I have seen two versions of a result called "Hecke Equidistribution" and I wanted to know if they were the same or different.




---


[**#1**](https://mathoverflow.net/questions/133410/hecke-equidistribution) Let $p = 4k+1 = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$. Then $\theta$ equidistributed as $p \to \infty$.


[**#2**](http://www.math.u-psud.fr/~ullmo/Publications/coursMontrealfinal.pdf) 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](http://people.reed.edu/~jerry/361/lectures/heckechar.pdf) and L-functions the second deals with [Hecke operators](https://mathoverflow.net/questions/19684/relation-between-hecke-operator-and-hecke-algebra) 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.



Equidistribution of Hecke points and $p = (a+bi)(a-bi) = e^{i\theta}\sqrt{a^2 + b^2}$
These are two very different results. Briefly, the first result depends on the nonvanishing of (certain) Hecke $L$-functions at the edge of the critical strip, while the second result depends on a nontrivial bound for Hecke eigenvalues of Maass forms on the modular surface. As you can see, both are related to objects named after Hecke, but the objects themselves are quite different.


**P.S.** I see now that you mentioned Hecke characters and Hecke operators in your post. At any rate, I don't know of a way of deducing one result from the other (apart from proving both results in their own way).


