TITLE: Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$
QUESTION [5 upvotes]: I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.

#1 Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$.  Then $\theta$ equidistributed as $p \to \infty$.
#2 Let $\displaystyle  T_n.z = \sum_{ad=n, 0 \leq b < d} \frac{az+b}{d}$ and extend to functions $\displaystyle [T_n ]f(y) = \sum_{y \in T_n.z} f(y)$.  Then we have equidistribution of Hecke points over the modular surface:
$$ \lim_{n \to \infty} [T_n f](z) = \int_{X(1)} f(z) \, dz$$

Both of these results have Hecke's name attached.  The first deals with Hecke characters and L-functions the second deals with Hecke operators and modular forms.  
Is result #1 considered an instance #2 of the equidistribution of Hecke points?
Possibly I consider the theta function:
$ \displaystyle \theta (z) = \sum_{(m, n) \in \mathbb{Z}^2} e^{2\pi i (m^2 + n^2) z} $
However I don't see how to act the Hecke operator in such a way as to prove the result.

REPLY [3 votes]: One common feature of these two equidistribution examples is that they have proofs (perhaps among others) thinking in terms of Weyl's equidistribution criterion: show that the corresponding measures (sums of point-mass measures) go to the constant function $1$ as distributions. On a nice (e.g., homogeneous) space, the question of this limit can be recast in terms of the appropriate harmonic analysis, which in interesting cases turns out to ask for estimates on $L$-functions or automorphic forms, as @GHfromMO noted.
In the first example, the relevant automorphic forms are harmonic binary theta series with a fixed binary form and varying grossencharacters. As @GH noted, the relevant estimate boils down to non-vanishing on the edge of the critical strip. In the second example, the relevant automorphic forms are Hecke-eigenfunction waveforms (of fixed level), along with corresponding Eisenstein series (for the continuous spectrum) and the relevant estimate reduces to a non-trivial estimate on the Fourier coefficients, since these mirror Hecke operators and eigenvalues.
(In my modular forms course 2013-14, I did some easier equidistribution examples: http://www.math.umn.edu/~garrett/m/mfms/, the least cliched of which might be "section 10" about equidistribution problems on spheres, treating higher-dimension, even-dimension cases already too easy to be of interest to Kloosterman and his contemporaries, but perhaps educational.)