# Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

## TL;DR

This paper proves that for Ramanujan's tau-function and Fourier coefficients of newforms, the absolute values can be ordered arbitrarily infinitely often by shifting indices, revealing a form of randomness in their distribution.

## Contribution

It establishes the existence of infinitely many shifts where the absolute values of these coefficients follow any prescribed permutation order.

## Key findings

- Existence of infinitely many m with ordered | au(m+s(i))| for Ramanujan's tau-function.
- Similar ordering results for Fourier coefficients of general newforms.
- Supports the conjecture that au(n) is nonzero for all n.

## Abstract

Let \tau(.) be the Ramanujan \tau-function, and let k be a positive integer such that \tau(n) is not 0 for n=1,...,[k/2]. (This is known to be true for k < 10^{23}, and, conjecturally, for all k.) Further, let s be a permutation of the set {1,...,k}. Then there exist infinitely many positive integers m such that |\tau(m+s(1))|<\tau(m+s(2))|<...<|\tau(m+s(k))|. We also obtain a similar result for Fourier-coefficients of general newforms.