Search ResultsShowing 1-20 of 22
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arXiv:2502.03464 (Published 2025-02-05)
Improving the trivial bound for $\ell$-torsion in class groups
Comments: 11 pagesCategories: math.NTFor any number field $K$ with $D_K=|\mathrm{Disc}(K)|$ and any integer $\ell \geq 2$, we improve over the commonly cited trivial bound $|\mathrm{Cl}_K[\ell]| \leq |\mathrm{Cl}_K| \ll_{[K:\mathbb{Q}],\varepsilon} D_K^{1/2+\varepsilon}$ on the $\ell$-torsion subgroup of the class group of $K$ by showing that $|\mathrm{Cl}_K[\ell]| = o_{[K:\mathbb{Q}],\ell}(D_K^{1/2})$. In fact, we obtain an explicit log-power saving. This is the first general unconditional saving over the trivial bound that holds for all $K$ and all $\ell$.
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arXiv:2410.06082 (Published 2024-10-08)
Explicit Deuring--Heilbronn phenomenon for Dirichlet $L$-functions
Comments: 17 PagesCategories: math.NTAssuming the existence of a Landau--Siegel zero, we establish an explicit Deuring-Heilbronn zero repulsion phenomenon for Dirichlet $L$-functions modulo $q$. Our estimate is uniform in the entire critical strip, and improves over the previous best known explicit estimate due to Thorner and Zaman.
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arXiv:2408.08309 (Published 2024-08-15)
Moments of random multiplicative functions over function fields
Comments: 32 pagesGranville-Soundararajan, Harper-Nikeghbali-Radziwill, and Heap-Lindqvist independently established an asymptotic for the even natural moments of partial sums of random multiplicative functions defined over integers. Building on these works, we study the even natural moments of partial sums of Steinhaus random multiplicative functions defined over function fields. Using a combination of analytic arguments and combinatorial arguments, we obtain asymptotic expressions for all the even natural moments in the large field limit and large degree limit, as well as an exact expression for the fourth moment.
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arXiv:2208.11123 (Published 2022-08-23)
An explicit version of Bombieri's log-free density estimate
Comments: 33 pagesCategories: math.NTWe make explicit Bombieri's refinement of Gallagher's log-free "large sieve density estimate near $\sigma = 1$" for Dirichlet $L$-functions. We use this estimate and recent work of Green to prove that if $N\geq 2$ is an integer, $A\subseteq\{1,\ldots,N\}$, and for all primes $p$ no two elements in $A$ differ by $p-1$, then $|A|\ll N^{1-1/10^{18}}$. This strengthens a theorem of S{\'a}rk{\"o}zy. Also, for integers $q\geq 3$ and $a$ such that $\gcd(a,q)=1$, we prove a highly uniform and explicit version of the prime number theorem that nontrivially counts primes $p\leq x$ such that $p\equiv a\pmod{q}$ when $x\geq \max\{q,411\}^{150\,000}$.
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arXiv:2108.11367 (Published 2021-08-25)
A conjectural asymptotic formula for multiplicative chaos in number theory
Comments: 18 pagesWe investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computational evidence that their result can likely be strengthened to an asymptotic result with a numerical estimate for the asymptotic constant. We also obtain similar conclusions when $A(N)$ is defined using standard real Gaussians or uniform $\pm 1$ random variables. However, our evidence suggests that the asymptotic constants do not possess a natural product structure.
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arXiv:2108.10878 (Published 2021-08-24)
Refinements to the prime number theorem for arithmetic progressions
Comments: 19 pagesCategories: math.NTWe prove a version of the prime number theorem for arithmetic progressions that is uniform enough to deduce the Siegel-Walfisz theorem, Hoheisel's asymptotic for intervals of length $x^{1-\delta}$, a Brun-Titchmarsh bound, and Linnik's bound on the least prime in an arithmetic progression as corollaries. Our proof uses the Vinogradov-Korobov zero-free region and a refinement of Bombieri's "repulsive" log-free zero density estimate. Improvements exist when the modulus is sufficiently powerful.
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arXiv:2108.08232 (Published 2021-08-18)
Sums of random multiplicative functions over function fields with few irreducible factors
Comments: 10 pagesCategories: math.NTWe establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This parallels work of Harper for random Rademacher multiplicative functions over the integers.
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arXiv:2108.07264 (Published 2021-08-16)
A model problem for multiplicative chaos in number theory
Comments: 25 pagesResolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version of Helson's conjecture. We develop and simplify the ideas of Harper in this context, with the hope that the simplified proof would be of use to readers seeking a gentle entry-point to this fascinating area.
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arXiv:2012.14422 (Published 2020-12-28)
An approximate form of Artin's holomorphy conjecture and non-vanishing of Artin $L$-functions
Categories: math.NTLet $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$. We prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region. Using this and related ideas, for example, we prove a strong effective prime number theorem for each faithful, irreducible Artin representation attached to almost all $K \in \mathfrak{F}_k^G$ in several natural cases, including when $G=S_n$ for any $n \geq 2$ and when $G$ is a transitive subgroup of $S_p$ for some prime $p \geq 2$. These results have several arithmetic applications, in particular to an effective variant of the Chebotarev density theorem, to bounds on $\ell$-torsion subgroups of class groups, to the resolution of a problem of Duke on the extremal order of class numbers, to the subconvexity problem for Dedekind zeta functions, and to the equidistribution of periodic torus orbits associated to ideal classes in prime degree fields.
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A zero density estimate for Dedekind zeta functions
Comments: Considerably streamlined, small refinements to Theorems 1.1 and 1.2. 14 pagesDOI: 10.1093/imrn/rnac015Categories: math.NTKeywords: dedekind zeta functions, first zero density estimate, nontrivial finite group, chebotarev density theorem, ideal class groupsTags: journal articleGiven a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/\mathbb{Q}$ with $\mathrm{Gal}(K/\mathbb{Q})\cong G$ that does not rely on unproven progress towards the strong form of Artin's conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butterbaugh, and Wood on the average error in the Chebotarev density theorem and $\ell$-torsion in ideal class groups.
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arXiv:1906.07717 (Published 2019-06-18)
The GL(n) large sieve
Comments: 15 pagesCategories: math.NTLet $\mathfrak{F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm{GL}_n$ over a number field with unitary central character. We prove an unconditional large sieve inequality for the Hecke eigenvalues of $\pi\in\mathfrak{F}_n$. This leads to the first unconditional zero density estimate for the family of $L$-functions $L(s,\pi)$ associated to $\pi\in\mathfrak{F}_n$, which we make log-free. As an application, we prove a subconvexity bound on $L(1/2,\pi)$ for almost all $\pi\in\mathfrak{F}_n$.
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arXiv:1804.06402 (Published 2018-04-17)
Zeros of Rankin-Selberg $L$-functions at the edge of the critical strip
Comments: 40 pagesCategories: math.NTLet $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields.
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arXiv:1803.02823 (Published 2018-03-07)
A unified and improved Chebotarev density theorem
We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem also interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors.
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arXiv:1710.08914 (Published 2017-10-24)
Primes represented by positive definite binary quadratic forms
Comments: 28 pagesCategories: math.NTLet $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH), and for almost all discriminants. The key feature of these estimates is that they hold whenever $x$ exceeds a small power of $D$ and, in some cases, this range of $x$ is essentially best possible. In particular, if $f$ is reduced then this optimal range of $x$ is achieved for almost all discriminants or by assuming GRH. We also exhibit an upper bound for the number of primes represented by $f$ in a short interval and a lower bound for the number of small integers represented by $f$ which have few prime factors.
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arXiv:1706.04173 (Published 2017-06-13)
The Density of Numbers Represented by Diagonal Forms of Large Degree
Let $s \geq 3$ be a fixed positive integer and $a_1,\dots,a_s \in \mathbb{Z}$ be arbitrary. We show that, on average over $k$, the density of numbers represented by the degree $k$ diagonal form \[ a_1 x_1^k + \cdots + a_s x_s^k \] decays rapidly with respect to $k$.
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arXiv:1704.03451 (Published 2017-04-11)
The least unramified prime which does not split completely
Let $K/F$ be a finite extension of number fields of degree $n \geq 2$. We establish effective field-uniform unconditional upper bounds for the least norm of a prime ideal of $F$ which is degree 1 over $\mathbb{Q}$ and does not ramify or split completely in $K$. We improve upon the previous best known general estimates due to X. Li when $F = \mathbb{Q}$ and Murty-Patankar when $K/F$ is Galois. Our bounds are the first when $K/F$ is not assumed to be Galois and $F \neq \mathbb{Q}$.
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arXiv:1606.09238 (Published 2016-06-29)
A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures
We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.
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arXiv:1604.01750 (Published 2016-04-06)
An explicit bound for the least prime ideal in the Chebotarev density theorem
Comments: 61 pages. This paper subsumes the contents of arXiv:1510.08086 but adds too much new material to be considered a revised versionCategories: math.NTWe 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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arXiv:1510.08086 (Published 2015-10-27)
Explicit results on the distribution of zeros of Hecke $L$-functions
Comments: 32 pagesCategories: math.NTWe prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem.
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arXiv:1508.00287 (Published 2015-08-02)
Bounding the least prime ideal in the Chebotarev Density Theorem
Comments: 21 pagesCategories: math.NTLet $L$ be a finite Galois extension of the number field $K$. We unconditionally bound the least prime ideal of $K$ occurring in the Chebotarev Density Theorem as a power of the discriminant of $L$ with an explicit exponent. We also establish a quantitative Deuring-Heilbronn phenomenon for the Dedekind zeta function.