Stewart (2013) proved that the biggest prime divisor of the $n$th term of a Lucas sequence of integers grows quicker than $n$, answering famous questions of ErdH{o}s and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewarts result.
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}. Th
en 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.
We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form $F$ of even integral weight on the Siegel modular group of arbitrary genus $ ggeq 2 $.
