usual, \(p\) denotes a prime number, \(\pi(N)\) is the number of primes less than or equal to \(N\) and \(\{x\}\) is the fractional part of the real number \(x\). They prove that for the specified range of \(\alpha\) the estimate \(D(N)= O(N^{1-\delta+\varepsilon})\) holds, where \(\delta= \delta(\alpha)\) is explicitly given. They combine the double large sieve inequality due to Bombieri and Iwaniec and Heath-Brown's method to decompose sums over primes.
scientific article; zbMATH DE number 1426379
scientific article