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Arithmetic properties of the Ramanujan function

Authors:

Institution:

(1) Instituto de Matemáticas, Universidad Nacional Autónoma de México, C.P. 58089 Morelia, Michoacán, México;(2) Department of Computing, Macquarie University, 2109 Sydney, NSW, Australia

Abstract:

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)^33/31+o(1) for infinitely many n, and

$P(\tau )(p)\tau (p^2 )\tau (p^3 )) > (1 + o(1))\frac{{\log \log p\log \log \log p}}{{\log \log \log \log p}}$

for every primep with τ(ρ) ≠ 0. Dedicated to T N Shorey on his sixtieth birthday

Keywords:

Ramanujan τ -function applications ofS-unit equations

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