Note on Ramanujan's arithmetical function τ (n)

1927 ◽  
Vol 23 (6) ◽  
pp. 675-680 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined byThis function is important in the theory of the representation of a number as a sum of 24 squares. In factwhere r24 (n) is the number of representations;where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

On Ramanujan's Arithmetical Function Σr, s(n)

1929 ◽  
Vol 25 (3) ◽  
pp. 255-264 ◽  
Author(s):  
J. R. Wilton
Keyword(s):  
Error Term ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position ◽  
The Right ◽  
Positive Integers

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

On the order of magnitude of Ramanujan's arithmetical function τ(n)

1951 ◽  
Vol 47 (4) ◽  
pp. 668-678 ◽  
Author(s):  
W. B. Pennington
Keyword(s):  
Asymptotic Formula ◽  
Arithmetical Function ◽  
Modular Functions ◽  
Arithmetical Functions ◽  
Order Of Magnitude ◽  
Image Position ◽  
Ramanujan’S Formula

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansionThis function appears in the discussion of an asymptotic formula for the functionand also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

A New Method in Arithmetical Functions and Contour Integration

1973 ◽  
Vol 16 (3) ◽  
pp. 381-387 ◽  
Author(s):  
Bruce C. Berndt
Keyword(s):  
Meromorphic Function ◽  
Closed Form ◽  
Meromorphic Functions ◽  
Arithmetical Function ◽  
Contour Integration ◽  
Partial Answer ◽  
Arithmetical Functions ◽  
Classical Technique ◽  
Limited Class ◽  
Image Position

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

A note on Ramanujan's arithmetical function τ(n)

1929 ◽  
Vol 25 (2) ◽  
pp. 121-129 ◽  
Author(s):  
J. R. Wilton
Keyword(s):  
Dirichlet Series ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Image Position

The function τ (n), defined by the equationis considered by Ramanujan* in his memoir “On certain arithmetical functions”The associated Diriċhlet seriesconverges when σ = > σ0, for a sufficiently large positive σ0.

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.

A further note on Ramanujan's arithmetical function τ(n)

1938 ◽  
Vol 34 (3) ◽  
pp. 309-315 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Arithmetical Function ◽  
Image Position

This note is a sequel to two in earlier volumes of the Proceedings, the first by myself and the second by Wilton†.Suppose thatfor |z| <1; that x > 0; thatforr ≥ 0; and thatwhere τ(x) is to mean 0 if x is not an integer. Thuswhere the dash shows that the last term is to be halved when x is an integer;forr > 1; and

A Note on the Distribution Function of Additive Arithmetical Functions in Short Intervals

1989 ◽  
Vol 32 (4) ◽  
pp. 441-445 ◽  
Author(s):  
Gutti Jogesh Babu ◽  
Paul Erdös
Keyword(s):  
Distribution Function ◽  
Arithmetical Function ◽  
Arithmetical Functions ◽  
Short Intervals ◽  
Additive Arithmetical Function

AbstractLet f be an additive arithmetical function having a distribution F. For any sequence letIn this note, we determine the slowest growing function b so that Qn{b, f) tends weakly to F, for various f.

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).

A Divisor Problem for Values of Polynomials

1992 ◽  
Vol 35 (1) ◽  
pp. 108-115
Author(s):  
Armel Mercier ◽  
Werner Georg Nowak
Keyword(s):  
Exponential Sums ◽  
Modern Method ◽  
Divisor Problem ◽  
Arithmetical Function ◽  
Asymptotic Result ◽  
Average Order ◽  
Equal Degree ◽  
Image Position ◽  
Classical Divisor ◽  
Hardy Littlewood Method

AbstractIn this article we investigate the average order of the arithmetical functionwhere p1(t), p2(t) are polynomials in Z [t], of equal degree, positive and increasing for t ≥ 1. Using the modern method for the estimation of exponential sums ("Discrete Hardy-Littlewood Method"), we establish an asymptotic result which is as sharp as the best one known for the classical divisor problem.

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

1940 ◽  
Vol 36 (2) ◽  
pp. 150-151 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Modular Form ◽  
Title I ◽  
Fundamental Region ◽  
Arithmetical Functions ◽  
Image Position

Suppose that is 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 The purpose of this note is to prove thatThe notation employed is that of my second paper under the same general title* I refer to this paper as II.

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