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Fourier Series and Gauss Sums
The basic functions that we will be concerned with in this chapter are for $$n...
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Quadratic Gauss Sums
We introduce Pell forms and show they lead us in a natural way to quadratic Gauss sums. We point out connections to the analytic class number formula...
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New identities involving certain Hardy sums and two-term exponential sums
In this paper, we present a new reciprocity formula for certain Hardy sums and study a computational problem of one type mean value containing...
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On the Mean Value of the Generalized Dedekind Sum and Certain Generalized Hardy Sums Weighted by the Kloosterman Sum
We study a hybrid mean-value problem related to the generalized Dedekind sum, certain generalized Hardy sums, and Kloosterman sum and obtain several...
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On inverted Kloosterman sums over finite fields
The classical n -variable Kloosterman sums over finite fields are well understood by Deligne’s theorem from complex point of view and by Sperber’s...
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A note on two-term exponential sum and the reciprocal of the quartic Gauss sums
The main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the...
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Hybrid mean value of the character sums over the short interval \([1,\frac{p}{8})\) and the other famous sums
The main purpose of this paper is utilizing the analytic methods to study Hybrid mean value including the character, the general trigonometric sums,...
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Matrix Kloosterman sums modulo prime powers
We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the first two authors on the case of prime moduli....
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Moments of Kummer sums weighted by L-functions
The main purpose of this article is to study higher order moments of Kummer sums weighted by L -functions using estimates for character sums and...
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Conditioning of Fourier sums on a quadratic curve
The stability property of representations of the Fourier sums on a quadratic curve is measured through a condition number. We analyze the...
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Reciprocity and the kernel of Dedekind sums
We use the action of Atkin–Lehner operators to generate a family of reciprocity formulas for newform Dedekind sums. This family of reciprocity...
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The size of wild Kloosterman sums in number fields and function fields
We study p -adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k ...
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Exponential Sums over Finite Fields
This is an expository paper on algebraic aspects of exponential sums over finite fields. This is a new direction. Various examples, results and open...
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Exact evaluations and reciprocity theorems for finite trigonometric sums
We evaluate in closed form several classes of finite trigonometric sums. Two general methods are used. The first method uses contour integration and...
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Bounds for theta sums in higher rank. I
Theta sums are finite exponential sums with a quadratic form in the oscillatory phase. This paper establishes new upper bounds for theta sums in the...
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On character sums with determinants
We estimate weighted character sums with determinants ad − bc of 2 × 2 matrices modulo a prime p with entries a , b , c and d varying over the interval...
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Three conjectures about character sums
We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the...
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Finite trigonometric sums arising from Ramanujan’s theta functions
Two classes of finite trigonometric sums, each involving only sines, are evaluated in closed form. The previous and original proofs arise from...