The distribution of Fourier coefficients of half-integral weight forms
be a newform of weight for the full modular group, and let
be a cusp form of weight and level 4 which is associated to by the Shimura map. We normalize by requiring that and we normalize by requiring that
Then for squarefree with
we have, by the formula of Kohnen and Zagier,
In particular, the Riemann Hypothesis for implies that
for an appropriate choice of .
If the bound
holds, then a similar bound for will hold (but with replaced by ).
The question here is to decide which (if either) of these bounds represents
the true state of affairs.
If has a square factor, then can be determined from values of where .
It is known, by Deligne's theorem, that
where is the number of divisors of . Of course, so
that we can write
where is real. The conjecture of Sato and Tate about the distribution of the asserts that
A consequence of the Sato-Tate conjecture is that there are infinitely many (actually a positive proportion of ) for which
for any given . The maximum order of is given by
The Sato-Tate conjecture implies the same assertion with replaced by
Thus, in particular, we find that
and we expect this bound to be sharp in the sense that the cannot be replaced by anything smaller.
Thus, if the ``smaller'' bound is true for
then there is a great variation between the distribution of the Fourier coefficients according to whether is square
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