[Paper Review] Estimates for character sums and Dirichlet $L$-functions to smooth moduli

A. J. Irving|arXiv (Cornell University)|Mar 24, 2015

Analytic Number Theory Research1 references3 citations

TL;DR

This paper establishes improved bounds for character sums and Dirichlet $L$-functions at $s = \frac{1}{2}$ when the modulus $q$ is squarefree and $q^\delta$-smooth, using a $q$-analogue of the van der Corput $ABA^3B$ process. The key result is $L\left(\frac{1}{2},\chi\right) \ll_\epsilon q^{\frac{27}{164} + O(\delta) + \epsilon}$, breaking the Weyl barrier of $q^{1/6}$ for sufficiently smooth $q$. This improves upon previous estimates for smooth moduli and extends the range of applicability of exponent pair methods to character sums with smooth conductors.

ABSTRACT

We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $χ$ is a primitive Dirichlet character modulo a squarefree, $q^δ$-smooth integer $q$ we show that $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$

Motivation & Objective

• To improve existing estimates for character sums and $L$-functions at $s=1/2$ when the modulus $q$ is squarefree and $q^\delta$-smooth.
• To extend the range of applicability of exponent pair methods to smooth moduli by developing a $q$-analogue of the van der Corput $ABA^3B$ process.
• To break the Weyl barrier of $q^{1/6}$ in the bound for $L(\frac{1}{2}, \chi)$, which had previously been the best known for smooth $q$.
• To provide a quantitative improvement over Heath-Brown's $q^{1/6 + O(\delta)}$ bound by achieving $q^{27/164 + O(\delta)}$, where $27/164 \approx 0.1646 < 1/6 \approx 0.1667$.

Proposed method

• The method employs a $q$-analogue of the van der Corput $ABA^3B$ process, adapted to character sums modulo smooth, squarefree $q$.
• The proof uses a factorization $q = q_0 q_1$ with $q_1 \in [q^{1/3}, q^{1/3 + \delta}]$, enabling the application of the $A$-process to reduce the modulus.
• It applies the $B$-process (completion of sums) after $A$-differencing, leveraging the smoothness of $q$ to control exponential sums via $q_0$-twisted Weyl sums.
• The analysis involves bounding $k$-fold exponential sums using $q_0$-twisted Weyl sums and estimating divisor-type sums via $\tau_k(h)$ and $\gcd(h, q_0)^{1/2}$, leading to a recursive bound in $k$.
• The method relies on a key lemma estimating the $2^k$-th power of a Weyl sum over a complete interval, using $q_0$-twisted character sums and averaging over shift parameters $h_i$.
• The final bound is derived by optimizing over $k$ and using summation by parts on the $L$-function, replacing the Burgess bound with the new character sum estimate from Theorem 1.1.

Experimental results

Research questions

• RQ1 Can the Weyl barrier of $q^{1/6}$ for $L(\frac{1}{2}, \chi)$ be broken for smooth moduli using improved character sum estimates?
• RQ2 What is the best possible exponent $\theta$ such that $L(\frac{1}{2}, \chi) \ll_\epsilon q^{\theta + \epsilon}$ holds uniformly for $q$ squarefree and $q^\delta$-smooth?
• RQ3 To what extent can the $q$-analogue of the van der Corput $ABA^3B$ process improve character sum estimates beyond the Burgess and Heath-Brown bounds?
• RQ4 How does the smoothness parameter $\delta$ affect the exponent in the bound for $L(\frac{1}{2}, \chi)$, and can the dependence be made explicit?

Key findings

• The paper establishes $L\left(\frac{1}{2}, \chi\right) \ll_\epsilon q^{\frac{27}{164} + O(\delta) + \epsilon}$ for primitive Dirichlet characters modulo a squarefree, $q^\delta$-smooth $q$, which improves upon the previous bound of $q^{1/6 + O(\delta)}$.
• The exponent $\frac{27}{164} \approx 0.1646$ is strictly less than $\frac{1}{6} \approx 0.1667$, thus breaking the Weyl barrier for smooth moduli.
• The character sum estimate $\sum_{M \leq n \leq M+N} \chi(n) \ll_\epsilon N^{23/41} q^{11/82 + O(\delta) + \epsilon}$ is proven for $N \leq q$, which underlies the $L$-function bound.
• The method achieves this by developing a $q$-analogue of the van der Corput $ABA^3B$ process, which generalizes classical exponent pair techniques to smooth moduli.
• The improvement is quantitatively significant for small $\delta$, as the $O(\delta)$ term allows the exponent to approach $27/164$ as $\delta \to 0$.
• The result demonstrates that stronger bounds are possible when the modulus is not just smooth but also squarefree, and that the $q$-analogue of the $ABA^3B$ method yields a nontrivial improvement over prior methods for smooth $q$.