The zeta Function, The Hurwitz zeta Function

The Hurwitz zeta Function

Let c be a real constant in the range (0,1]. Define the hurwitz (biography) zeta function ζ(s,c) as the sum, as n runs from 0 to infinity, of 1/(n+c)^s. Note that c = 0 would leave the first term (n = 0) undefined. When c = 1, the riemann zeta function appears.

For s > 1, apply the integral test, and the hurwitz zeta function converges absolutely.

The series is locally uniformly convergent, hence it is analytic to the right of 1.

Hurwitz and the l Series

Consider l_χ,m, and regroup terms mod m. (We can only do this to the right of 1, where the series is absolutely convergent.)

Let j be a unit mod m, hence j is relatively prime to m. Add up the terms whose indexes are j mod m.

∑{k=0,∞} χ(km+j) / (km+j)^s

χ(j) × ∑{k=0,∞} 1 / (km+j)^s

χ(j)/m^s × ∑{k=0,∞} 1 / (k+j/m)^s

χ(j)/m^s × ζ(s,j/m)

Do this for each j in Z_m^*. Thus the l series is a linear combination of hurwitz zeta functions.