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We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In particular, we expand on ideas from some previous work on arithmetic series and the RH. To accomplish this, Mellin inversion is applied to certain infinite series over arithmetic functions to apply Cauchy's residue theorem, and then the remainder of terms is estimated according to the assumption of the RH. In the last section, we use simple properties...

A note on character sums over finite fields.

D.A. Burgess (1972)

Journal für die reine und angewandte Mathematik

A note on exponential sums.

L. Carlitz (1978)

Mathematica Scandinavica

A Note on prime k-th power nonresidues.

Richard H. Hudson (1983)

Manuscripta mathematica

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

A note on the Dirichlet characters of polynomials

Wenpeng Zhang, Weili Yao (2004)

Acta Arithmetica

A note on the distribution of angles associated to indefinite integral binary quadratic forms

Dragan Đokić (2019)

Czechoslovak Mathematical Journal

To each indefinite integral binary quadratic form $Q$ , we may associate the geodesic in $ℍ$ through the roots of quadratic equation $Q (x, 1)$ . In this paper we study the asymptotic distribution (as discriminant tends to infinity) of the angles between these geodesics and one fixed vertical geodesic which intersects all of them.

A note on the mean value of the general Kloosterman sums

Weili Yao (2011)

Czechoslovak Mathematical Journal

The main purpose of this paper is to use the analytic method to study the calculating problem of the general Kloosterman sums, and give an exact calculating formula for it.

A note on the zeros of exponential polynomials

A. J. Van der Poorten (1975)

Compositio Mathematica

A notion of diaphony based on p-adic arithmetic

Peter Hellekalek (2010)

Acta Arithmetica

A problem concerning a character sum.

Teske, Edlyn, Williams, Hugh C. (1999)

Experimental Mathematics

A problem on semicubical powers

N. Watt (1989)

Acta Arithmetica

A projective invariant comparing rings of integers in wildly ramified extensions.

S.M.J. Wilson (1990)

Journal für die reine und angewandte Mathematik

A purely analytical lower bound for $L (1, χ)$

Olivier Ramaré (2009)

Annales mathématiques Blaise Pascal

We give a simple proof of $L (1, χ) \sqrt{q} ≫ 2^{ω (q)}$ when $χ$ is an odd primitiv quadratic Dirichlet character of conductor $q$ . In particular we do not use the Dirichlet class-number formula.

A question of the distribution of points on a curve over a finite field.

Cobeli, Cristian, Zaharescu, Alexandru (2002)

Acta Universitatis Apulensis. Mathematics - Informatics

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