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Third Italian Number Theory Meeting

Sieve functions in almost all short intervals

speaker: Giovanni Coppola (Università di Salerno)

abstract: An arithmetic function \(f\) is called a sieve function of range \(Q\), if it is the convolution product of the constantly \(1\) function and \(g\) such that \(g(q)\ll_{\varepsilon} q^{\varepsilon}\), \(\forall \varepsilon> 0\), for \(q\le Q\), and \(g(q)=0\) for \(q>Q\), i.e. \[ f(n):=\sum_{q
n,q\le Q}g(q). \] For example, the GPY truncated divisor sum \(\Lambda_R\) involves sieve functions of range \(R\). In a joint work with Maurizio Laporta (1, 2), we have started the study of the distribution of \(f\) in short intervals by analyzing its so-called weighted Selberg integral \[ J_{w,f}(N,H):=\sum_{N\sum_{x-H\le x\le x+H}w(n-x)f(n)-M_f(x,w)\right
^2 \] where w is a complex valued weight, that is bounded and supported in \([−H,H]\) with \(H=o(N)\), as \(N\to \infty\), while \(M_f(x,w):=\sum_a w(a)\sum_{q\le Q}g(q)/q\) is the expected (short intervals) mean value of the weighted \(f\) (see 1 for more general considerations when \(f\) has its Dirichlet series in the Selberg Class). We’ll give a short panorama of our methods to estimate \(J_{w,f}(N,H)\), including the recent applications of our results on the distribution of f over short arithmetic bands 3 \[ \cup_{1\le a\le H}\{n\in(N,2N]:n\equiv a(\bmod m)\} \]


1 G. Coppola and M. Laporta, Generations of correlation averages, J. of Numbers 2014 (2014), 13pp. (and draft http:/arxiv.orgabs1205.1706, v3)

2 G. Coppola and M. Laporta, Symmetry and short interval mean-squares, submit- ted, http:/arxiv.orgabs1312.5701

3 G. Coppola and M. Laporta, Sieve functions in arithmetic bands, submitted, http:/arxiv.orgabs1503.07502

Mon 21 Sep, 17:00 - 17:25, Aula Russo
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