Ramanujan's $\tau$ conjecture states that $$\tau(n)\sim O(n^{\frac{11}2+\epsilon})$$ which is a consequence of Deligne's proof of Weil conjectures. Answers in Status of $\tau(n)$ before Deligne tell that best exponent before Deligne reached $\frac{29}5$.

What was it that cohomological approach gave that broke the traditional approach barrier to reach $5.5$ in exponent?

  • $\begingroup$ This is way beyond me. But I once tried to get an overview on Deligne's proof, and I found a survey/expository article by Nick Katz. In his Leitfaden the Ramanujan $\tau$ is one of the nodes. IANA an algebraic geometer, but I would guess that $\tau$ makes an appearance in 11th cohomology of some variety - this is probably not accurate. Anyway, Katz' survey may help you more than it ever helped me. $\endgroup$ – Jyrki Lahtonen Mar 25 '15 at 20:51
  • $\begingroup$ please post your survey links. $\endgroup$ – T.... Mar 25 '15 at 20:57
  • $\begingroup$ Katz' survey appeared in print some time in 70s-80s. Deadwood produce was the default media at the time. I don't have a link. I searched for it with methods fitting the early 90s. I trust you have command of more efficient search methods. Don't count on it being available on-line. $\endgroup$ – Jyrki Lahtonen Mar 25 '15 at 20:59
  • $\begingroup$ While there are people here who can probably help you, there are possibly more such people on MathOverflow, since it devoted strictly towards research-level questions. $\endgroup$ – Thomas Andrews Mar 25 '15 at 21:14
  • 1
    $\begingroup$ Indeed, for the same question on MO with answers see here. $\endgroup$ – Dietrich Burde Mar 27 '15 at 9:03

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