Benoit Louvel
Using the theory of metaplectic forms,we study the asymptotic behavior of cubic exponential sums over the ring of Eisenstein integers. In the first part of the paper, some non-trivial estimates on average over arithmetic progressions are obtained. In the second part of the paper, we prove that the sign of cubic exponential sums changes infinitely often, as the modulus runs over almost prime integers.
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Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.
A recent paper of A. Sofo proves some results about sums of products of quadratic alternating harmonic numbers and reciprocal binomial coefficients. In this paper, we extend his result to cubic alternating harmonic number sums and develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial coefficients. Some inter- esting (known or new) illustrative special cases as well as immediate consequences of the main results are also considered.
Variance of point-counts for families of cubic curves over $\mathbb{F}_p$ and Jacobsthal sums
本文显式计算了单参数三次曲线族上的点数方差，并重点展示了涉及 Jacobsthal 和的方差求值结果。
An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.
P. Flajolet and B. Salvy \cite{FS1998} prove the famous theorem that a nonlinear Euler sum $S_{i_1i_2\cdots i_r,q}$ reduces to a combination of sums of lower orders whenever the weight $i_1+i_2+\cdots+i_r+q$ and the order $r$ are of the same parity. In this article, we develop an approach to evaluate the cubic sums $S_{1^2m,p}$ and $S_{1l_1l_2,l_3}$. By using the approach, we establish some relations involving cubic, quadratic and linear Euler sums. Specially, we prove the cubic sums $S_{1^2m,m}$ and $S_{1(2l+1)^2,2l+1}$ are reducible to zeta values, quadratic and linear sums. Moreover, we pro…
An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its value is $-(-2)^{s/2}$).
We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $αx^3+βx$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type.
We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet character sums.
Evaluation of eight different families of cubic Euler sums
研究四种、五种和六种三次欧拉和，通过三种不同的分母类型计算非线性欧拉和，利用zeta值和polylogarithmic值明确表达结果。
In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as "Birch sums". Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower …
For each positive integer k, we investigate the L-function attached to the k-th symmetric power of the F-crystal associated to the family of cubic exponential sums of x^3 + λx. We explore its rationality, field of definition, degree, trivial factors, functional equation, and Newton polygon. The paper is essentially self-contained, due to the remarkable and attractive nature of Dwork's p-adic theory. A novel feature of this paper is an extension of Dwork's effective decomposition theory when k < p. This allows for explicit computations in the associated p-adic cohomology. In particular, the act…
Let K be a cubic number field. In this paper, we study the Ramanujan sums c_{J}(I), where I and J are integral ideals in O_{K}. The asymptotic behaviour of sums of c_{J}(I) over both I and J is investigated.
On partial T-adic exponential sums and partial exponential sums with p-power conductor
本文推广了Liu和Wan的T进指数和理论到部分情形，证明了相关L函数的T进亚纯性，从而给出p幂导子特征部分L函数有理性的p进证明，并给出Newton-over-Hodge估计。
Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is the paper of Livné and Patterson \cite{LP} on sums of cubic exponential sums over $\Q[ω], ω^3=1.$ The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of $GL_2,$ and an identity on exponential sums relating Kloosterman sums and quartic exponential sums. To synthesize the spectral theory and …
We improve an existing result on exponential quadrilinear sums in the case of sums over multiplicative subgroups of a finite field and use it to give a new bound on exponential sums with quadrinomials.
We study averages over squarefree moduli of the size of exponential sums with polynomial phases. We prove upper bounds on various moments of such sums, and obtain evidence of un-correlation of exponential sums associated to different suitably unrelated and generic polynomials. The proofs combine analytic arguments with the algebraic interpretation of exponential sums and their monodromy groups.
We establish bounds of triple exponential sums with mixed exponential and linear function. The method we use is by Shparlinski together with a bound of additive energy from Roche-Newton, Rudnev and Shkredov.
This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.
We extend some methods of bounding exponential sums of the type $\displaystyle\sum_{n\le N}e^{2πiag^n/p}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.
In this paper, we completely determine the slopes and weights of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems. One consequence of our main result is a sharp estimate of these exponential sums. Another consequence is to obtain an explicit counterexample of Adolphson-Sperber's conjecture on weights of toric exponential sums.
On quadratic character sums over quartics
本文研究了多项式变元的二次特征和问题，推导出了四次和三次多项式变元的二次特征和的变换公式。
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
Bias in cubic Gauss sums: Patterson's conjecture
本研究在广义黎曼假设下证明了关于三次高斯和的渐近公式，证实了Patterson1978年的猜想，解释了Kummer观测到的数值偏差，还得到显式Voronoi求和公式，证明三次大筛法是紧的。
In this paper we introduce two nonlinear functionals, the H and J functional, in the theory of complete exponential sums.
Cubic Polynomials and Sums of Two Squares
研究三次不可约monic多项式作为两个平方和的表达频率下界，提出量化回答Grechuk关于此类值无限性的疑问，利用二维单位论和六次数域算术证明主要结论。
In this paper, we compute the $q$-adic slopes of the L-functions of an important class of exponential sums arising from analytic number theory. Our main tools include Adolphson-Sperber's work on toric exponential sums and Wan's decomposition theorems.
In this paper we expose our main results about rank problems concerning persymmetric matrices over F_2 associated to some exponential sums.
We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.
In this paper we obtain some new estimates for multilinear exponential sums in prime fields with a more general class of weights than previously considered. Our techniques are based on some recent progress of Shkredov on multilinear sums which has roots in Rudnev's point plane incidence bound. We apply our estimates to obtain new results concerning exponential sums with sparse polynomials and Weyl sums over small generalized arithmetic progressions.
We use Bourgain's recent bound for short exponential sums to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.
Barban-Davenport-Halberstam type theorems for exponential sums and Piatetski-Shapiro primes
该研究在数论领域建立了三类Barban-Davenport-Halberstam型定理，分别适用于素数指数和、Piatetski-Shapiro素数及该类素数上的指数和，拓展了相关定理的应用范围。
Two Kinds Hybrid Power Mean Involving Two-Term Exponential Sums and Dedekind Sums
本文利用解析方法、二次剩余变换及 Dedekind 和性质，研究两类含两项指数和与 Dedekind 和的混合幂平均计算问题，并给出两个渐近公式，推广了已有结论。
We derive a new bound for some bilinear sums over points of an elliptic curve over a finite field. We use this bound to improve a series of previous results on various exponential sums and some arithmetic problems involving points on elliptic curves.
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
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
The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski's bound for exponential sums attached to certain recurrence sequences over finite fields.
Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.
Difference sets and positive exponential sums II: cubic residues in cyclic groups
本文通过构造非负指数和，给出了循环群中差集避免三次剩余的集合基数的上界。
In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.
We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators.
We investigate the size of L^p-integrals for exponential sums over k-free numbers and prove essentially tight bounds.
Moment of double exponential sums
本文研究有限域中任意集合与区间上含单项式的双指数和的矩，延续了Heath-Brown关于算术级数中最小无平方因子数的经典工作。
We give a new bound on colinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.
We improve the known upper bound for short exponential sums and increase the range on which a sharp upper bound is known.
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.
Under the generalized Lindelöf Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.
We obtain using exponential quadratic sums, explicit expressions for the number of triple persymmetric matrices over F_2 of given rank. (A matrix [a(i,j)] is persymmetric if a(i,j) = a(r,s) for i+j = r+s)
We obtain, using exponential quadratic sums, explicit expressions for the number of double persymmetric matrices with entries in F_2 of given rank. (A matix [a(i,j)) is persymmetric if a(i,j) = a(r,s) for i+j = r+s)
We prove the remaining part of the conjecture by Denef and Sperber [Denef, J. and Sperber, S., \textit{Exponential sums mod $p^n$ and {N}ewton polyhedra}, Bull. Belg. Math. Soc., {\bf{suppl.}} (2001) 55-63] on nondegenerate local exponential sums modulo $p^m$. We generalize Igusa's conjecture of the introduction of [Igusa, J., \textit{Lectures on forms of higher degree}, Lect. math. phys., Springer-Verlag, {\bf{59}} (1978)] from the homogeneous to the quasi-homogeneous case and prove the nondegenerate case as well as the modulo $p$ case. We generalize some results by Katz of [Katz, N. M., \tex…
We show that automatic sequences are asymptotically orthogonal to periodic exponentials of type $e_q(f(n))$, where $f$ is a rational fraction, in the Pólya-Vinogradov range. This applies to Kloosterman sums, and may be used to study solubility of congruence equations over automatic sequences. We obtain this as consequence of a general result, stating that sums over automatic sequences can be bounded effectively in terms of two-point correlation sums over intervals.
We give a new bound on the number of collinear triples for two arbitrary subsets of a finite field. This improves on existing results which rely on the Cauchy inequality. We then us this to provide a new bound on trilinear and quadrilinear exponential sums.
Exponential sums over singular binary quintics
本文利用Waring分解，将二元四次型上指数和的方法推广至奇异二元五次型，给出无特征显式公式。
In this paper, a new method for generation of infinite series of symmetric identities written for exponential sums in real numbers is proposed. Such systems have numerous applications in theory of numbers, chaos theory, algorithmic complexity, dynamic systems, etc. Properties of generated identities are studied. Relations of the introduced method for generation of symmetric exponential sums to the Morse-Hedlund sequence and to the theory of magic squares are established.
We define finite field $A$-hypergeometric functions and show that they are Fourier expansions of families of exponential sums on the torus. For an appropriate choice of $A$, our finite field $A$-hypergeometric function can be specialized to the finite field ${}_kF_{k-1}$-hypergeometric function defined by McCarthy.
We obtain resonances for short exponential sums involving Fourier coefficients of Maass forms for $\mathrm{SL}(n,\mathbb Z)$. This involves deriving asymptotics for the integrals appearing in the $\mathrm{GL}(n)$ Voronoi summation formula. As an application, we also prove an $Ω$-result for short sums of Fourier coefficients.
We consider a class of double exponential sums studied in a paper of Sinai and Ulcigrai. They proved a linear bound for these sums along the sequence of denominators in the continued fraction expansion of $α$, provided $α$ is badly-approximable. We provide a proof of a result, which includes a simple proof of their theorem, and which applies for all irrational $α$.
We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the multiplicative characters modulo a prime $p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.
A survey paper on some recent results on additive problems with prime powers.
Improving and extending recent results of the author, we conditionally estimate exponential sums with Dirichlet coefficients of L-functions, both over all integers and over all primes in an interval. In particular, we establish new conditional results on exponential sums with Hecke eigenvalues and squares of Hecke eigenvalues over primes. We employ these estimates to improve our recent result on squares of Hecke eigenvalues at Piatetski-Shapiro primes under the Riemann Hypothesis for symmetric square L-functions for Hecke eigenforms.
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