scholarly journals Mean square of the error term in the asymmetric multidimensional divisor problem

2016 ◽  
Vol 54 (2) ◽  
pp. 173-193 ◽  
Author(s):  
Xiaodon Cao ◽  
Yoshio Tanigawa ◽  
Wenguang Zhai
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Mean Square

scholarly journals On some mean value results for the zeta-function and a divisor problem

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2315-2327
Author(s):  
Aleksandar Ivic
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Mean Value ◽  
Divisor Problem ◽  
Mean Square ◽  
Asymptotic Formulae ◽  
Dirichlet Divisor Problem ◽  
The Mean

Let ?(x) denote the error term in the classical Dirichlet divisor problem, and let the modified error term in the divisor problem be ?*(x) = -?(x) + 2?(2x)-1/2?(4x). We show that ?T+H,T ?*(t/2?)|?(1/2+it)|2dt<< HT1/6log7/2 T (T2/3+? ? H = H(T) ? T), ?T,0 ?(t)|?(1/2+it)|2dt << T9/8(log T)5/2, and obtain asymptotic formulae for ?T,0 (?*(t/2?))2|?( 1/2+it)|2 dt, ?T0 (?*(t/2?))3|?(1/+it)|2 dt. The importance of the ?*-function comes from the fact that it is the analogue of E(T), the error term in the mean square formula for |?(1/2+it)|2. We also show, if E*(T) = E(T)-2??*(T/(2?)), ?T0 E*(t)Ej(t)|?(1/2+it)|2 dt << j,? T7/6+j/4+? (j=1,2,3).

scholarly journals On the mean square formula of the error term in the Dirichlet divisor problem

2009 ◽  
Vol 146 (2) ◽  
pp. 277-287 ◽  
Author(s):  
YUK-KAM LAU ◽  
KAI-MAN TSANG
Keyword(s):  
Remainder Term ◽  
Error Term ◽  
Divisor Problem ◽  
Mean Square ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean ◽  
Upper Estimate

AbstractLet F(x) be the remainder term in the mean square formula of the error term Δ(t) in the Dirichlet divisor problem. We improve on the upper estimate of F(x) obtained by Preissmann around twenty years ago. The method is robust, which applies to the same problem for the error terms in the circle problem and the mean square formula of the Riemann zeta-function.

scholarly journals The asymptotic behaviour of the mean-square of fractional part sums

2000 ◽  
Vol 43 (2) ◽  
pp. 309-323 ◽  
Author(s):  
Manfred Kühleitner ◽  
Werner Georg Nowak
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Behaviour ◽  
Error Term ◽  
Fractional Part ◽  
Mean Square ◽  
The Mean

AbstractIn this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

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]).

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

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