# 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} }

We give a short survey of old and new results in the theory of Ramanujan expansions for arithmetical functions.

## 31 Citations

### Ramanujan expansions of arithmetic functions of several variables

- Mathematics
- 2017

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…

### Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions

- Mathematics
- 2016

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…

### On Ramanujan expansions and primes in arithmetic progressions

- Mathematics
- 2022

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.…

### A note on arithmetical functions with absolutely convergent Ramanujan expansions

- Mathematics
- 2016

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…

### On partial sums of arithmetical functions of two variables with absolutely convergent Ramanujan expansions

- MathematicsProceedings - Mathematical Sciences
- 2018

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…

### On the error term in a Parseval type formula in the theory of Ramanujan expansions II

- Mathematics
- 2015

### Twin Primes In Quadratic Arithmetic Progressions

- Mathematics
- 2017

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…

### Spectral Methods And Prime Numbers Counting Problems

- Mathematics
- 2015

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 smooth shift approach for a Ramanujan expansion

- Mathematics
- 2019

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…

### Partial sums of arithmetical functions with absolutely convergent Ramanujan expansions

- MathematicsProceedings - Mathematical Sciences
- 2016

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}…

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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…

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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…

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