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 $\cup_{1\le a\le H}\{n\in(N,2N]:n\equiv a(\bmod m)\}$

References

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

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