Giovanni Coppola

Generations of Correlation Averages

We give a general link between weighted Selberg integrals of any arithmetic function and averages of correlations in short intervals, proved by the elementary dispersion method (our version of Linnik’s method). We formulate conjectural bounds for the so-called modified Selberg integral of the divisor functions , gauged by the Cesaro weight in the short interval and improved by these some recent results by Ivic. The same link provides, also, an unconditional improvement. Then, some remarkable conditional implications on the 2th moments of Riemann zeta function on the critical line are derived. We also give general requirements on that allow our treatment for weighted Selberg integrals.

On the error term in a Parseval type formula in the theory of Ramanujan expansions II

Abstract Given two arithmetical functions f , g we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums ∑ n ≤ N f ( n ) g ( n + h ) , building on an earlier work of Gadiyar, Murty and Padma. A key role in our method is played by the theory of Ramanujan expansions for arithmetical functions.

Symmetry and Short Interval Mean-Squares

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.

Sieve Functions in Arithmetic Bands, II

An arithmetic function f is called a sieve function of range Q if its Eratosthenes transform g = f * μ is supported in [1,Q] ∩ N, where g(q) ≪εqε(∀ε > 0). We continue our study of the distribution of f(n) over short arithmetic bands, n ≡ ar + b (mod q), with n ∈ (N,2N] ∩ N, 1 ≤ a ≤ H = o(N) and r,b ∈ Z such that g:c:d:(r,q) = 1. In particular, the optimality of some results is discussed.

Finite Ramanujan expansions and shifted convolution sums of arithmetical functions

Abstract We continue our study of convolution sums of two arithmetical functions f and g , of the form ∑ n ≤ N f ( n ) g ( n + h ) , in the context of heuristic asymptotic formulae. Here, the integer h ≥ 0 is called, as usual, the shift of the convolution sum. We deepen the study of finite Ramanujan expansions of general f , g for the purpose of studying their convolution sum. Also, we introduce another kind of Ramanujan expansion for the convolution sum of f and g , namely in terms of its shift h and we compare this ‘‘shift Ramanujan expansion’’, with our previous finite expansions in terms of the f and g arguments. Last but not least, we give examples of such shift expansions, in classical literature, for the heuristic formulae.