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AFFINE CONVOLUTIONS, RAMANUJAN–FOURIER EXPANSIONS AND SOPHIE GERMAIN PRIMES

Published online by Cambridge University Press:  25 October 2022

ELIEZER FUENTES*
Affiliation:
Mathematics Department, Pontificia Universidad Católica de Chile, Santiago, Chile

Abstract

For a fixed integer h, the standard orthogonality relations for Ramanujan sums $c_r(n)$ give an asymptotic formula for the shifted convolution $\sum _{n\le N} c_q(n)c_r(n+h)$ . We prove a generalised formula for affine convolutions $\sum _{n\le N} c_q(n)c_r(kn+h)$ . This allows us to study affine convolutions $\sum _{n\le N} f(n)g(kn+h)$ of arithmetical functions $f,g$ admitting a suitable Ramanujan–Fourier expansion. As an application, we give a heuristic justification of the Hardy–Littlewood conjectural asymptotic formula for counting Sophie Germain primes.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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References

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AFFINE CONVOLUTIONS, RAMANUJAN–FOURIER EXPANSIONS AND SOPHIE GERMAIN PRIMES
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