Plot of Ramanujan tau function

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

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

• 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). Oct 28 '15 at 1:42
• @GHfromMO You are right, I see my mistake now. The result only holds infinitely often, much like Hardy's proof that $\tau (n) > n^{11/2}$. Oct 28 '15 at 2:04