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 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 EXPLICIT REPRESENTATIONS OF THE INTEGRAL CONTAINING THE ERROR TERM IN THE DIVISOR PROBLEM II

2011 ◽  
Vol 54 (1) ◽  
pp. 133-147
Author(s):  
JUN FURUYA ◽  
YOSHIO TANIGAWA
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Explicit Representation ◽  
Integral Sign ◽  
Integral Formulas ◽  
Alternative Approach ◽  
Bernoulli Functions ◽  
Dirichlet Divisor Problem ◽  
Explicit Representations ◽  
Riemann Zeta

AbstractIn our previous paper [2], we derived an explicit representation of the integral ∫1∞t−θΔ(t)logjtdt by differentiation under the integral sign. Here, j is a fixed natural number, θ is a complex number with 1 < θ ≤ 5/4 and Δ(x) denotes the error term in the Dirichlet divisor problem. In this paper, we shall reconsider the same formula by an alternative approach, which appeals to only the elementary integral formulas concerning the Riemann zeta- and periodic Bernoulli functions. We also study the corresponding formula in the case of the circle problem of Gauss.

scholarly journals On the mean of the shifted error term in the theory of the Dirichlet divisor problem

2016 ◽  
Vol 46 (1) ◽  
pp. 105-124
Author(s):  
Xiaodong Cao ◽  
Jun Furuya ◽  
Yoshio Tanigawa ◽  
Wenguang Zhai
Keyword(s):  
Error Term ◽  
Divisor Problem ◽  
Dirichlet Divisor Problem ◽  
The Mean

scholarly journals On some upper bounds for the zeta-function and the Dirichlet divisor problem

2016 ◽  
Vol 12 (08) ◽  
pp. 2231-2239
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Upper Bounds ◽  
Mean Value ◽  
Divisor Problem ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Riemann Zeta

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

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).

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