The Dirichlet Divisor Problem of Arithmetic Progressions

2016 ◽  
Vol 59 (3) ◽  
pp. 592-598
Author(s):  
H. Q. Liu
Keyword(s):  
Asymptotic Formula ◽  
Divisor Problem ◽  
Arithmetic Progressions ◽  
Elementary Method ◽  
Dirichlet Divisor Problem ◽  
Better Than

AbstractWe present an elementary method for studying the problem of getting an asymptotic formula that is better than Hooley’s and Heath-Brown’s results for certain cases.

scholarly journals Remark on Smith’s result on a divisor problem in arithmetic progressions

1985 ◽  
Vol 98 ◽  
pp. 37-42 ◽  
Author(s):  
Kohji Matsumoto
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Divisor Problem ◽  
Arithmetic Progressions ◽  
Euler Function

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds: where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).

scholarly journals Estimates of convolutions of certain number-theoretic error terms

2004 ◽  
Vol 2004 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Abelian Groups ◽  
Divisor Problem ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Convolution Function

Several estimates for the convolution functionC [f(x)]:=∫1xf(y) f(x/y)(dy/y)and its iterates are obtained whenf(x)is a suitable number-theoretic error term. We deal with the case of the asymptotic formula for∫0T|ζ(1/2+it)|2kdt(k=1,2), the general Dirichlet divisor problem, the problem of nonisomorphic Abelian groups of given order, and the Rankin-Selberg convolution.

On the seventh power moment of Δ(x)

2017 ◽  
Vol 13 (03) ◽  
pp. 571-591
Author(s):  
Jinjiang Li
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Divisor Problem ◽  
Power Moment ◽  
Dirichlet Divisor Problem

Let [Formula: see text] be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text] which improves the previous result.

scholarly journals On a Theorem of Bombieri, Friedlander, and Iwaniec

2012 ◽  
Vol 64 (5) ◽  
pp. 1019-1035 ◽  
Author(s):  
Daniel Fiorilli
Keyword(s):  
Type Theorem ◽  
Divisor Problem ◽  
Arithmetic Progressions

Abstract In this article, we show to what extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.

scholarly journals On the Dirichlet divisor problem in short intervals

2013 ◽  
Vol 33 (3) ◽  
pp. 447-465 ◽  
Author(s):  
Aleksandar Ivić ◽  
Wenguang Zhai
Keyword(s):  
Divisor Problem ◽  
Short Intervals ◽  
Dirichlet Divisor Problem

scholarly journals Elliptic curves with a given number of points over finite fields

2012 ◽  
Vol 149 (2) ◽  
pp. 175-203 ◽  
Author(s):  
Chantal David ◽  
Ethan Smith
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Distribution Of Primes ◽  
Primes In Arithmetic Progressions ◽  
Interval Distribution ◽  
Better Than

AbstractGiven an elliptic curve E and a positive integer N, we consider the problem of counting the number of primes p for which the reduction of E modulo p possesses exactly N points over 𝔽p. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short-interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.

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