Archiv der Mathematik

, Volume 89, Issue 5, pp 411–418 | Cite as

A note on the Ramanujan τ-function

  • M. Z. Garaev
  • V. C. Garcia
  • S. V. Konyagin


Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
$$\# \{ \tau (n):n\, \leqslant\, x \} \gg x^{1/2} e^{ - 4\log x/ {\rm log} \log x}$$
We also show that
$$ \omega \left( {\mathop \prod \limits_{\begin{array}{*{20}c} {p \leqslant x} \\ {\tau (p) \ne 0} \\ \end{array} } \tau (p)\tau (p^2 )} \right) \gg \frac{{(\log x)^{13/11} }} {{\log \log x}}, $$
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9].

Mathematics Subject Classification (2000).

11N25 11F30 


Ramanujan τ-function value set prime factors 


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Copyright information

©  2007

Authors and Affiliations

  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMoreliaMéxico
  2. 2.Department of Mechanics and MathematicsMoscow State UniversityMoscowRussia

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