On the symmetry of primes in almost all short intervals (Q2504963)

Denote by \(\Lambda(n)\) von Mangoldt function. Given positive integers \(h, N\), where \(h=h(N) \to\infty\), the author proves for a given \(A>2\) the mean-square estimate \[ \sum_{x \sim N} \left| \sum_{j\leq 2h} \; \sum_{d;\;| d - \frac xj| \leq \frac hj} \Lambda(d) \cdot\text{sgn} \left( d - \frac xj\right) \right|^2 \ll \frac{N \, h^2}{\log^A (N)}, \] as long as \[ N^{\frac13} \log^B(N) \leq h(N) \ll \frac N{\log^A(N)}, \] where \(B=B(A)\) can be explicitly computed. A similar result is given for the generalized von Mangoldt function \(\displaystyle \Lambda_K(n) = \sum_{d\mid n}\mu(d) \cdot \log^K\left(\frac nd\right)\). For the proof, the sum \[ \sum_{d\leq x+h} \Lambda(d) \cdot \left( - \sum_{{|n-h| \leq x}\atop{ n\equiv 0\bmod d}}\text{sgn}(n-x)\right) \] is \(\Bigl[\)with some parameter \(M\), which later is chosen as \(N^{1/6} \log^{4/3}(N)\Bigr]\) is split into two ranges \(d \leq M \cdot\sqrt N\) and \(M \sqrt N < d \leq x+h\). The first sum is dealt with the Large Sieve; for the second sum a lemma from a paper of \textit{G. Coppola} and \textit{S. Salerno} [Acta Arith. 113, No. 2, 189--201 (2004; Zbl 1122.11062)] is important.