Geodesic
Symmetry and short interval mean-squares
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 62-85

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The weighted Selberg integral is a discrete mean-square that generalizes 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*\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.
Keywords: mean square, short interval, symmetry, correlation. 
@article{TM_2017_299_a3,
     author = {Giovanni Coppola and Maurizio Laporta},
     title = {Symmetry and short interval mean-squares},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {62--85},
     publisher = {mathdoc},
     volume = {299},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a3/}
}
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Giovanni Coppola; Maurizio Laporta. Symmetry and short interval mean-squares. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 62-85. http://geodesic.mathdoc.fr/item/TM_2017_299_a3/