taken over all primes also diverges, as first shown by Euler in 1737 (Nagell 1951,
p. 59; Hardy and Wright 1979, pp. 17 and 22; Wells 1986, p. 41; Havil
2003, pp. 28-31), although it does so very slowly. The sum exceeds 1, 2, 3,
... after 3, 59, 361139, ... (OEIS A046024)
primes.
Its asymptotic behavior is given by
(2)
where
(3)
(OEIS A077761) is the Mertens constant (Hardy and Wright 1979, p. 351; Hardy 1999, p. 50; Havil 2003,
p. 64).