Symmetry and Short Interval Mean-Squares
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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 * μ. 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.
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