Amazingly, the
prime counting function is related to
by the
Möbius transform (5)
where
is the
Möbius function (Riesel 1994, p. 49; Havil 2003, p. 196; Derbyshire 2004, p. 302). More amazingly still,
is connected with the
Riemann zeta function by " width=170 height=37
(Riesel 1994, p. 47; Edwards 2001, p. 23; Derbyshire 2004, p. 309).
is also given by " width=211 height=38> (7)
where
is the
Riemann zeta function, and (
6) and (
7) form a
Mellin transform pair.