In a previous paper with Schmid (math.NT/0402382) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound sum_{n < T} a_n exp(2 pi i n alpha) = O_ε(T^{3/4+ε}) uniformly for alpha real, for a_n the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions.
Let $λ_g (n)$ be the Fourier coefficients of a holomorphic cusp modular form $g$ for $\mathrm{SL}_2 (\mathbb{Z})$. The aim of this article is to get non-trivial bound on non-linearly additively twisted sums of the Fourier coefficients $λ_g (n)$. Precisely, we prove for any $3/4 < β< 3/2$, $β\neq 1 $, the following non-trivial estimate $$ \sum_{n \leq N}λ_g(n)\,e(α\, n^β)\ll_{g, α, β, \varepsilon} N^{\frac{1}{2}+ \fracβ{3} +\varepsilon} + N^{\frac{3}{2}-\frac {2β}{3} + \varepsilon}, $$ for any $\varepsilon > 0$. This is the first time that non-trivial estimate for such sums is achieved for $1 <…
We derive a truncated Voronoi identity for rationally additively twisted sums of Fourier coefficients of Maass forms for $\mathrm{SL}(3,\mathbb Z)$, and as an application obtain a pointwise estimate and a second moment estimate for the sums in question.
We obtain a nontrivial bound for cancellations between the Kloosterman sums modulo a large prime power with a prime argument running over very short interval, which in turn is based on a new estimate on bilinear sums of Kloosterman sums. These results are analogues of those obtained by various authors for Kloosterman sums modulo a prime. However the underlying technique is different and allows us to obtain nontrivial results starting from much shorter ranges.
We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel, and Milićević (2014) and Fouvry, Kowalski and Michel (2014).
We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a…
We present new estimates for sums of the divisor function, and other similar arithmetic functions, in short intervals over function fields. (When the intervals are long, one obtains a good estimate from the Riemann hypothesis.) We obtain an estimate that approaches square root cancellation as long as the characteristic of the finite field is relatively large. This is done by a geometric method, inspired by work of Hast and Matei, where we calculate the singular locus of a variety whose $\mathbb F_q$-points control this sum. This has applications to highly unbalanced moments of $L$-functions.
We establish cancellation in short sums of certain special trace functions over $\mathbb{F}_q[u]$ below the Pólya-Vinogradov range, with savings approaching square-root cancellation as $q$ grows. This is used to resolve the $\mathbb{F}_q[u]$-analog of Chowla's conjecture on cancellation in Möbius sums over polynomial sequences, and of the Bateman-Horn conjecture in degree $2$, for some values of $q$. A final application is to sums of trace functions over primes in $\mathbb{F}_q[u]$.
In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of $\mathrm{GL}_2$ or $\mathrm{GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for $\mathrm{GL}_2$ and the Miller type bound for $\mathrm{GL}_3$ which are uniform in terms of the additive character.
We prove sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$. We place Bourgain's progress on the Lindel"of hypothesis in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.
In this paper we establish the function field analogue of Bateman-Horn conjecture in short interval in the limit of a large finite field. Hence we start with counting prime polynomials generated by primitive quadratic functions in short intervals. To this end we further work out on function field analogs of cancellation of Mobius sums and its correlations(Chowla type sums) and confirm that square root cancellation in Mobius sums is equivalent to square root cancellation in Chowla type sums.
We prove estimates for the level of distribution of the Möbius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $\infty$, and the power savings converges to square-root cancellation as $q$ goes to $\infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
We give explicit expressions for the Fourier coefficients of Eisenstein series twisted by Dirichlet characters and modular symbols on $Γ_0(N)$ in the case where $N$ is prime and equal to the conductor of the Dirichlet character. We obtain these expressions by computing the spectral decomposition of automorphic functions closely related to these Eisenstein series. As an application, we then evaluate certain sums of modular symbols in a way which parallels past work of Goldfeld, O'Sullivan, Petridis, and Risager. In one case we find less cancellation in this sum than would be predicted by the co…
We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.
We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.
We obtain several estimates for bilinear form with exponential sums with binomials $mx^k + nx^\ell$. In particular we show the existence of nontrivial cancellations between such sums when the coefficients $m$ and $n$ vary over rather sparse sets of general nature
We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang.
We give a general version of cancellation in exponential sums that arise as sums of products of trace functions satisfying a suitable independence condition related to the Goursat-Kolchin-Ribet criterion, in a form that is easily applicable in analytic number theory.
We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as cancellation in sums of function field analogs of the Möbius $μ$ function and its correlations (similar to sums appearing in Chowla's conjecture). For generic $f$, i.e., for $f$ a Morse polynomial, the error terms are roughly of size $O(\sqrt{p})$ (with typical main terms of order $p$). For non-generic $f$ we prove that independence still holds for "generic" set of …
We consider partial sums of a weighted Steinhaus random multiplicative function and view this as a model for the Riemann zeta function. We give a description of the tails and high moments of this object. Using these we determine the likely maximum of $T \log T$ independently sampled copies of our sum and find that this is in agreement with a conjecture of Farmer--Gonek--Hughes on the maximum of the Riemann zeta function. We also consider the question of almost sure bounds. We determine upper bounds on the level of squareroot cancellation and lower bounds which suggest a degree of cancellation …
We obtain several estimates for bilinear form with Kloosterman sums. Such results can be interpreted as a measure of cancellations amongst with parameters from short intervals. In particular, for certain ranges of parameters we improve some recent results of Blomer, Fouvry, Kowalski, Michel and Milićević and also of Fouvry, Kowalski and Michel. In particular, we improve the bound on the error term in the asymptotic formula for mixed moments of $L$-series associated with Hecke eigenforms.
Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentio…
We prove conjecturally sharp upper bounds for the Dirichlet character moments $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)|^{2q}$, where $r$ is a large prime, $1 \leq x \leq r$, and $0 \leq q \leq 1$ is real. In particular, if both $x$ and $r/x$ tend to infinity with $r$ then $\frac{1}{r-1} \sum_{χ\; \text{mod} \; r} |\sum_{n \leq x} χ(n)| = o(\sqrt{x})$, and so the sums $\sum_{n \leq x} χ(n)$ typically exhibit "better than squareroot cancellation". We prove analogous better than squareroot bounds for the moments $\frac{1}{T} \int_{0}^{T} |\sum_{n \leq x} n^{it}|^{2q} dt$ of…
We show that sums of the SL(3,Z) long element Kloosterman sum against a smooth weight function have cancellation due to the variation in argument of the Kloosterman sums, when each modulus is at least the square root of the other. Our main tool is Li's generalization of the Kuznetsov formula on SL(3,R), which has to date been prohibitively difficult to apply. We first obtain analytic expressions for the weight functions on the Kloosterman sum side by converting them to Mellin-Barnes integral form. This allows us to relax the conditions on the test function and to produce a partial inversion fo…
Following attempts at an analytic proof of the Pentagonal Number Theorem, we report on the discovery of a general principle leading to an unexpected cancellation of oscillating sums. After stating the motivation, and our theorem, we apply it to prove several results on the Prouhet-Tarry-Escott Problem, integer partitions, and the distribution of prime numbers. Regarding the Prouhet-Tarry-Escott problem, we show that \begin{align*} \sum_{|\ell|\leq x}(4x^2-4\ell^2)^{2r}-\sum_{|\ell|<x}(4x^2-(2\ell+1)^2)^{2r}=\text{polynomial w.r.t. } x \text{ with degree }2r-1. \end{align*} This can perhaps be …
Let $M=M_1 M_2 M_3$ be the product of three distinct primes and let $χ=χ_1 χ_2 χ_3$ be a Dirichlet character of modulus $M$ such that each $χ_i$ is a primitive character modulo $M_i$ for $i=1,2,3$. In this paper, we provide a $δ$-symbol method for obtaining non-trivial cancellation in smooth character sums of the form $\sum_{n=1}^\infty χ(n) W(n/N)$, with $N$ roughly of size $\sqrt M$ and $W$ a smooth compactly supported weight function on $(0, \infty)$. As a corollary, we establish hybrid subconvexity bounds for the associated Dirichlet $L$-function.
Higher order uniformity of the primes and cancellation of the Möbius function in shorter intervals
University of Turku（图尔库大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）；Universidad Autónoma de Madrid（马德里自治大学）；University of Cambridge（剑桥大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.
We consider large values of long linear exponential sums involving Fourier coefficients of holomorphic cusp forms. The sums we consider involve rational linear twists $e(nh/k)$ with sufficiently small denominators. We prove both pointwise upper bounds and bounds for the frequency of large values. In particular, the $k$-aspect is treated. As an application we obtain upper bounds for all the moments of the sums in question. We also give the asymptotics with the right main term for fourth moments. We also consider the mean square of very short sums, proving that on average short linear sums with …
Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{λ_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $λ_f(n)$, we investigate the range of $x$ (in terms of $k$) for which there are cancellations in the sum $S_f(x)=\sum_{n\leq x} λ_f(n)$. We first show that $S_f(x)=o(x\log x)$ implies that $λ_f(n)<0$ for some $n\leq x$. We also prove that $S_f(x)=o(x\log x)$ in the range $\log x/\log\log k\to \infty$ assuming the Riemann hypothesis for $L(s, f)$, and furthermore that this range is best …
Let $f$ and $g$ be weight $k$ holomorphic cusp forms and let $S_f(n)$ and $S_g(n)$ denote the sums of their first $n$ Fourier coefficients. Hafner and Ivic [HI], building on Chandrasekharan and Narasimhan [CN], proved asymptotics for $\sum_{n \leq X} \lvert S_f(n) \rvert^2$ and proved that the Classical Conjecture, that $S_f(X) \ll X^{\frac{k-1}{2} + \frac{1}{4} + ε}$, holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series $D(s, S_f \times S_g) = \sum S_f(n)\overline{S_g(n)} n^{-(s+k-1)}$ and $D(s, S_f \times \overline{S_…
Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with Fourier coefficients $λ_f(n)$ and $λ_g(n)$, respectively. For real $α\neq0$ and $0<β\leq1$, consider a smooth resonance sum $S_X(f,g;α,β)$ of $λ_f(n)λ_g(n)$ against $e(αn^β)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;α,β)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. By taking…
This thesis contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field. We investigate the decay rate of the cutoff function and its derivatives in terms of the analytic conductor introduced by Iwaniec and Sarnak. We also see that the truncation can be made uniformly explicit at the cost of an error term. The results extend to products of central values. We establish, via the Hardy-Littlewood circle method, a nontrivia…
We examine correlations of the Möbius function over $\mathbb{F}_q[t]$ with linear or quadratic phases, that is, averages of the form \begin{equation} \label{eq:average} \frac{1}{q^n}\sum_{\text{deg }f<n} μ(f)χ(Q(f)) \end{equation} for an additive character $χ$ over $\mathbb{F}_q$ and a polynomial $Q\in\mathbb{F}_q[x_0,\ldots,x_{n-1}]$ of degree at most 2 in the coefficients $x_0,\ldots, x_{n-1}$ of $f=\sum_{i< n}x_i t^i$. Like in the integers, it is reasonable to expect that, due to the random-like behaviour of $μ$, such sums should exhibit considerable cancellation. In this paper we show that…
