IJPAM: Volume 119, No. 2 (2018)

Title

SOME ARITHMETIC FUNCTIONS AND THEIR MEANS

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
(Deemed to be University under section 3 of UGC Act, 1956), INDIA
$^3$CTW, Military College
Secundrerabad, 500015, INDIA

Abstract

We give four formulas for $d(n^{2})$ in terms of arithmetical functions $\omega$, $\mu$, $d_{3}$ and $d$:

\begin{displaymath}d(n^{2})=\sum\limits_{k\vert n}d(k) \mu^{2}(\frac{n}{k})=\sum... ...mits_{m\vert n}d(m))\mu(k)=\sum\limits_{k\vert n}2^{\omega(k)}.\end{displaymath}

We estimate the partial sum

\begin{displaymath}\sum\limits_{n\leq x}d(n^{2})=(\frac{1}{\zeta(2)}+\circ(1))xlog^{2}x\end{displaymath}

using Tauberian theorem. Similarly, we estimate the partial sums for $\frac{1}{d(n)}$, $\frac{logn}{d(n)}$, and $\frac{\sigma(n)}{d(n)}.$

History

Received: April 24, 2017
Revised: July 6, 2018
Published: July 6, 2018

AMS Classification, Key Words

AMS Subject Classification: 11M06, 11M41, 11M45
Key Words and Phrases: arithmetical functions, divisor functions, Tauberian theorem

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Bibliography

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

1
Richard Bellman, Analytic Number Theory an Introduction, Benjamin-Cummins, 1980.

1
G.H. Hardy, Ramanujan Chelsea, 1959.

2
M. Ram Murty, Topics in Number Theory, Mehta Research Institute, 1993.

3
G. Sudhaamsh Mohan Reddy, S. Srinivas Rau and B. Uma, Some Dirichlet series and means of their coefficients, Southeast Asian Bull. Math., 40 (2016), 1-7.

3
G. Sudhaamsh Mohan Reddy, S. Srinivas Rau and B. Uma, Means of certain arithmetic functions, Submitted.

4
G. Sudhaamsh Mohan Reddy and S. Srinivas Rau, Different approches to calculate means of arithmetic functions, In: Proceeding of National Seminar on Emerging Research Trends in Mathematics and Computation, 13-14 June, 2014; ISBN: 978-93-82163-01-5, PARAMOUNT Publishing House, New Delhi, 82-85.

3
G. Sudhaamsh Mohan Reddy, S. Srinivas Rau and B. Uma, A remark on Hardy-Ramanujan's approximation of divisor functions, In: RMS Annual Conference, Tiruchi, India, 18-21 June 2016.

How to Cite?

DOI: 10.12732/ijpam.v119i2.9 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: 119
Issue: 2
Pages: 369 - 374


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