arXiv subjects

Nawapan Wattanawanichkul

Publications and source records attributed to Nawapan Wattanawanichkul.

8 recordsLinked to original sources

arXiv2026

Zeros of Polynomials in Derivatives of Automorphic $L$-functions

Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbolπ)$ be a polynomial in the derivatives of $L$-functions associated with representations $π\in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbolπ)$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the dimensions, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbolπ)$ lie near the critical line $\operatorname{Re}(s)=1/2$.

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arXiv2026

A metric approach to zero-free regions for $L$-functions

For integers $m, m' \ge 1$, let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(m)$ and $\mathrm{GL}(m')$, respectively. We present a new proof of zero-free regions for $L(s, π)$ and for $L(s, π\times π')$ under the assumption that $π, π'$ or $L(s,π\times π')$ is self-dual. Our approach builds on ideas of "pretentious" multiplicative functions due to Granville and Soundararajan (as presented by Koukoulopoulos) and the notion of a positive semi-definite family of automorphic representations due to Lichtman and Pascadi.

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arXiv2026

The $θ= \infty$ Conjecture and the Riemann Hypothesis for Automorphic $L$-functions

The $θ=\infty$ conjecture asserts that the mollified second moments of the Riemann zeta function remain bounded for mollifiers of arbitrary polynomial length. We investigate an analogue of this conjecture for automorphic $L$-functions associated with cuspidal representations of $\text{GL}_m(\mathbb{A}_{\mathbb{Q}})$, exploring its implications for the distribution of their nontrivial zeros. Extending the framework of Bettin and Gonek, we prove that if the mollified second moments of these $L$-functions remain suitably bounded for mollifiers of arbitrary polynomial length, then the $L$-functions are non-vanishing in corresponding regions of the critical strip. Furthermore, we establish a version of this criterion for families of $L$-functions, demonstrating that the $θ= \infty$ conjecture for a family of $L$-functions implies a quasi-Riemann Hypothesis for that family.

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arXiv2026

Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions

Let $f$ and $g$ be spectrally normalized holomorphic newforms of even weight $k \geq2$ on $Γ_0(q)$. If $f\neq g$, then assume that $q$ is squarefree. For a nice test function $ψ$ supported on $Γ_0(1)\backslash\mathbb{H}$, we establish the best known bounds (uniform in $k$, $q$, and $ψ$) for \[ \int_{Γ_0(q)\backslash\mathbb{H}}ψ(z)f(z)\overline{g(z)}y^{k}\frac{dxdy}{y^2}-\mathbf{1}_{f = g}\frac{3}π\int_{Γ_0(1)\backslash\mathbb{H}}ψ(z)\frac{dx dy}{y^2}.\] When $f=g$, our results yield an effective holomorphic variant of quantum unique ergodicity, refining work of Holowinsky-Soundararajan and Nelson-Pitale-Saha. When $f \neq g$, our results extend and improve the effective decorrelation result of Huang for $q=1$. To prove our results, we refine Soundararajan's weak subconvexity bound for Rankin-Selberg $L$-functions.

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arXiv2024

Walking to Infinity Along Some Number Theory sequences

An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to primes. We introduce stochastic models and analyze the walks' expected length and frequency of digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences or on primes in different bases.

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arXiv2023

An Alternative Approach to Computing $β(2k+1)$

This paper presents a new approach to evaluating the special values of the Dirichlet beta function, $β(2k+1)$, where $k$ is any nonnegative integer. Our approach relies on some properties of the Euler numbers and polynomials, and uses basic calculus and telescoping series. By a similar procedure, we also yield an integral representation of $β(2k)$. The idea of our proof adapts from a previous study by Ciaurri et al., where the authors introduced a new proof of Euler's formula for $ζ(2k)$.

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arXiv2022

Walking to Infinity on the Fibonacci Sequence

An interesting open problem in number theory asks whether it is possible to walk to infinity on primes, where each term in the sequence has one more digit than the previous. In this paper, we study its variation where we walk on the Fibonacci sequence. We prove that all walks starting with a Fibonacci number and the following terms are Fibonacci numbers obtained by appending exactly one digit at a time to the right have a length of at most two. In the more general case where we append at most a bounded number of digits each time, we give a formula for the length of the longest walk.

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arXiv2022

Modeling Random Walks to Infinity on Primes in $\mathbb{Z}[\sqrt{2}]$

An interesting question, known as the Gaussian moat problem, asks whether it is possible to walk to infinity on Gaussian primes with steps of bounded length. Our work examines a similar situation in the real quadratic integer ring $\mathbb{Z}[\sqrt{2}]$ whose primes cluster near the asymptotes $y = \pm x/\sqrt{2}$ as compared to Gaussian primes, which cluster near the origin. We construct a probabilistic model of primes in $\mathbb{Z}[\sqrt{2}]$ by applying the prime number theorem and a combinatorial theorem for counting the number of lattice points whose absolute values of their norms are at most $r^2$. We then prove that it is impossible to walk to infinity if the walk remains within some bounded distance from the asymptotes. Lastly, we perform a few moat calculations to show that the longest walk is likely to stay close to the asymptotes; hence, we conjecture that there is no walk to infinity on $\mathbb{Z}[\sqrt{2}]$ primes with steps of bounded length.

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