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We investigate the problem of the distribution of sums of functions of primenumbers located on an arithmetic progression. This problem is closely relatedto the problem of the distribution of prime numbers on an arithmeticprogression. Based on this distribution, a general formula was obtained for theasymptotic estimate of the sums of functions of primes, and also asymptoticswas found for the sums of various functions of primes on a geometricprogression. Several assertions about asymptotics of sums of functions of primenumbers on a geometric progression are proved. Necessary and sufficientconditions for the existence of these asymptotics are also proved.

Asymptotics of sums of functions of primes located on an arithmetic progression