There is a picture on wikipedia of Ramanujan tau function. At first I noticed that there are exceptional red point (where the red points are sparse in the lower part), this should be due to Sato-Tate conjecture. But it seems strange since where red points are dense blue points are sparse! Is there any explanation about these?

(Wiki: The blue line picks only the values of n that are multiples of 121.) enter image description here

By the way, is there any conjecture about the growth (a lower bound) of this function? (Surely stronger than the Lehmer's conjecture.)

  • 2
    $\begingroup$ The probabilistic distribution and growth of $\tau$ is now understood because the Sato-Tate conjecture is proved. $\endgroup$
    – eric
    Oct 21 '15 at 20:37
  • $\begingroup$ You can find some explanation on www2.math.ou.edu/~rschmidt/satotate. $\endgroup$ Oct 24 '15 at 1:25
  • $\begingroup$ Is S-T proved for modular forms? SO what's the S-T group of $\Delta$? How to explain these sparse points? $\endgroup$
    – user42690
    Oct 25 '15 at 5:01

The best lower bound known seems to be due to Ram Murty:

$$\tau(n)=\Omega (n^{11/2}e^{c \log n / \log \log n})$$

for some $c>0$ absolute and effective (note: thanks to the Sato-Tate conjecture might be able to take $c<\log 2$).

This result is proved in:

  • Ram Murty, Some Omega results for Ramanujan's tau function (1982)

In their book on Ramanujan, Ram and Kummar mention:

This result is essentially best possible since we know that

$$d(n)<e^{c' \log n / \log \log n}$$

It might be worth mention that this holds for arbitrary cusp forms, for some $c>0$,

$$a(n)=\Omega (n^{(k-1)/2}e^{c \log n / \log \log n})$$

The exceptional values you observe seem to be related to congruences modulo $11$ and $121$, but I have no idea about that. Ramanujan himself certainly had much to say on the matter:

Hopefully someone can answer that part of your question better.

  • 5
    $\begingroup$ The first bound is definitely not proven, since we do not even know that $\tau(n)=0$ only has finitely many solutions. Perhaps you meant it as an upper bound. Then, the Ramanujan conjecture (not Sato-Tate) implies that any $c>\log 2$ is admissible, but this is not due to Murty but Deligne and Ramanujan himself (Deligne proved the Ramanujan conjecture, and Ramanujan determined the order of the divisor function). $\endgroup$
    – GH from MO
    Oct 28 '15 at 1:42
  • 5
    $\begingroup$ Similarly, the fourth bound would imply the first bound, so that one is unproven as well. There is some big confusion here... $\endgroup$
    – GH from MO
    Oct 28 '15 at 1:44
  • $\begingroup$ @GHfromMO You are right, I see my mistake now. The result only holds infinitely often, much like Hardy's proof that $\tau (n) > n^{11/2}$. $\endgroup$
    – Myshkin
    Oct 28 '15 at 2:04
  • $\begingroup$ (It's fixed now) $\endgroup$
    – Myshkin
    Oct 28 '15 at 3:05
  • 1
    $\begingroup$ I had posted the same thing and then deleted it out of serious confusion. If you look at the Murty paper the theorem is stated as |\tau(n)| > \Omega(...). But the meaning of \Omega is never stated! What a perfect example of needing to define all notation! (I could be remembering wrong, of course --- I'm unfortunately only on my phone at the moment.) $\endgroup$
    – alpoge
    Oct 28 '15 at 3:34

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