Jitendra Bajpai, Subham Bhakta, Victor C. Garcia
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.
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The main results extend to sums over primes in a short interval earlier estimates by the author for "long" Weyl sums over primes.
We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank.
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 prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.
Sums of Kloosterman sums formed with modular symbols
研究由模符号形成的克洛斯特曼和的和，利用陶伯型方法给出相关拉马努金和的和的估计，定义类似塞尔伯格zeta函数的函数并确定其解析延拓，证明和的抵消结论，解释与特征值问题联系并给出数值证据。
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。
We obtain new results on additive properties of the set $$ I^{-1}= \{x^{-1}: \quad x\in I\} $$ where $I$ is an arbitrary interval in the field of residue classes modulo a large prime $p$. We combine our results with multilinear exponential sum estimates and obtain new results on incomplete multilinear Kloosterman sums.
The paper solves the problems of determining the asymptotics of the number of primes and the sums of functions of primes in a subset of the natural series that satisfies the conditions that the asymptotic density of the number of primes in this subset is constant and not equal to zero.
We prove two estimates for averages of sums of Kloosterman fractions over primes. The first of these improves previous results of Fouvry-Shparlinski and Baker.
Triple sums of Kloosterman sums and the discrepancy of modular inverses
研究大区间内模正整数$c$的模逆分布，给出其盒、球及各向同性偏差的上下界以展示与随机点集的偏离，分析基于克洛斯特曼和三重和的新界。
This work gives a general approach to the determination of the asymptotic behavior of the sums of functions of primes based on the distribution of primes. It refines the estimate of the remainder term of the asymptotic expansion of the sums of functions of primes. Also, the necessary and sufficient conditions for the existence of these asymptotics are proved in the paper.
We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.
In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of these approximations
In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/γ}]$ with $\frac{865}{886}<γ< 1$.
In this note we study Kloosterman sums twisted by a multiplicative characters modulo a prime power. We show, by an elementary calculation, that these sums become equidistributed on the real line with respect to a suitable measure.
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.
We show that an asymptotic property of the determinants of certain matrices whose entries are finite sums of cotangents with rational arguments is equivalent to the GRH for odd Dirichlet characters. This is then connected to the existence of certain quantum modular forms related to Maass Eisenstein series.
We obtain the general k-correlations for a short divisor sum related to primes.
On certain bilinear sums with modular square roots and applications
扩展了Dunn等人关于模平方根加法能量的界，并将结果应用于带模平方根的双线性指数和，在平方模大筛法上取得部分进展。
Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.
We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.
The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain algebraic and arithmetic characterizations for Jacobi sums of order 2l^2 .
We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficien…
We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{α,β}$, where $α>1$ is irrational and of finite type.
A survey paper on some recent results on additive problems with prime powers.
We prove explicit upper bounds for weighted sums over prime numbers in arithmetic progressions with slowly varying weight functions. The results generalize the well-known Brun-Titchmarsh inequality.
Under the assumption of the Riemann Hypothesis (RH), we prove explicit quantitative relations between hypothetical error terms in the asymptotic formulae for truncated mean-square average of exponential sums over primes and in the mean-square of primes in short intervals. We also remark that such relations are connected with a more precise form of Montgomery's pair-correlation conjecture.
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
On sums of two primes and six Fibonacci numbers
该研究运用圆法证明，足够大的正整数可表示为两个素数与六个斐波那契数之和、两个素数与七个卢卡斯数之和，还可表示为两个素数、三个斐波那契数与三个卢卡斯数之和。
The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.
We give some theoretical and computational results on "random" harmonic sums with prime numbers, and more generally, for integers with a fixed number of prime factors.
We obtain the triple correlations for a truncated divisor sum related to primes. We also obtain the mixed correlations for this divisor sum when it is summed over the primes, and give some applications to primes in short intervals.
In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.
Quartic Gauss sums over primes and metaplectic theta functions
本文改进了四次高斯和的估计，并引入了新的数学工具，同时推测了相关矩的渐进行为。
On bilinear sums with modular square roots and applications III
本文改进了之前方法，针对素数平方模，通过限制二次高斯和到简化剩余类，实现显著消去，提升双线性和的分析。
We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit new phenomena concerning Chebyshev-type biases of such sums when the number of irreducible factors is very large.
On bilinear sums with modular square roots and applications II
本文进一步研究了带有模平方根的双线性和和加性能量，改进了变量范围较大时的结果，并在平方模数的大筛理论中取得进展。
Bounds for moments of quadratic Dirichlet character sums of prime moduli
本文在广义黎曼假设下，先对模8p的二次Dirichlet L-函数平移矩建立精确上界，再据此证明素数模二次Dirichlet特征和矩的界。
We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cramér's model as well. We also discuss generalizations to the Gaussian integers.
Character sums to prime power moduli evaluated at binary quadratic forms
研究素数幂模下二元二次型的特征和估计，采用p进分析方法，补充Heath-Brown对平方自由模的估计。
Problems 66 and 67 on sums of residue classes and primes of András Sárközy's collection of unsolved problems
本文讨论András Sárközy 2001年未解决问题集中的第66、67题，证明了素数稀疏子集和集的最优结果。
Let $k\ge 1$ be an integer. We prove that a suitable asymptotic formula for the average number of representations of integers $n=p_{1}^{k}+p_{2}^{2}+p_{3}^{2}$, where $p_1,p_2,p_3$ are prime numbers, holds in intervals shorter than the ones previously known.
For $q$ prime, $X \geq 1$ and coprime $u,v \in \mathbb{N}$ we estimate the sums \begin{equation*} \sum_{\substack{p \leq X \substack p \equiv u \hspace{-0.25cm} \mod{v} p \text{ prime}}} \text{Kl}_2(p;q), \end{equation*} where $\text{Kl}_2(p;q)$ denotes a normalised Kloosterman sum with modulus $q$. This is a sparse analogue of a recent theorem due to Blomer, Fouvry, Kowalski, Michel and Milićević showing cancellation amongst sums of Kloosterman sums over primes in short intervals. We use an optimisation argument inspired by Fouvry, Kowalski and Michel. Our argument compares three different bo…
In this paper we show that if $A$ is a subset of Chen primes with positive relative density $α$, then $A+A$ must have positive upper density at least $cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$ in the natural numbers.
Corentin Perret-Gentil proved, under some very general conditions, that short sums of $\ell$-adic trace functions over finite fields of varying center converges in law to a Gaussian random variable or vector. The main inputs are P.~Deligne's equidistribution theorem, N.~Katz' works and the results surveyed in \cite{MR3338119}. In particular, this applies to $2$-dimensional Kloosterman sums $\mathsf{Kl}_{2,\mathbb{F}_q}$ studied by N.~Katz in \cite{MR955052} and in \cite{MR1081536} when the field $\mathbb{F}_q$ gets large. \par This article considers the case of short sums of normalized classic…
Sums of four generalized polygonal numbers of almost prime length
研究四类受限质数因子数的广义多边形数之和，证明大数可表示为此类和，参数限制最多988个质因子。
Matrix Kloosterman sums modulo prime powers
本文研究素数幂模的矩阵 Kloosterman 和，将既有素数模结果推广到素数幂情形，给出最优界，并指出其与 $\mathrm{GL}_n$ 上 horocyclic 流的联系。
Sums of products of Kloosterman sums to prime power moduli
本文针对素幂模下的Kloosterman和乘积和，证明了新的强幂节省界，通过利用p进可微性的初始项估计完全指数和的方法实现，为相关数论问题提供了量化的对齐分析。
Shifted bilinear sums of Salié sums and the distribution of modular square roots of shifted primes
研究Salié和的移位双线性和的上界，并分析移位素数的模平方根分布，揭示$b\not \equiv 0 \bmod q$时的新困难。
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。
We express a family of Hecke--Appell-type sums of Hikami and Lovejoy in terms of mixed mock modular forms; in particular, we express the sums in terms of Appell functions and theta functions. Hikami and Lovejoy's family of Hecke--Appell-type sums was obtained by considering certain duals of generalized quantum modular forms.
In this paper, we consider sums of class numbers of the type $\sum_{m\equiv a\pmod{p}} H(4n-m^2)$, where $p$ is an odd prime, $n\in \mathbb{N},$ and $a\in \mathbb{Z}$. By showing that these are coefficients of mixed mock modular forms, we obtain explicit formulas. Using these formulas for $p=5$ and $7$, we then prove a conjecture of Brown et al. in the case that $n=\ell$ is prime.
Assuming that the Generalized Riemann Hypothesis (GRH) holds, we prove an explicit formula for the number of representations of an integer as a sum of $k\geq 5$ primes. Our error terms in such a formula improve by some logarithmic factors an analogous result by Friedlander-Goldston.
For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $δx (\log x)^{-1}$ with $δ\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, δx (\log\log x)^{-1}$.
We show that the binomial and related multiplicative character sums $$ \sum_{\stackrel{x=1}{(x,p)=1}}^{p^m} χ(x^l(Ax^k +B)^w),\hspace{3ex} \sum_{x=1}^{p^m} χ_1 (x)χ_2(Ax^k +B), $$ have a simple evaluation for large enough $m$ (for $m\geq 2$ if $p\nmid ABk$).
For integer $q$, let $χ$ be a primitive multiplicative character$\pmod q.$ For integer $a$ coprime to $q$, we obtain a new bound for the sums $$\sum_{n\le N}Λ(n)χ(n+a),$$ where $Λ(n)$ is the von Mangoldt function. This bound improves and extends the range of a result of Friedlander, Gong and Shparlinski
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$, where $p$ is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components $F_γ$ of the vector-valued modular form are antisymmetric in the sense that $F_γ = -F_{-γ}$ for all $γ\in A$. As an application, we compute restrictions of …
Quantum Modular Forms from Real Quadratic Double Sums
本文研究Lovejoy和Osburn发现的十二个$q$-超几何级数，证明其傅里叶系数构成群$Γ_0(2)$上的三个向量值量子模形式。
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.
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…
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