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 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.

scholarly journals On the fourth power moment of the error term for the divisor problem with congruence conditions

2018 ◽  
Vol 14 (06) ◽  
pp. 1525-1546 ◽  
Author(s):  
Jinjiang Li ◽  
Min Zhang
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Congruence Class ◽  
Divisor Problem ◽  
Fourth Power ◽  
Power Moment ◽  
Higher Power ◽  
Power Moments

Let [Formula: see text] denote the number of factorizations [Formula: see text], where each of the factors [Formula: see text] belongs to a prescribed congruence class [Formula: see text]. Let [Formula: see text] be the error term of the asymptotic formula of [Formula: see text]. In this paper, we establish an asymptotic formula of the fourth power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text], which improves the previous value [Formula: see text] of Liu [On higher-power moments of the error term for the divisor problem with congruence conditions, Monatsh. Math. 163(2) (2011) 175–195].

ON THE FIFTH-POWER MOMENT OF Δ(x)

2011 ◽  
Vol 07 (01) ◽  
pp. 71-86 ◽  
Author(s):  
DEYU ZHANG ◽  
WENGUANG ZHAI
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Power Moment ◽  
Dirichlet Divisor Problem

Let Δ(x) denote the error term in the Dirichlet divisor problem. In this paper, we study the fifth-power moment of Δ(x) and prove that [Formula: see text] with δ5 = 3/80, which improves the previous results.

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 On a generalized divisor problem I

2002 ◽  
Vol 165 ◽  
pp. 71-78 ◽  
Author(s):  
Yuk-Kam Lau
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Sign Changes ◽  
Oscillatory Nature ◽  
Dirichlet Divisor Problem

We give a discussion on the properties of Δa(x) (− 1 < a < 0), which is a generalization of the error term Δ(x) in the Dirichlet divisor problem. In particular, we study its oscillatory nature and investigate the gaps between its sign-changes for −½ ≤ a < 0.

scholarly journals On the periodic zeta-function with rational parameter

2011 ◽  
Vol 52 ◽  
Author(s):  
Sondra Černigova
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Error Term ◽  
Fourth Power ◽  
Power Moment ◽  
Rational Parameter ◽  
Estimated Error

We obtain an asymptotic formula with estimated error term for the fourth power moment of the periodic zeta-function with rational parameter.

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