arXiv subjects

Kyle Pratt

Publications and source records attributed to Kyle Pratt.

29 records · Page 2Linked to original sources

arXiv2020

Exceptional characters and nonvanishing of Dirichlet $L$-functions

Let $ψ$ be a real primitive character modulo $D$. If the $L$-function $L(s,ψ)$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then we say the character $ψ$ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values $L(1/2,χ)$ of the Dirichlet $L$-functions $L(s,χ)$ are nonzero, where $χ$ ranges over primitive characters modulo $q$ and $q$ is a large prime of size $D^{O(1)}$. Under the same hypothesis we also show that, for almost all $χ$, the function $L(s,χ)$ has at most a simple zero at $s = 1/2$.

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arXiv2019

More than five-twelfths of the zeros of $ζ$ are on the critical line

The second moment of the Riemann zeta-function twisted by a normalized Dirichlet polynomial with coefficients of the form $(μ\star Λ_1^{\star k_1} \star Λ_2^{\star k_2} \star \cdots \star Λ_d^{\star k_d})$ is computed unconditionally by means of the autocorrelation of ratios of $ζ$ techniques from Conrey, Farmer, Keating, Rubinstein and Snaith (2005), Conrey, Farmer and Zirnbauer (2008) as well as Conrey and Snaith (2007). This in turn allows us to describe the combinatorial process behind the mollification of \[ ζ(s) + λ_1 \frac{ζ'(s)}{\log T} + λ_2 \frac{ζ''(s)}{\log^2 T} + \cdots + λ_d \frac{ζ^{(d)}(s)}{\log^d T}, \] where $ζ^{(k)}$ stands for the $k$th derivative of the Riemann zeta-function and $\{λ_k\}_{k=1}^d$ are real numbers. Improving on recent results on long mollifiers and sums of Kloosterman sums due to Pratt and Robles (2017), as an application, we increase the current lower bound of critical zeros of the Riemann zeta-function to slightly over five-twelfths.

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arXiv2018

Dirichlet $L$-functions of quadratic characters of prime conductor at the central point

We prove that more than nine percent of the central values $L(\frac{1}{2},χ_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $χ_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},χ_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.

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arXiv2018

Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

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arXiv2018

Primes from sums of two squares and missing digits

Let $\mathcal{A}'$ be the set of integers missing any three fixed digits from their decimal expansion. We produce primes in a thin sequence by proving an asymptotic formula for counting primes of the form $p = m^2 + \ell^2$, with $\ell \in \mathcal{A}'$. The proof draws on ideas from the work of Friedlander-Iwaniec on primes of the form $p = x^2+y^4$, as well as ideas from the work of Maynard on primes with restricted digits.

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arXiv2018

Perturbed moments and a longer mollifier for critical zeros of $ζ$

Let $A(s)$ be a general Dirichlet polynomial and $Φ$ be a smooth function supported in $[1,2]$ with mild bounds on its derivatives. New main terms for the integral $I(α,β)=\int_{\mathbb{R}} ζ(\frac{1}{2}+α+it)ζ(\frac{1}{2}+β+it)|A(\frac{1}{2}+it)|^2 Φ(\frac{t}{T})dt$ are given. For the error term, we show that the length of the Feng mollifier can be increased from $θ< \frac{17}{33}$ to $θ< \frac{6}{11}$ by decomposing the error into Type I and Type II sums and then studying the resulting sums of Kloosterman sums. As an application, we slightly increase the proportion of zeros of $ζ(s)$ on the critical line.

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arXiv2016

A lower bound for the least prime in an arithmetic progression

Fix $k$ a positive integer, and let $\ell$ be coprime to $k$. Let $p(k,\ell)$ denote the smallest prime equivalent to $\ell \pmod{k}$, and set $P(k)$ to be the maximum of all the $p(k,\ell)$. We seek lower bounds for $P(k)$. In particular, we show that for almost every $k$ one has $P(k) \gg ϕ(k) \log k \log_2 k \log_4 k / \log_3 k,$ answering a question of Ford, Green, Konyangin, Maynard, and Tao. We rely on their recent work on large gaps between primes. Our main new idea is to use sieve weights to capture not only primes, but also small multiples of primes. We also give a heuristic which suggests that $\liminf_{k} \frac{P(k)}{ ϕ(k) \log^2 k} = 1.$

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arXiv2014

Coefficient Bounds for Level 2 Cusp Forms and Modular Functions

We give explicit upper bounds for the coefficients of arbitrary weight $k$, level 2 cusp forms, making Deligne's well-known $O(n^{\frac{k-1}{2}+ε})$ bound precise. We also derive asymptotic formulas and explicit upper bounds for the coefficients of certain level 2 modular functions.

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arXiv2014

Zeros of Dirichlet L-functions over Function Fields

Random matrix theory has successfully modeled many systems in physics and mathematics, and often the analysis and results in one area guide development in the other. Hughes and Rudnick computed $1$-level density statistics for low-lying zeros of the family of primitive Dirichlet $L$-functions of fixed prime conductor $Q$, as $Q \to \infty$, and verified the unitary symmetry predicted by random matrix theory. We compute $1$- and $2$-level statistics of the analogous family of Dirichlet $L$-functions over $\mathbb{F}_q(T)$. Whereas the Hughes-Rudnick results were restricted by the support of the Fourier transform of their test function, our test function is periodic and our results are only restricted by a decay condition on its Fourier coefficients. We show the main terms agree with unitary symmetry, and also isolate error terms. In concluding, we discuss an $\mathbb{F}_q(T)$-analogue of Montgomery's Hypothesis on the distribution of primes in arithmetic progressions, which Fiorilli and Miller show would remove the restriction on the Hughes-Rudnick results.

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arXiv2013

Special Sets of Primes in Function Fields

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.

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arXiv2013

Interlacing of zeros of weakly holomorphic modular forms

We prove that the zeros of a family of extremal modular forms interlace, settling a question of Nozaki. Additionally, we show that the zeros of almost all forms in a basis for the space of weakly holomorphic modular forms of weight $k$ for $\SL_2(\mathbb{Z})$ interlace on most of the lower boundary of the fundamental domain.

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