(Hardy and Wright 1979, p. 268), plotted as the red curve above. The first values of
are 1, 2, 4, 6, 10, 12, 18, 22, 28, ... (OEIS A002088).
has the asymptotic series
(5)
(6)
where
is the Riemann zeta function (Perrot 1881;
Nagell 1951, p. 131; Hardy and Wright 1979, p. 268; blue curve above).
An improved asymptotic estimate due to Walfisz (1963) is given by
plotted as the red curve above. For , 2, ..., the first few terms are 1, 2, 5/2, 3, 13/4, 15/4,
47/12, 25/6, ... (OEIS A028415 and A048049).
The sum diverges as , but Landau (1900) showed that the asymptotic behavior
is given by
(OEIS A082695), is the Möbius function,
is the Riemann zeta function, and is the th prime (Landau 1900; Halberstam and Richert 1974, pp. 110-111;
DeKoninck and Ivić 1980, pp. 1-3; Finch 2003, p. 116; Havil 2003,
p. 115; Dickson 2005).
and
can also be written as
(16)
(17)
and
(18)
(19)
respectively, making these constants similar in form to Artin's
constant (Finch 2003, pp. 116-117).
The sum
(20)
(21)
(22)
(OEIS A118262) is sometimes known as the totient
constant (Niklasch), where
(23)
(OEIS A065483) and the products are taken over the primes .