Topics in analytic number theory
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Date
2016
Authors
Aryan, Farzad
University of Lethbridge. Faculty of Arts and Science
Journal Title
Journal ISSN
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Publisher
Lethbridge, Alta : University of Lethbridge, Dept. of Mathematics and Computer Science
Abstract
In this thesis, we investigate three topics belonging
to the probabilistic, classical and modern branches
of analytic number theory.
Our first result concerns the probabilistic
distribution of squares modulo a composite number,
and of tuples of reduced residues, in short intervals.
We obtain variance upper bounds generalizing
those of Montgomery and Vaughan, as well as new
lower bounds.
Our second work, joint with Nathan Ng, concerns the estimation
of discrete mean values of Dirichlet polynomials,
where summation is over the zeros of an $L$-function
attached to an automorphic representation.
Conditionally on strong bounds on autocorrelations of
the coefficients of $L$-functions, a corollary of our results
is that the gaps between consecutive zeros of the Riemann zeta function
are infinitely often smaller than half of the average gap.
Our last work concerns
the additive and quadratic divisor problem.
We study shifted convolution sums
for the divisor function, Fourier coefficients of a cusp form
and the representation function of integers as sums of two squares.
For convolution sums of a certain type,
we improve several estimates available in the literature,
by expanding the delta-method of Duke, Friedlander and Iwaniec. Also by using a smooth variant of Dirichlet hyperbola method, we improve the error term obtained by Duke, Friedlander and Iwaniec in the quadratic divisor problem.
Description
doi:10.1093/imrn/rnv061
Keywords
analytic number theory , divisor function , gaps between zeros , squares , zeta function