On Dirichlet Series for Sums of Squares

Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions _k (n) and _k ²(n) in the terms of Riemann Zeta function (s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f _i and g _i are completely multiplicative, then we have where L _f(s) := _{n = 1} f(n)n ^–s is the Dirichlet series corresponding to f. Let r _N(n) be the number of solutions of x ₁ ² + ··· + x _N ² = n and r _2,P (n) be the number of solutions of x ² + Py ² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of (s) and Dirichlet L-functions, for the generating functions of r _N(n), r _N ²(n), r _2,P (n) and r _2,P (n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.     

Sums of Five Almost Equal Prime Squares

Let Pi, 1≤i≤5, be prime numbers. It is proved that every integer N that satisfies N=5 （mod 24） can be written as N=p1^2＋p2^2＋P3^2＋p4^2 ＋p5^2, where │√N5-Pi│≤N^1/2-19/850＋∈.     

Results of Hurwitz Type for Five or More Squares

Let r _k(n) denote the number of representations of n as a sum of k squares. We give many cases (for k = 5, 6, 7) in which the generating function _n0 r _k(an + b)q ⁿ is a simple infinite product. For instance,

Estimating Additive Character Sums for Fuchsian Groups

In the usual construction of non-holomorphic Eisenstein series, for a general Fuchsian group, a multiplicative character may be included. The properties of these series are well known. Here we instead include an additive character and develop the properties of the resulting series. We pay particular attention to additive characters that are non-cuspidal, i.e., that are not zero on some parabolic generators. These series may be used to estimate certain additive character sums. For example, asymptotics for a weighted sum over group elements that counts the number of appearances of a fixed generator of the Fuchsian group are obtained.     

Exponential Sums and Coding Theory: A Review

This article is a survey of several recent applications of methods from analytic number theory to research in coding theory, including results on Kloosterman codes, binary Goppa codes, and prime phase shift sequences. The mathematical methods focus on exponential sums, in particular Kloosterman sums. The interrelationships with the Weil–Carlitz–Uchiyama bound, results on Hecke operators, theorems of Bombieri and Deligne and the Eichler–Selberg trace formula are reviewed.     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

Representations of Integers as Sums of 32 Squares

In this paper, we derive a new explicit formula for r ₃₂(n), where r _k(n) is the number of representations of n as a sum of k squares. For a fixed integer k, our method can be used to derive explicit formulas for r _8k (n). We conclude the paper with various conjectures that lead to explicit formulas for r _2k (n), for any fixed positive integer k > 4.     

Sums of squares in octonion algebras

Sums of squares in composition algebras are investigated using methods from the theory of quadratic forms. For any integer octonion algebras of level and of level are constructed.

     

On Shimura, Shintani and Eichler-Zagier correspondences

In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

     

Hecke Algebras and Automorphic Forms

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G ₂(F₅) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL₂ to PGSp₄.     

Estimates for Sums of Coefficients of Dirichlet Series with Functional Equation

Suppose we have a Dirichlet series L(s) = _{n = 1} a _n n ^–s such that it, and its twists by Dirichlet characters have analytic continuation and a functional equation of a specific kind. Suppose also that the root numbers of the twists are equidistributed on the unit circle. The purpose of this note is to get an estimate for the quantity for a prime modulus p.We use a modification of the method of Chandrasekharan and Narasimhan and we use in an essential way a Rankin-Selberg type estimate for the average of |a _n|².     

A note on biquadratic forms

It is shown that semidefinite quadratic forms in two by n variables are sums of squares of bilinear forms.     

Exponential Sums and the Riemann Zeta Function V

A Van der Corput exponential sum is S = exp (2 i f(m)) wherem has size M, the function f(x) has size T and = (log M) / log T < 1. There are different bounds for S in differentranges for . In the middle range where is near 1/over 2, . This bounds the exponent of growthof the Riemann zeta function on its critical line Re s = 1/over2. Van der Corput used an iteration which changed at each step.The Bombieri–Iwaniec method, whilst still based on meansquares, introduces number-theoretic ideas and problems. TheSecond Spacing Problem is to count the number of resonancesbetween short intervals of the sum, when two arcs of the graphof y = f'(x) coincide approximately after an automorphism ofthe integer lattice. In the previous paper in this series [Proc.London Math. Soc. (3) 66 (1993) 1–40] and the monographArea, lattice points, and exponential sums we saw that coincidenceimplies that there is an integer point close to some ‘resonancecurve’, one of a family of curves in some dual space,now calculated accurately in the paper ‘Resonance curvesin the Bombieri–Iwaniec method’, which is to appearin Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatisedstep: an upper bound for the number of integer points closeto a plane curve gives a bound in the Second Spacing Problem,and a small improvement in the bound for S. Ends and cusps ofresonance curves are treated separately. Bounds for sums oftype S lead to bounds for integer points close to curves, andanother branching iteration. Luckily Swinnerton-Dyer's methodis stronger. We improve from 0.156140... in the previous paperand monograph to 0.156098.... In fact (32/205 + , 269/410 +) is an exponent pair for every > 0. 2000 Mathematics SubjectClassification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).     

指数和与特征和的混合均值

本文利用三角和估计及其特征和的性质研究一类二项指数和与多项式特征和的混合均值的计算问题,并给出两个有趣的计算公式.     

多项式的特征和与二项指数和的混合均值

本文主要利用解析方法以及二项指数和与Dirichlet特征的性质,研究多项式的特征和与二项三次指数和的混合均值的计算问题,并得到一个较强的渐近公式.     

General Kloosterman Sums and the Difference Between an Integer and Its Inverse Modulo Q

The main purpose of this paper is to use the generalized Bernoulli numbers, Gauss sums and the mean value theorems of Dirichlet L-functions to study the asymptotic property of one class of number-theoretic functions, and to give four interesting hybrid mean value formulae.     

On the Representation of Numbers by Positive Diagonal Quadratic Forms with Five Variables of Level 16

A general formula is derived for the number of representations r(n; f) of a natural number n by diagonal quadratic forms f with five variables of level 16. For f belonging to one-class series, r(n; f) coincides with the sum of a singular series, while in the case of a many-class series an additional term is required, for which the generalized theta-function introduced by T. V. Vepkhvadze [4] is used.