Let $\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$-function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)\neq 0$ for $1\leqslant n\leqslant k/2$. (This is known to be true for $k<10^{23}$, and, conjecturally, for all $k$.) Further, let $\unicode[STIX]{x1D70E}$ be a permutation of the set $\{1,\ldots ,k\}$. We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

The original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate amodular analogue of the Sato–Tate Conjecture and prove that the angles arising from non-$\text{CM}$ holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate–Tatemeasure for non-$\text{CM}$ elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform.

We formulate and prove a Sato–Tate equidistribution law for Drinfeld modules.