The zeta Function, The Line of Absolute Convergence

The Line of Absolute Convergence

For a given function f and complex number s, assume ζ(f,s) is absolutely convergent.  Apply the norm equivalence principle, and any other series of complex numbers having the same norms is absolutely convergent.  In particular, add any multiple of i to s, and watch what this does to the nth term.  The denominator use to be ns.  When we add i to s we multiply the denominator by ni, which is a point on the unit circle in the complex plane.  Thus the norm of the denominator, in fact the norm of the nth term, is unchanged.  The norms still converge absolutely, hence we still have an absolutely convergent series.  If ζ(f,s) is absolutely convergent then we have absolute convergence along the vertical line passing through s.

If ζ(f,s) is absolutely convergent, discard the imaginary component, so that s is a real number.  Then consider any larger value of s.  The norms of the terms of ζ(f,s) decrease as s increases.  In other words, the terms get smaller and the series is still absolutely convergent.  The entire half plane to the right of the vertical line passing through s is absolutely convergent.

Let sa be the greatest lower bound of the real numbers s where ζ(f,s) converges absolutely.  Everything to the right of sa converges absolutely, and nothing to the left of sa converges absolutely.  The vertical line passing through sa is the line of absolute convergence.

Some Dirichlet zeta functions don't have a line of absolute convergence, because they converge everywhere or nowhere.  But again, this is atypical.  Most Dirichlet functions have sa somewhere between 0 and 2.

What is sa for the Riemann zeta function?  We know that s = 1 produces the harmonic series, which does not converge, and s > 1 produces an absolutely convergent series by the integral test.  Thus sa = 1.