The classical Lindelof hypothesis is equivalent to a certain estimate for the sums \(\displaystyle \sum_{n\leq x}n^{it}\). We propose that a more general
form of the Lindelof hypothesis is true, one involving similar estimates for sums of the type \(\sum_{ \substack{n\leq x \\ n\in \mathcal N }}n^{it},\) where \(\mathcal N\) can be a rather general sequence of integers. We support this with various examples and show that when \(\mathcal N\) is the sequence of prime numbers, the truth of our conjecture is equivalent to the Riemann hypothesis. Moreover, if our conjecture holds when \(\mathcal N\) is the sequence of primes congruent to \(a (\bmod q)\), with \(a\) coprime to \(q\), then the Riemann hypothesis holds for all Dirichlet \(L\)-functions with characters modulo \(q\), and conversely. These results suggest that a general form of the Lindelof hypothesis may be true that is in some sense more fundamental than either the classical Lindelof hypothesis or the Riemann hypothesis. This is joint work with Sid Graham and Yoonbok Lee.