On the General Divergent Arithmetic Sums over the Primes and the Symmetries of Riemann’s Zeta Function

In this paper, we address the problem of the divergent sums of general arithmetic functions over the set of primes. In classical analytic number theory, the sum of the logarithm of the prime numbers plays a crucial role. We consider the sums of powers of the logarithm of primes and its connection with Riemann’s zeta function (z.f.). This connection is achieved through the second Chebyshev function of order n, which can be estimated by exploiting the symmetry properties of Riemann’s zeta function. Finally, a heuristic approach to evaluating more general sums is also given.