[Paper Review] A note on character sums over short moving intervals

Adam J. Harper|arXiv (Cornell University)|Mar 17, 2022

Analytic Number Theory Research4 citations

TL;DR

This paper investigates the distribution of character sums over short moving intervals, showing that Lamzouri's conjecture—Gaussian limit laws for normalized sums $ S_{ ho,H}(X)/\sqrt{H} $ under $ H = o(q/\log q) $—fails for some Dirichlet characters, even when $ H = q/\log^A q $. However, it holds for 'almost all' characters in the range $ q^{1-o(1)} \leq H = o(q) $, using moment methods and Pólya’s Fourier expansion.

ABSTRACT

We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H ightarrow \infty$ and $(\log H)/\log q ightarrow 0$ as $q ightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true for "almost all" characters on the range $q^{1-o(1)} \leq H = o(q)$. Using Pólya's Fourier expansion, these results may be reformulated as statements about the distribution of certain Fourier series with number theoretic coefficients. Tools used in the proofs include the existence of characters with large partial sums on short initial segments, and moment estimates for trigonometric polynomials with random multiplicative coefficients.

Motivation & Objective

• To investigate the validity of Lamzouri’s conjecture on the Gaussian distribution of normalized character sums over short moving intervals.
• To determine whether the central limit theorem for character sums holds under the weaker condition $ H = o(q/\log q) $, as conjectured by Lamzouri.
• To analyze the statistical behavior of $ S_{\chi,H}(X)/\sqrt{H} $ when $ X $ is uniformly random over $ \{0, \dots, q-1\} $, with $ \chi $ a non-principal Dirichlet character modulo a prime $ q $.
• To identify the set of characters for which the central limit theorem holds, particularly in the regime $ H = o(q) $, and quantify the proportion of such characters.

Proposed method

• Use Pólya’s Fourier expansion to re-express character sums as trigonometric polynomials with number-theoretic coefficients.
• Apply moment methods to analyze convergence in distribution, focusing on the $ j $-th and $ k $-th moments of the normalized sum.
• Estimate moments of trigonometric polynomials with random multiplicative coefficients using probabilistic tools and orthogonality of characters.
• Establish bounds on the $ L^2 $-norm of the difference between moment integrals and Gaussian moments, using Steinhaus random multiplicative functions as a model.
• Prove that for $ j,k \leq \min\{ \log^{1/4}(q/H)/50, \log q / (4\log(q/H)) \} $, the moment difference decays exponentially.
• Use the union bound over low-order moments to show that the set of 'bad' characters has exponentially small density, implying 'almost all' characters satisfy the central limit theorem.

Experimental results

Research questions

• RQ1 Does the central limit theorem for character sums over short intervals hold under the conjectured bound $ H = o(q/\log q) $, as proposed by Lamzouri?
• RQ2 Are there specific Dirichlet characters for which the normalized sum $ S_{\chi,H}(X)/\sqrt{H} $ fails to converge to a Gaussian distribution even when $ H = q/\log^A q $ for fixed $ A > 0 $?
• RQ3 What proportion of characters modulo $ q $ satisfy the central limit theorem for $ H \in [q^{1-o(1)}, o(q)] $?
• RQ4 Can the moment method be extended beyond the $ (\log H)/\log q \to 0 $ regime to cover larger $ H $, and if so, under what conditions?
• RQ5 How do the Fourier coefficients in Pólya’s expansion influence the distributional behavior of short character sums?

Key findings

• Lamzouri’s conjecture fails for some Dirichlet characters even when $ H = q/\log^A q $ for any fixed $ A > 0 $, contradicting the expected Gaussian behavior.
• For $ H \in [q^{1-o(1)}, o(q)] $, the central limit theorem holds for 'almost all' characters modulo $ q $, with the proportion of 'bad' characters decaying as $ \exp(-c \log^{3/4}(q/H)) $.
• The moment method can be extended to larger $ H $ by controlling the $ L^2 $-norm of moment differences via probabilistic estimates on random multiplicative functions.
• The key technical step is proving that the $ (j,k) $-th moment of the normalized sum converges to the Gaussian moment $ k! \mathbf{1}_{j=k} $ for all fixed $ j,k $, provided $ \chi $ is in a 'good' set of characters.
• The proof relies on the fact that for $ j,k $ up to $ \log^{1/4}(q/H)/50 $, the character sums involved are supported on integers $ \leq q^{0.51} $, allowing orthogonality to be applied.
• The result is established by showing that the expected $ L^2 $-distance between the moment integrals and their Gaussian limits decays faster than any power of $ q $, implying convergence in measure and thus almost sure convergence for a large proportion of characters.