Symmetry and short interval mean-squares

Abstract

The weighted Selberg integral is a discrete mean-square, that is a generalization of the classical Selberg integral of primes to an arithmetic function $f$, whose values in a short interval are suitably attached to a weight function. We give conditions on $f$ and select a particular class of weights, in order to investigate non-trivial bounds of weighted Selberg integrals of both $f$ and $f\ast \mu$. In particular, we discuss the cases of the symmetry integral and the modified Selberg integral, the latter involving the Cesaro weight. We also prove some side results when $f$ is a divisor function.

Through an optimal Lemma 3 we correct our Theorem 1 proof

References in corpus (3)

• On The Symmetry Of Arithmetical Functions In Almost All Short Intervals, V
• A generalization of Gallagher's lemma for exponential sums
• On the symmetry of the Liouville function in almost all short intervals

Cited by in corpus (4)

• Finite Ramanujan expansions and shifted convolution sums of arithmetical functions
• A generalization of Gallagher's lemma for exponential sums
• Some optimal links between generations of correlation averages
• Sieve functions in arithmetic bands, II