  We investigate the problem of the distribution of sums of functions of prime
numbers located on an arithmetic progression. This problem is closely related
to the problem of the distribution of prime numbers on an arithmetic
progression. Based on this distribution, a general formula was obtained for the
asymptotic estimate of the sums of functions of primes, and also asymptotics
was found for the sums of various functions of primes on a geometric
progression. Several assertions about asymptotics of sums of functions of prime
numbers on a geometric progression are proved. Necessary and sufficient
conditions for the existence of these asymptotics are also proved.
