Polynomial Gauss sums

A recent bound for exponential sums by Friedlander, Hansen and Shparlinski is extended to twisted exponential sums with general polynomial arguments. As a by-product a new result about perfect powers in certain products of polynomials is established.

     

On twisted character sums

In this paper we give a simple proof of a result by Burgess about short sums involving Dirichlet characters and exponentials. Indeed we establish a slightly stronger and more general bound that applies to sums of the form \({\sum_{n=M+1}^{M+N}f(\alpha n)\chi(n)}\), where χ is a non-principal character to the modulus p and f is a smooth 1-periodic function.     

On Weil's proof of the bound for Kloosterman sums

While most proofs of the Weil bound on one-variable Kloosterman sums over finite fields are carried out in all characteristics, the original proof of this bound, by Weil, assumes the characteristic is odd. We show how to make Weil's argument work in even characteristic, for both ordinary Kloosterman sums and sums twisted by a multiplicative character.     

Twisted exponential sums of polynomials in one variable

The twisted T-adic exponential sums associated to a polynomial in one variable are studied. An explicit arithmetic polygon in terms of the highest two exponents of the polynomial is proved to be a lower bound of the Newton polygon of the C-function of the twisted T-adic exponential sums. This bound gives lower bounds for the Newton polygon of the L-function of twisted p-power order exponential sums.     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.     

Ricci-positive metrics on connected sums of projective spaces

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed Ricci positive metrics on arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct Ricci positive metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.     

Short character sums for composite moduli

We establish new estimates on short character sums for arbitrary composite moduli with small prime factors. Our main result improves on the Graham-Ringrose bound for square-free moduli and also on the result due to Gallagher and Iwaniec when the core q′ = Π _p|q p of the modulus q satisfies log q′ ～ log q. Some applications to zero free regions of Dirichlet L-functions and the Pólya and Vinogradov inequalities are indicated.     

Exponential sums and singular hypersurfaces

We give upper bounds for the absolute value of exponential sums in several variables attached to certain polynomials with coefficients in a finite field. This bounds are given in terms of invariants of the singularities of the projective hypersurface defined by its highest degree form. For exponential sums attached to the reduction modulo a power of a large prime of a polynomial f with integer coefficients and veryfying a certain condition on the singularities of its highest degree form, we give a bound in terms of the dimension of the Jacobian quotient . Received: 3 November 1997     

A remark on LSL for weighted sums of i.i.d random elements

In the paper, the upper bound and lower bound of the law of the single logarithm (LSL) are established under the condition that the sequence of the normalized weighted sums of random elements is bounded in probability. The main result improves the upper bound in [Sung, S.H., 2009. A law of the single logarithm for weighted sums of i.i.d. random elements. Statist. Probab. Lett., 79, 1351–1357] and hence extends the result in [Chen, P., Gan, S., 2007. Limiting behavior of weighted sums of i.i.d. random variables. Statist. Probab. Lett., 77, 1589–1599].     

Decompositions of modules lacking zero sums

A module over a semiring lacks zero sums (LZS) if it has the property that v +w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics.A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly projective” module is a finite direct sum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonal primitive idempotents of R). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.     

Irrationality of the sums of zeta values

In this paper, we establish a lower bound for the dimension of the vector spaces spanned over ? by 1 and the sums of the values of the Riemann zeta function at even and odd points. As a consequence, we obtain numerical results on the irrationality and linear independence of the sums of zeta values at even and odd points from a given interval of the positive integers.     

Estimates from above of certain double trigonometric sums

In this paper we consider double trigonometric sums. Expressions of this type appear in some problems of quantum chaos and number theory. We are interested in rotation numbers of bounded type. We prove a uniform linear bound on double trigonometric sums along the subsequence of denominators of the continued fraction. The proof uses elementary techniques and the analysis of cancellations in sums of certain oscillatory functions over rotations. We also include a proof of a result on discrepancy for rotations of bounded type and in the Appendix we give an elementary proof of a result by Hardy and Littlewood.     

Exponential sums with Catalan numbers and middle binomial coefficients

We estimate the number of solutions of certain congruences with Catalan numbers and middle binomial coefficients modulo a prime. We use these results to bound double exponential sums with products of two Catalan numbers and two middle binomial coefficients, respectively, which in turn lead us to upper bounds on single exponential sums.     

Bounds of trilinear sums with Kloosterman fractions

Archiv der Mathematik - We obtain a new bound for trilinear exponential sums with Kloosterman fractions which in some ranges of parameters improves that of S. Bettin and V. Chandee (2018). We also...     

Explicit estimates of sums related to the Nyman–Beurling criterion for the Riemann Hypothesis

We give an estimate for sums appearing in the Nyman–Beurling criterion for the Riemann Hypothesis. These sums contain the Möbius function and are related to the imaginary part of the Estermann zeta function. The estimate is remarkably sharp in comparison to other sums containing the Möbius function. The bound is smaller than the trivial bound – essentially the number of terms – by a fixed power of that number. The exponent is made explicit. The methods intensively use tools from the theory of continued fractions and from the theory of Fourier series.     

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.     

Generalized Cochrane sums and Cochrane-Hardy sums

In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.     

Bound for mixed exponential sums associated to binary good cyclic codes

In this paper, we give an improved bound for mixed exponential sums associated to good cyclic codes.     

New estimates of short Kloosterman sums

In this paper, we obtain new, sharper, estimates of short Kloosterman sums.