scholarly journals Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem

1961 ◽  
Vol 5 (2) ◽  
pp. 67-75 ◽  
Author(s):  
Eckford Cohen
Keyword(s):  
Remainder Term ◽  
Substantial Reduction ◽  
Divisor Problem ◽  
Greatest Common Divisor ◽  
Arithmetical Functions ◽  
Precise Statement ◽  
Elementary Methods ◽  
Dirichlet Divisor Problem ◽  
Riemann Ζ Function ◽  
Image Position

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also placewhere γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

Square-Reduced Residue Systems (Mod r) and Related Arithmetical Functions

1979 ◽  
Vol 22 (2) ◽  
pp. 207-220 ◽  
Author(s):  
R. Sivaramakrishnan
Keyword(s):  
Symmetric Functions ◽  
Arithmetic Function ◽  
Exponential Sum ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Class Division ◽  
Arithmetical Functions ◽  
New Class ◽  
Image Position ◽  
Residue Systems

AbstractWe define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that(1)(2)In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined bywhere the summation is over a square-reduced residue system (mod r).B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and(3)As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

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.

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.

Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions

1939 ◽  
Vol 35 (3) ◽  
pp. 357-372 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Form ◽  
Fundamental Region ◽  
Arithmetical Functions ◽  
Image Position

Suppose thatis an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Thenwhere a, b, c, d are integers such that ad − bc = 1.

scholarly journals On a Conjecture of Graham Concerning a Sequence of Integers

1978 ◽  
Vol 21 (3) ◽  
pp. 285-287 ◽  
Author(s):  
E. Z. Chein
Keyword(s):  
Greatest Common Divisor ◽  
Image Position ◽  
Interesting Theorem

Let 0<a1<…<an be integers and (a, b) denotes the greatest common divisor of a, b. R. L. Graham [1] has conjectured thatfor some i and j. In a recent paper Weinstein [2] has improved Winterle's result [3] and has proven the following interesting theorem:If A is the sequence a1< … <an where ak = P, a prime for some k and , then.

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