IJPAM: Volume 118, No. 4 (2018)

Title

A REMARK ON HARDY-RAMANUJAN'S APPROXIMATION
OF DIVISOR FUNCTIONS

Authors

G. Sudhaamsh Mohan Reddy$^1$, S. Srinivas Rau$^2$, B. Uma$^3$
$^{1,2}$Department of Mathematics
Faculty of Science and Technology
The Icfai Foundation for Higher Education
Hyderabad, INDIA
$^3$CTW, Military College
Secundrerabad, 500015, INDIA

Abstract

We obtain asymptotic estimates for the partial sums $\sum\limits_{1<n\leq x}a_{n}$ for $a_{n}$ = (i)   $\frac{(logn)^{log2}}{d(n)}$     (ii)     $\frac{2^{\omega(n)}}{(logn)^{log2}}$     (iii)   $\frac{1}{d(n)}$ (due to Ramanujan-Wilson). These are based on Delange's Tauberian theorem. We deduce that the normal order of Hardy-Ramanujan's approximation to $d(n)$ is more often lower than $d(n)$ as asserted by them.

History

Received: February 1, 2017
Revised: May 8, 2018
Published: May 23, 2018

AMS Classification, Key Words

AMS Subject Classification: 11A25, 11B99, 11N56
Key Words and Phrases: divisor functions, normal order, Abel summation, Delange's Tauberian theorem

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Bibliography

1
Tom Apostol, Introduction to Analytic Number Theory, Springer 1976

1
G H Hardy, Ramanujan's Twelve lectures; Chelsea 1959

2
G.Tenenbaum, Introduction to Analytic and Probabilitic Number Theory, Cmbridge University Press 1995.

2
Paul Bateman, Paul Erdos, Carl Pomerance and E G Strauss, The Arithmetic Mean of Divisors of an integer(preprint).

How to Cite?

DOI: 10.12732/ijpam.v118i4.12 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 118
Issue: 4
Pages: 997 - 1000


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