Arithmetic Properties of the Ramanujan Function

[摘要] We study some arithmetic properties of the Ramanujan function 𝜏(𝑛), such as the largest prime divisor 𝑃(𝜏 (𝑛)) and the number of distinct prime divisors 𝜔(𝜏(𝑛)) of 𝜏(𝑛) for various sequences of 𝑛. In particular, we show that 𝑃(𝜏 (𝑛)) ≥ $(log n)^{33/31+𝜎(1)}$ for infinitely many 𝑛, and$$P(𝜏(p)𝜏(p^2)𝜏(p^3))>(1+𝜎(1))frac{loglog plogloglog p}{loglogloglog p}$$for every prime 𝑝 with $𝜏(p)≠ 0$.

[学科分类] 数学（综合）

[关键词] Ramanujan 𝜏-function;applications of $mathcal{S}-unit equations.