On the Correlations, Selberg Integral and Symmetry of Sieve Functions in Short Intervals, III
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 $q\leq Q$, and $g(q)=0$ for $q>Q$. Here we establish a new result on the autocorrelation of $f$ by using a famous theorem on bilinear forms of Kloosterman fractions by Duke, Friedlander and Iwaniec. In particular, for such correlations we obtain nontrivial asymptotic formulæ that are actually unreachable by the standard approach of the distribution of $f$ in the arithmetic progressions. Moreover, we apply our asymptotic formulæ to obtain new bounds for the socalled Selberg integral and symmetry integral of $f$, which are basic tools for the study of the distribution of $f$ in short intervals.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.0302
 Bibcode:
 2010arXiv1003.0302C
 Keywords:

 Mathematics  Number Theory;
 11N37;
 11N25
 EPrint:
 This is a much expanded version ! (Already submitted)