[Paper Review] Character sums over products of prime polynomials

Samuel Porritt|arXiv (Cornell University)|Mar 26, 2020

Analytic Number Theory Research7 references4 citations

TL;DR

This paper derives explicit formulae for character sums over monic polynomials in $\mathbb{F}_q[t]$ with a prescribed number $k$ of irreducible factors, linking them to the zeros of Dirichlet $L$-functions over function fields. It reveals a novel bias phenomenon: when $k(n)$ grows rapidly with $n$, the bias term in the sum can dominate oscillatory terms, contradicting the expectation that bias diminishes as $k \to \infty$, with a critical threshold at $k/\log n \approx \gamma \approx 1.2021$ for strong bias persistence.

ABSTRACT

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.

Motivation & Objective

• To extend explicit formulae for character sums over polynomials with $k$ irreducible factors from fixed $k$ to $k = k(n)$ growing with $n$.
• To analyze the behavior of Chebyshev-type biases in function field $L$-functions when $k$ is not fixed but increases with $n$.
• To identify conditions under which the bias term in the sum remains significant despite increasing $k$, contrary to the intuition that bias should vanish as $k \to \infty$.
• To establish uniform asymptotic expansions for the normalized sum $\widetilde{\pi}_k(n,\chi)$ in terms of $L$-function zeros, valid for $k$ up to $q^{1/2 - \epsilon} \log n$.

Proposed method

• Derives an explicit formula for $\widetilde{\pi}_k(n,\chi)$ using the inverse Mellin transform and Perron-type summation, expressing the sum in terms of the zeros of $L(u,\chi)$.
• Applies complex analysis techniques, including the use of the Mellin transform and the residue theorem, to express the sum as a sum over the zeros of $L(u,\chi)$.
• Uses the factorization of $L(u,\chi)$ into terms associated with zeros $\alpha_j(\chi)$, $\beta_{j'}(\chi)$, and the real zeros at $\pm q^{-1/2}$, distinguishing cases based on whether $\chi^2 = \chi_0$.
• Applies Lemma 2 and Lemma 3 (from the appendix) to handle the asymptotic behavior of the sum as $k$ grows with $n$, treating each zero contribution separately.
• Employs Stirling's formula and asymptotic analysis of the gamma function to evaluate the main terms in the expansion, particularly for the $\rho = \pm q^{-1/2}$ and non-real $\rho$ cases.
• Derives a critical threshold $\gamma \approx 1.2021$ such that if $k/\log n > \gamma$, the bias term dominates the oscillatory terms for all large even $n$.

Experimental results

Research questions

• RQ1 How do character sums over polynomials with $k$ irreducible factors behave when $k$ grows with $n$, rather than being fixed?
• RQ2 What is the role of the zeros of the $L$-function $L(u,\chi)$ in determining the asymptotic behavior of $\pi_k(n,\chi)$ for growing $k$?
• RQ3 Does the Chebyshev-type bias in the sign of $\pi_k(n,\chi)$ persist when $k \to \infty$, or does it vanish as expected?
• RQ4 Under what conditions on $k(n)$ does the bias term in the explicit formula for $\pi_k(n,\chi)$ dominate the oscillatory terms arising from non-real zeros?
• RQ5 Is there a critical value of $k/\log n$ beyond which the bias remains strong, even as $k$ increases?

Key findings

• For non-real $\chi$, the normalized sum $\widetilde{\pi}_k(n,\chi)$ is asymptotically a sum over the zeros $\alpha_j(\chi)^{-1}$ of $L(u,\chi)$, with terms $m_j^k e^{in\gamma_j}$, and a main term $m_+^k + (-1)^n m_-^k$, valid for $1 \leq k \leq q^{1/2 - \epsilon} \log n$.
• When $\chi$ is real ($\chi^2 = \chi_0$), the sum includes additional contributions from the real zeros at $\pm q^{-1/2}$, which are treated via Lemma 3 and contribute terms involving $m_\pm + 1/2$.
• The bias term persists and can dominate the oscillatory terms if $k/\log n > \gamma \approx 1.2021$, contradicting the intuition that bias should vanish as $k \to \infty$.
• For $m_j = 1$ and $m_\pm = 0$, the main term in the asymptotic expansion is $h_j(\alpha) = F_{-\alpha}(\rho,\chi) c_\rho^{-\alpha} / \Gamma(1 + \alpha)$, with $\rho = \alpha_j(\chi)^{-1}$, and $\alpha = (k-1)/\log n$.
• When $\chi^2 = \chi_0$ and $m_\pm = 0$, the contribution from $\rho = \pm q^{-1/2}$ is asymptotically $h_\pm(\alpha) \sqrt{1 + r^2/\alpha}$, where $r$ solves $r^2 + r/2 = (k-1)/\log n$, and this term dominates if $k/\log n > \gamma$.
• The paper establishes that the bias term remains significant when $k/\log n > \gamma$, and that the sign of the bias depends on the sign of $b = \frac{1}{2} \left( \frac{s-1}{2} - \log(2s) \right)$, where $s$ is the positive root of $s^2 + \frac{s}{4u} - \frac{1}{4u} = 0$, with $u = (k-1)/\log n$.