Is there a Generalized Wiener Ikehara theorem for exp[1/2](n) instead of n ?

While considering Dirichlet series I wonder what would happen if we take a(0) + a(1)/exp[1/2](1)^s + a(2)/exp[1/2](2)^s + ... + a(n)/exp[1/2](n)^s instead of a(0) + a(1)/1^s + a(2)/2^s + ... a(n)/n^s.

Since exp^[1/2](n) grows faster than any polynomial P(n) we cannot apply the normal Wiener Ikehara theorem in most cases.

Im hoping to 'bridge' analytic number theory and tetration in this way ...

regards

tommy1729

While considering Dirichlet series I wonder what would happen if we take a(0) + a(1)/exp[1/2](1)^s + a(2)/exp[1/2](2)^s + ... + a(n)/exp[1/2](n)^s instead of a(0) + a(1)/1^s + a(2)/2^s + ... a(n)/n^s.

Since exp^[1/2](n) grows faster than any polynomial P(n) we cannot apply the normal Wiener Ikehara theorem in most cases.

Im hoping to 'bridge' analytic number theory and tetration in this way ...

regards

tommy1729