For an integer \(m>2\), call a prime \(p\) to be “\(m\)-admissible” if some power of it is congruent to \(-1\) modulo \(m\). Such primes \(p\) are precisely the ones for which the Jacobian variety of the Fermat curve \(X^m+Y^m=Z^m\) considered in characteristic \(p\) is supersingular. Let \(\delta(m)\) denote the asymptotic density of the set of \(m\) admissible primes. The purpose of this paper is to give a sharp asymptotic formula for the summatory function \(\sum_{2<m\leq x}\delta(m)\). The asymptotic formula says that for every positive integer \(K\) one has
\[ \sum_{2<m\leq x} \delta(m)=x\sum_{k=0}^K A_k (\log x)^{2^{-k}/3-1}+O\left(x(\log x)^{2^{-K}/4-1}\right). \]
An analogous result is obtained for the \(r\)th moment of \(\delta(m)\) for any fixed positive integer \(r\). An important tool in the proofs is a very accurate Tauberian theorem proved by R. W. K. Odoni in [J. Lond. Math. Soc., II. Ser. 44, No. 2, 203–217 (1991; Zbl 0757.11014)].