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arXiv · 2409.03428

Sums of two squares and the tau-function: Ramanujan's trail

Bruce C. Berndt·Pieter Moree
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Abstract

Ramanujan, in his famous first letter to Hardy, claimed a very precise estimate for the number of integers that can be written as a sum of two squares. Far less well-known is that he also made further claims of a similar nature for the non-divisibility of the Ramanujan tau-function for certain primes. In this survey, we provide more historical details and also discuss related later developments. These show that, as so often, Ramanujan was an explorer in a fascinating wilderness, leaving behind him a beckoning trail.

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Bruce C. Berndt, Pieter Moree. 2024-09-05. Sums of two squares and the tau-function: Ramanujan's trail. https://arxiv.org/abs/2409.03428

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