Modified L-functions

Abstract

The sequence of generalized prime numbers q₀ = 1, q_n = p^k_n+1 -1, n ∈ N, and the corresponding zeta-function Z_k(s) = \prod_p>2( 1 - (p^k - 1)^-s)^-1 , s = σ + it, are analyzed. The analyticity of Z_k(s) in the domain σ > 0, except for a simple pole s = 1/k , is proved.