• Corpus ID: 8544189

# Ramanujan series for arithmetical functions

@article{Murty2013RamanujanSF,
title={Ramanujan series for arithmetical functions},
author={M. Ram Murty},
journal={Hardy–Ramanujan Journal},
year={2013},
volume={36}
}
• M. Murty
• Published 2013
• Mathematics
• Hardy–Ramanujan Journal
We give a short survey of old and new results in the theory of Ramanujan expansions for arithmetical functions.
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of
For an arithmetical function f with absolutely convergent Ramanujan expansion, we derive an asymptotic formula for the sum ∑n≤Nf(n)${\sum }_{n \le N} f(n)$ with explicit error term. As a corollary we
A celebrated theorem of Delange gives a suﬃcient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coeﬃcients provided by a previous result of Wintner.
Arithmetical functions with absolutely convergent Ramanujan expansions have been recently studied in certain contexts, by the present author, Murty and many others. In this article, we aim to weaken
In this article, we derive an asymptotic formula for the sums of the form $${\sum }_{n_{1},n_{2}\le N}f(n_1,n_2)$$∑n1,n2≤Nf(n1,n2) with an explicit error term, for any arithmetical function f of two
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the
A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the
All arithmetical functions $F$ satisfying Ramanujan Conjecture, i.e., $F(n)\ll_{\varepsilon}n^{\varepsilon}$, and with $Q-$smooth divisors, i.e., with Eratosthenes transform $F':=F\ast \mu$ supported
For an arithmetical function f with absolutely convergent Ramanujan expansion, we derive an asymptotic formula for the sum ∑n≤Nf(n)\documentclass[12pt]{minimal} \usepackage{amsmath}

## References

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Using a functional analytic idea, I will show that every bounded arithmetic function possesses a Ramanujan expansion which is pointwise absolutely convergent.
This volume focuses on the classical theory of number-theoretic functions emphasizing algebraic and multiplicative techniques. It contains many structure theorems basic to the study of arithmetic
This paper summarizes the development of Ramanujan expansions of arithmetic functions since Ramanujan's paper in 1918, following Carmichael's mean-value-based concept from 1932 up to 1994. A new
• Mathematics
• 2012
The theory of supercharacters, recently developed by Diaconis-Isaacs and André, is used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line
The Wintner theorem is generalized and the multiplicative case is considered by expressing the mean-value as an infinite product over all prime numbers by the Riemann zeta function.
• Mathematics
• 2013
These sums are obviously of very great interest, and a few of their properties have been discussed already∗. But, so far as I know, they have never been considered from the point of view which I
Problems.- Arithmetic Functions.- Primes in Arithmetic Progressions.- The Prime Number Theorem.- The Method of Contour Integration.- Functional Equations.- Hadamard Products.- Explicit Formulas.- The
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• Mathematics
Nature
• 1946
THIS book must be welcomed most warmly into X the select class of Oxford books on pure mathematics which have reached a second edition. It obviously appeals to a large class of mathematical readers.
• Mathematics
• 1988
AbstractStill another proof is given for Parseval's well-known equation \sum\limits_{1 \leqslant r< \infty } {|a_r |^2 \cdot \varphi (r) = \parallel f\parallel \begin{array}{*{20}c} 2 \\ 2 \\