  Consider words of length $n$. The set of all periods of a word of length $n$
is a subset of $\{0,1,2,\ldots,n-1\}$. However, any subset of
$\{0,1,2,\ldots,n-1\}$ is not necessarily a valid set of periods. In a seminal
paper in 1981, Guibas and Odlyzko have proposed to encode the set of periods of
a word into an $n$ long binary string, called an autocorrelation, where a one
at position $i$ denotes the period $i$. They considered the question of
recognizing a valid period set, and also studied the number of valid period
sets for length $n$, denoted $\kappa_n$. They conjectured that $\ln(\kappa_n)$
asymptotically converges to a constant times $\ln^2(n)$. If improved lower
bounds for $\ln(\kappa_n)/\ln^2(n)$ were proposed in 2001, the question of a
tight upper bound has remained opened since Guibas and Odlyzko's paper. Here,
we exhibit an upper bound for this fraction, which implies its convergence and
closes this long standing conjecture. Moreover, we extend our result to find
similar bounds for the number of correlations: a generalization of
autocorrelations which encodes the overlaps between two strings.
