The H and J functionals in the theory of complete exponential sums

On Evaluation of Nonlinear Exponential Sums

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.

Incomplete exponential sums over exponential functions

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.

L-functions of Certain Exponential Sums over Finite Fields

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.

Closed Forms Evaluations Of Some Exponential Sums

This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.

L-functions of certain exponential sums over finite fields II

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.

Results about persymmetric matrices over F_2 and related exponential sums

In this paper we expose our main results about rank problems concerning persymmetric matrices over F_2 associated to some exponential sums.

On Character Sums and Exponential Sums over Generalized Arithmetic Progressions

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.

Multilinear Exponential Sums With A General Class Of Weights

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.

On Bourgain's bound for short exponential sums and squarefree numbers

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.

指数和与Piatetski-Shapiro素数的Barban-Davenport-Halberstam型定理

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。

涉及两项指数和与 Dedekind 和的两类混合幂平均

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。

Exponential Sums over Points of Elliptic Curves

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.

A Bombieri -- Vinogradov type result for exponential sums over Piatetski-Shapiro primes

In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.

Bilinear Forms with Exponential Sums with Binomials

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

Asymptotics and formulas for cubic exponential sums

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.

Exponential sums in prime fields for modular forms

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.

Exponential sums over Piatetski-Shapiro primes in arithmetic progressions

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.

Exponential Sums of Witt Towers over Affinoids

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.

On Fourth and Higher Moments of Short Exponential Sums Related to Cusp Forms

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.

Moment estimates for exponential sums over k-free numbers

We investigate the size of L^p-integrals for exponential sums over k-free numbers and prove essentially tight bounds.

双指数和的矩

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。

Multiplicative Energy of Shifted Subgroups and Bounds On Exponential Sums with Trinomials in Finite Fields

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.

An improvement on the upper bound of exponential sums connected to holomorphic cusp forms

We improve the known upper bound for short exponential sums and increase the range on which a sharp upper bound is known.

On the distribution of cubic exponential sums

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.

Bounds on exponential sums over small multiplicative subgroups

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.

Exponential sums with coefficients of certain Dirichlet series

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.

Exponential sums and rank of triple persymmetric matrices over F_2

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)

Exponential sums and rank of double persymmetric matrices over F_2

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)

Exponential sums: questions by Denef, Sperber, and Igusa

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…

Exponential sums with automatic sequences

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.

Incidence Results and Bounds of Trilinear and Quadrilinear Exponential Sums

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.

奇异二元五次型上的指数和

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。

Generation of symmetric exponential sums

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.

Exponential sums and finite field $A$-hypergeometric functions

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.

Resonances and $Ω$-results for Exponential Sums Related to Maass Forms for $\mathrm{SL}(n,\mathbb Z)$

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.

Upper bounds for double exponential sums along a subsequence

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 $α$.

Multiple Exponential and Character Sums with Monomials

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.

Applications of some exponential sums on prime powers: a survey

A survey paper on some recent results on additive problems with prime powers.

Exponential sums with Dirichlet coefficients of L-functions

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.

L-functions of symmetric powers of the generalized Airy family of exponential sums

This paper looks at the L-function of the k-th symmetric power of the ell-adic sheaf over the affine line associated to the generalized Airy family of exponential sums. We compute the degree of this rational function and the local factors at infinity.

Exponential sums with multiplicative coefficients and applications

We show that if an exponential sum with multiplicative coefficients is large then the associated multiplicative function is "pretentious". This leads to applications in the circle method, and a natural interpretation of the local-global principle.

Exponential sums over finite fields and the large sieve

By using a variant of Kowalski's large sieve for Frobenius in compatible systems, we obtain zero-density estimates for arguments of $\ell$-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

On unit root formulas for toric exponential sums

Starting from a classical generating series for Bessel functions due to Schlomilch, we use Dwork's relative dual theory to broadly generalize unit-root results of Dwork on Kloosterman sums and Sperber on hyperkloosterman sums. In particular, we express the (unique) p-adic unit root of an arbitrary exponential sum on the torus in terms of special values of the p-adic analytic continuation of a ratio of A-hypergeometric functions. In contrast with the earlier works, we use noncohomological methods and obtain results that are valid for arbitrary exponential sums without any hypothesis of nondegen…

$T$-adic Exponential Sums over Affinoids

We introduce and develop $(π,p)$-adic Dwork theory for $L$-functions of exponential sums associated to one-variable rational functions, interpolating $p^k$-order exponential sums over affinoids. Namely, we prove a generalization of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic continuation of the $C$-function $C_f(s,π)$. We compute the lower $(π,p)$-adic bound, the Hodge polygon, for this $C$-function. Along the way, we also show why a strictly $π$-adic theory will not work in this case.

Exponential sums with sparse polynomials over finite fields

We obtain new bounds of exponential sums modulo a prime $p$ with sparse polynomials $a_0x^{n_0} + \cdots + a_νx^{n_ν}$. The bounds depend on various greatest common divisors of exponents $n_0, \ldots, n_ν$ and their differences. In particular, two new bounds for binomials are obtained, improving previous results in broad ranges of parameters.

Exponential Sums Related to Maass Forms

We estimate short exponential sums weighted by the Fourier coefficients of a Maass form. This requires working out a certain transformation formula for non-linear exponential sums, which is of independent interest. We also discuss how the results depend on the growth of the Fourier coefficients in question. As a byproduct of these considerations, we can slightly extend the range of validity of a short exponential sum estimate for holomorphic cusp forms. The short estimates allow us to reduce smoothing errors. In particular, we prove an analogue of an approximate functional equation previously …

On the mean square of short exponential sums related to cusp forms

The purpose of the article is to estimate the mean square of a squareroot length exponential sum of Fourier coefficients of a holomorphic cusp form.

通过指数和估计$|ζ'(1/2+it)|$的有效上界

Effective Upper Bound Estimates for $|ζ'(1/2+it)|$ via Exponential Sums

本研究针对黎曼ζ函数导数在临界线上的有效上界估计问题，利用指数和方法推导对应估计公式，且可通过选取不同参数得到不同的有效上界。

关于短区间中素数指数和的一个条件性结果

A conditional result on exponential sums over primes in short intervals

本文在广义 Riemann 假设及 Dirichlet L 函数密度假设同时成立的条件下，给出短区间中素数指数和的一个条件性结果。

Rational exponential sums over the divisor function

We consider a problem posed by Shparlinski, of giving nontrivial bounds for rational exponential sums over the arithmetic function $τ(n)$, counting the number of divisors of $n$. This is done using some ideas of Sathe concerning the distribution in residue classes of the function $ω(n)$, counting the number of prime factors of $n$, to bring the problem into a form where, for general modulus, we may apply a bound of Bourgain concerning exponential sums over subgroups of finite abelian groups and for prime modulus some results of Korobov and Shkredov.

Three-dimensional exponential sums under constant perturbation

In this paper, by generalizing Bombieri and Iwaniec's double large sieve inequality, we obtain an estimation on a type of three-dimensional exponential sums with constant perturbation. As an application of our estimation, we obtain an asymptotic formula for a sum involving the Mangoldt function and the integral part function. This type of sum was originally studied by Bordelles-Dai-Heyman-Pan-Shparlinski.

Decoupling, exponential sums and the Riemann zeta function

We establish a new decoupling inequality for curves in the spirit of [B-D1], [B-D2] which implies a new mean value theorem for certain exponential sums crucial to the Bombieri-Iwaniec method as developed further in [H]. In particular, this leads to an improved bound $|ζ(\frac 12+it)|\ll t^{53/342+\varepsilon}$ for the zeta function on the critical line

On Estimates of Exponential Sums

This paper introduces a general technique for estimating the absolute value of pure Gaussian sums of order k over a prime p for a class of composite order k. The new estimate improves the classical estimate by a factor of about 2 or better (depending on k) for all sufficiently large primes p.

正多边形涡丝演化与指数和

Regular Polygonal Vortex Filament Evolution and Exponential Sums

本文通过证明一类具算术内涵的指数和为纯虚数，严格推导了M边正多边形在涡丝方程演化有理时刻出现的挠多边形的相邻边夹角表达式。