Let A be a finite nonempty subset of an additive abelian group G, and let Σ(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |Σ(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of Σ(A). Our result implies that Σ(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erdős and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.
This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polyloga- rithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.
In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. As applications of these formulae, we give new closed form representations of several quadratic Euler sums through Riemann zeta function and linear sums. The given representations are new.
In this paper, we obtain some formulas for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers. By using these formulas, we give new closed form sums of several quadratic Euler series through Riemann zeta values, polylogarithm functions and linear sums. Furthermore, some relationships between Euler sums and integrals of polylogarithm functions are established.
The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.
In this paper we study Ramanujan sums $c_{\bf m}(\bf n)$, where $ {\bf m}$ and ${\bf n}$ are integral ideals in an arbitrary quadratic number field. We give some new results about the asymptotic behavior of sums of $c_{\bf m}(\bf n)$ over both $ {\bf m}$ and $ {\bf n}$.
Large values of quadratic character sums revisited
Tongji University（同济大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科学大学）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；Hunan University（湖南大学）
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.
We study smoothed character sums involving $\sum_{m,n} ( \frac{m}{n} )_2$, where $( \frac{m}{n} )_2$ denotes the quadratic symbol in the Gaussian field. We extend previously known results to obtain asymptotic formulas for the sums considered to larger ranges of $m$ and $n$.
For suitable pairs of diagonal quadratic forms in 8 variables we use the circle method to investigate the density of simultaneous integer solutions and relate this to the problem of estimating linear correlations among sums of two squares.
Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\mid Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+\varepsilon})$, for any fixed $\varepsilon>0$. Without the coprimality condition on the determinant one could not achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.
The main purpose of this article is to study higher power mean values of generalized quadratic Gauss sums using estimates for character sums, analytic method and algebraic geometric methods. In this article, we prove two conjectures which were proposed in \cite{BLZ}.
It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.
In this paper we define the generalized q-analogues of Euler sums and present a new family of identities for q-analogues of Euler sums by using the method of Jackson q-integral rep- resentations of series. We then apply it to obtain a family of identities relating quadratic Euler sums to linear sums and q-polylogarithms. Furthermore, we also use certain stuffle products to evaluate several q-series with q-harmonic numbers. Some interesting new results and illustrative examples are considered. Finally, we can obtain some explicit relations for the classical Euler sums when q approaches to 1.
Soochow University（苏州大学）；Hainan Bielefeld University of Applied Sciences（海南比勒费尔德应用科技大学）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；Hunan University（湖南大学）；Tongji University（同济大学）
In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.
Let $\mathcal L^+$ be the set of all primes $p$ for which the sums of $\left(\frac{n}{p}\right)$ over the interval $[1,N]$ are non-negative for all $N$. We prove that the estimate \[ |\mathcal L^+\cap [1,x]|\ll \frac{x}{\ln x(\ln\ln x)^{c-o(1)}} \] holds for $c\approx 0.0368$
We study real quadratic fields $\mathbb{Q}(\sqrt{D})$ such that, for a given rational integer $m$, all $m$-multiples of totally positive integers are sums of squares. We prove quite sharp necessary and sufficient conditions for this to happen. Further, we give a fast algorithm that solves this question for specific $m$, $D$ and we give complete results for $m \leq 5000$.
Sarnak obtained the asymptotic formula of the sum of the class numbers of indefinite binary quadratic forms from the prime geodesic theorem for the modular group. In the present paper, we show several asymptotic formulas of partial sums of the class numbers by using the prime geodesic theorems for the congruence subgroups of the modular group.
Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn + tn^2$. Suppose that $|S(x,t)|$ attains its maximum for given $x$ at $t = t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of $S(x,t(x))$.
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.
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.
In this paper we have estimated one multi-variable character sum \begin{align*} \sum_{a=2}^{p-2}\sum_{b=1}^{p-1}\sum_{c=2}^{p-2}\sum_{d=1}^{p-1}\left(\frac{a^2-b^2}{p}\right)\left(\frac{b^2-1}{p}\right) \left(\frac{c^2-d^2}{p}\right)\left(\frac{d^2-1}{p}\right)\left(\frac{a^2c^2-1}{p}\right), \end{align*} for odd prime $p$. With the help of our estimate of the above character sum, we have studied the tenth power mean value of generalized quadratic Gauss sums using estimates for character sums and analytic methods.
In this methodological paper, we first review the classic cubic Diophantine equation $a^3 + b^3 + c^3 = d^3$, and consider the specific class of solutions $q_1^3 + q_2^3 + q_3^3 = q_4^3$ with each $q_i$ being a binary quadratic form. Next we turn our attention to the familiar sums of powers of the first $n$ positive integers, $S_k = 1^k + 2^k + \cdots + n^k$, and express the squares $S_k^2$, $S_m^2$, and the product $S_k S_m$ as a linear combination of power sums. These expressions, along with the above quadratic-form solution for the cubic equation, allows one to generate an infinite number o…
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.
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 $c\, δx (\log\log x)^{-1}$.
A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are generalized.
Let F_q be the finite field of q elements. Let H be a multiplicative subgroup of F_q^*. For a positive integer k and element b\in F_q, we give a sharp estimate for the number of k-element subsets of H which sum to b.
For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$, then there is a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that $P(A)=\mathbb{N}\setminus B$. This result shows that the answer to the problem of Chen and Wu [`The inverse problem on subset sums', European. J. Combin. 34(2013), 841-845] is negative.
For $1$-periodic functions $f$ satisfying only a weak local regularity assumption of Dini's type at rational points of $]0,1[$, we study the Farey sums $$F_n(f)= \sum_{\frac{\k}ł\in \F_n} f\big(\frac{\k}ł\big),\qq F_{n,\s}(f)= \sum_{\frac{\k}ł\in \F_n} \frac{1}{\k^\sł^\s}f\big(\frac{\k}ł\big),\qq 1/2\le \s<1 , $$ where $\F_n$ is the Farey series of order $n\ge 1$. We obtain sharp estimates of $F_{n,\s}(f)$, for all $0< \s\le1$. We prove similar results for the corresponding Riemann quadratic sums $$ S_{n,\s}(f) \ =\ \sum_{1\le k\le \ell \le n}\frac{1}{(k\ell)^{\s }}\, f\big( \frac{k}{\ell}\big…
We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo--Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted varian…
Let f(z) = sum_n a(n) n^{(k-1)/2} e(nz) be a cusp form for Gamma_0(N), character chi and weight k geq 4. Let q(x) = x^2 + sx + t be a polynomial with integral coefficients. It is shown that sum_{n \leq X} a(q(n)) = cX + O(X^{6/7+eps}) for some constant c depending on f and q. The constant vanishes in many cases, for example if k is even. On the way a Kuznetsov formula for half-integral weight and entries having different sign is derived.
Let $τ_3(n)$ be the triple divisor function which is the number of solutions of the equation $d_1d_2d_3=n$ in natural numbers. It is shown that $$ \sum_{1\leq n_1,n_2,n_3\leq \sqrt{x}}τ_3(n_1^2+n_2^2+n_3^2)=c_1x^{\frac{3}{2}}(\log x)^2+ c_2x^{\frac{3}{2}}\log x +c_3x^{\frac{3}{2}} +O_{\varepsilon}(x^{\frac{11}{8}+\varepsilon}) $$ for some constants $c_1$, $c_2$ and $c_3$.
Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2π/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2π/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bi…
We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel's mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.
In the past two decades, many researchers have studied {\it index $2$} Gauss sums, where the group generated by the characteristic $p$ of the underling finite field is of index $2$ in the unit group of ${\mathbb Z}/m{\mathbb Z}$ for the order $m$ of the multiplicative character involved. A complete solution to the problem of evaluating index $2$ Gauss sums was given by Yang and Xia~(2010). In particular, it is known that some nonzero integral powers of the Gauss sums in this case are in quadratic fields. On the other hand, Chowla~(1962), McEliece~(1974), Evans~(1977, 1981) and Aoki~(1997, 2004…
