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Displaying 81 – 100 of 169

On Linnik's theorem on Goldbach numbers in short intervals and related problems

Alessandro Languasco, Alberto Perelli (1994)

Annales de l'institut Fourier

Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ > 0$ , the interval $[N, N + {log}^{3 + ϵ} N]$ contains a number which is the sum of two primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C {log}^{2} N$ , the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s...

On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II

R. Nair (1993)

Studia Mathematica

On some divisor problems

Hong-Quan Liu (1994)

Acta Arithmetica

On sum-product representations in $ℤ_{q}$

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$ , by performing sum and product set operations starting from a given subset $A$ of $ℤ_{q}$ . We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $| A | \sim log q$ ). Roughly speaking we show that all residue classes are obtained from a $k$ -fold sum of an $r$ -fold product set of $A$ , where $r ≪ log q$ and $log k ≪ log q$ , provided the residue sets...

On sums and differences of two coprime kth powers

Wenguang Zhai (1999)

Acta Arithmetica

On sums and products of residues modulo p

A. Sárközy (2005)

Acta Arithmetica

On sums of five almost equal prime squares

Jianya Liu, Tao Zhan (1996)

Acta Arithmetica

On sums of two kth powers: a mean-square asymptotics over short intervals

Manfred Kühleitner, Werner Georg Nowak (2001)

Acta Arithmetica

On sums of two kth powers: an asymptotic formula for the mean square of the error term

M. Kühleitner (2000)

Acta Arithmetica

On the $2 k$ -th power mean of $\frac{L^{'}}{L} (1, χ)$ with the weight of Gauss sums

Dongmei Ren, Yuan Yi (2009)

Czechoslovak Mathematical Journal

The main purpose of this paper is to study the hybrid mean value of $\frac{L^{'}}{L} (1, χ)$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum_{χ \neq χ_{0}} | τ (χ) | | \frac{L^{'}}{L} {(1, χ) |}^{2 k}$ of $\frac{L^{'}}{L}$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.