arXiv subjects

Jesse Thorner

Publications and source records attributed to Jesse Thorner.

45 records · Page 3Linked to original sources

arXiv2016

Effective log-free zero density estimates for automorphic $L$-functions and the Sato-Tate conjecture

Let $K/\mathbb{Q}$ be a number field. Let $π$ and $π^\prime$ be cuspidal automorphic representations of $\mathrm{GL}_d(\mathbb{A}_K)$ and $\mathrm{GL}_{d^\prime}(\mathbb{A}_K)$, and suppose that either both $d$ and $d'$ are at most 2 or at least one of $π$ and $π^\prime$ is self-dual. When $d=d^\prime=2$, we prove an unconditional and effective log-free zero density estimate for the Rankin-Selberg $L$-function $L(s,π\otimesπ^\prime,K)$. For other choices of $d$ and $d^\prime$, we obtain similar results assuming that either $π$ or $π^\prime$ satisfies the generalized Ramanujan conjecture. With these density estimates, we make effective the Hoheisel phenomenon of Moreno regarding primes in short intervals and extend it to the context of the Sato-Tate conjecture; additionally, we bound the least prime in the Sato-Tate conjecture in analogy with Linnik's theorem on the least prime in an arithmetic progression. When $K=\mathbb{Q}$, we also prove an effective log-free density estimate for $L(s,π\otimesπ^\prime,\mathbb{Q})$ averaged over twists by Dirichlet characters. With this second density estimate, we prove an averaged form of the prime number theorem in short intervals for $L(s,π\otimes\tildeπ,\mathbb{Q})$ when $π$ is a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$.

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arXiv2016

The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (Γ_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $θ_p\in[0,π]$ to be the angle for which $a(p)=2p^{( k -1)/2}\cos θ_p $. Let $I\subset[0,π]$ be a closed subinterval, and let $dμ_{ST}=\frac{2}π\sin^2θdθ$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then \[ \left|\#\{p\leq x:θ_p\in I\} -μ_{ST}(I)\int_2^x\frac{dt}{\log t}\right|\ll\frac{x^{3/4}\log(N k x)}{\log x} \] with an implied constant of $3.34$. By letting $I$ be a short interval centered at $\fracπ{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a(n)\neq0$. In particular, if $τ$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that \[ \lim_{x\to\infty}\frac{\#\{n\leq x:τ(n)\neq0\}}{x}>1-1.54\times10^{-13}. \] We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.

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arXiv2016

An explicit bound for the least prime ideal in the Chebotarev density theorem

We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of $\mathbb{F}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

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arXiv2016

Explicit estimates for the zeros of Hecke $L$-functions

Let $K$ be a number field and, for an integral ideal $\mathfrak{q}$ of $K$, let $χ$ be a character of the narrow ray class group modulo $\mathfrak{q}$. We establish various new and improved explicit results, with effective dependence on $K$, $\mathfrak{q}$ and $χ$, regarding the zeros of the Hecke L-function $L(s,χ)$, such as zero-free regions, Deuring-Heilbronn phenomenon, and zero density estimates.

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arXiv2015

A variant of the Bombieri-Vinogradov theorem in short intervals with applications

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a splitting condition in a Galois extension $L/K$ of number fields. Using this result in conjunction with recent work of Maynard, we prove that rational primes in short intervals with a given splitting condition in a Galois extension $L/\mathbb{Q}$ exhibit dense clusters in short intervals. We explore several arithmetic applications related to questions of Serre regarding the nonvanishing Fourier coefficients of cuspidal modular forms, including finding dense clusters of fundamental discriminants $ d $ in short intervals for which the central values of $d$-quadratic twists of modular $L$-functions are non-vanishing.

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arXiv2015

The Error Term in the Sato-Tate Conjecture

Let $f(z)=\sum_{n=1}^\infty a(n)e^{2πi nz}\in S_k^{new}(Γ_0(N))$ be a newform of even weight $k\geq2$ that does not have complex multiplication. Then $a(n)\in\mathbb{R}$ for all $n$, so for any prime $p$, there exists $θ_p\in[0,π]$ such that $a(p)=2p^{(k-1)/2}\cos(θ_p)$. Let $π(x)=\#\{p\leq x\}$. For a given subinterval $I\subset[0,π]$, the now-proven Sato-Tate Conjecture tells us that as $x\to\infty$, \[ \#\{p\leq x:θ_p\in I\}\sim μ_{ST}(I)π(x),\quad μ_{ST}(I)=\int_{I} \frac{2}π\sin^2(θ)~dθ. \] Let $ε>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to\infty$, \[ \#\{p\leq x:θ_p\in I\}=μ_{ST}(I)π(x)+O\left(\frac{x}{(\log x)^{9/8-ε}}\right), \] where the implied constant is effectively computable and depends only on $k,N,$ and $ε$.

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arXiv2015

Bounded Gaps Between Primes in Chebotarev Sets

A new and exciting breakthrough due to Maynard establishes that there exist infinitely many pairs of distinct primes $p_1,p_2$ with $|p_1-p_2|\leq 600$ as a consequence of the Bombieri-Vinogradov Theorem. In this paper, we apply his general method to the setting of Chebotarev sets of primes. We study applications of these bounded gaps with an emphasis on ranks of prime quadratic twists of elliptic curves over $\mathbb{Q}$, congruence properties of the Fourier coefficients of normalized Hecke eigenforms, and representations of primes by binary quadratic forms.

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arXiv2015

Bounded Gaps Between Primes in Multidimensional Hecke Equidistribution Problems

Using Duke's large sieve inequality for Hecke Gr{ö}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application, for any fixed $0<ε<\frac{1}{2}$, we prove the existence of infinitely many bounded gaps between primes of the form $p=a^2+b^2$ such that $|a|<ε\sqrt{p}$. Furthermore, for certain diagonal curves $\mathcal{C}:ax^α+by^β=c$, we obtain infinitely many bounded gaps between the primes $p$ such that $|p+1-\#\mathcal{C}(\mathbb{F}_p)|<ε\sqrt{p}$.

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arXiv2014

Benford's Law for Coefficients of Newforms

Let $f(z)=\sum_{n=1}^\infty λ_f(n)e^{2πi n z}\in S_{k}^{new}(Γ_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $Γ_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence $\{λ_f(p)\}_{p\in\mathbb{P}}$ does not satisfy Benford's Law in any base $b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base $b$, the set \[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of $λ_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal to $\log_b(1+S^{-1})$. Thus $\{λ_f(p)\}_{p\in\mathbb{P}}$ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.

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