We derive new classes of infinite products taken over the primes, for example expressing ∏ p ( 1 1 − p − n ) ( 1 − p − m ) − 1 as an infinite produce of Riemann zeta functions, this product being taken over the set of rational numbers α / β geater than zero with a relatively prime to β ζ ( n ) ∏ α , β ζ ( α m + β n ) 1 / β .