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Register a new userThe Limiting Distribution of Character Sums
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Authors
Ayesha Hussain
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In this paper, we consider the distribution of the continuous paths of Dirichlet character sums modulo prime $q$ on the complex plane. We also find a limiting distribution as $q rightarrow infty$ using Steinhaus random multiplicative functions, stating properties of this random process. This is motivated by Kowalski and Sawins work on Kloosterman paths.
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Let $f(n)$ be a multiplicative function with $|f(n)|leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, chi$ be a non-principal Dirichlet character modulo $q$. Let $varepsilon$ be a sufficiently small positive constant, $A$ be a large constant, $q^{frac12+varepsilon}ll Nll q^A$. In this paper, we shall prove that $$ sum_{nleq N}f(n)chi(n+a)ll Nfrac{loglog q}{log q} $$ and that $$ sum_{nleq N}f(n)chi(n+a_1)cdotschi(n+a_t)ll Nfrac{loglog q}{log q}, $$ where $tgeq 2, a_1, ldots, a_t$ are distinct integers modulo $q$.
Given a finite set of integers $A$, its sumset is $A+A:= {a_i+a_j mid a_i,a_jin A}$. We examine $|A+A|$ as a random variable, where $Asubset I_n = [0,n-1]$, the set of integers from 0 to $n-1$, so that each element of $I_n$ is in $A$ with a fixed probability $p in (0,1)$. Recently, Martin and OBryant studied the case in which $p=1/2$ and found a closed form for $mathbb{E}[|A+A|]$. Lazarev, Miller, and OBryant extended the result to find a numerical estimate for $text{Var}(|A+A|)$ and bounds on the number of missing sums in $A+A$, $m_{n,;,p}(k) := mathbb{P}(2n-1-|A+A|=k)$. Their primary tool was a graph-theoretic framework which we now generalize to provide a closed form for $mathbb{E}[|A+A|]$ and $text{Var}(|A+A|)$ for all $pin (0,1)$ and establish good bounds for $mathbb{E}[|A+A|]$ and $m_{n,;,p}(k)$. We continue to investigate $m_{n,;,p}(k)$ by studying $m_p(k) = lim_{ntoinfty}m_{n,;,p}(k)$, proven to exist by Zhao. Lazarev, Miller, and OBryant proved that, for $p=1/2$, $m_{1/2}(6)>m_{1/2}(7)<m_{1/2}(8)$. This distribution is not unimodal, and is said to have a divot at 7. We report results investigating this divot as $p$ varies, and through both theoretical and numerical analysis, prove that for $pgeq 0.68$ there is a divot at $1$; that is, $m_{p}(0)>m_{p}(1)<m_{p}(2)$. Finally, we extend the graph-theoretic framework originally introduced by Lazarev, Miller, and OBryant to correlated sumsets $A+B$ where $B$ is correlated to $A$ by the probabilities $mathbb{P}(iin B mid iin A) = p_1$ and $mathbb{P}(iin B mid i otin A) = p_2$. We provide some preliminary results using the extension of this framework.
We develop a new method for studying sums of Kloosterman sums related to the spectral exponential sum. As a corollary, we obtain a new proof of the estimate of Soundararajan and Young for the error term in the prime geodesic theorem.
Using $ell$-adic cohomology of tensor inductions of lisse $overline{mathbb Q}_ell$-sheaves, we study a class of incomplete character sums.
A $k$-sum of a set $Asubseteq mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $kwedge A$ for the set of $k$-sums of $A$ we prove that [ frac{|(k+1)wedge A|}{|kwedge A|}, le , frac{|A|-k}{k+1} ] whenever $|A|ge (k^{2}+7k)/2$. The inequality is tight -- the above ratio being attained when $A$ is a geometric progression. This answers a question of Ruzsa.
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