the abscissa of convergence
/ ab-SISS-uh /
Where exactly does a Dirichlet series sum a_n / n^s converge? The answer has a beautifully simple shape: there is a single real number sigma_0, called the abscissa of convergence, that acts as a threshold on the real part of s. For every s with Re s > sigma_0 the series converges, and for every s with Re s < sigma_0 it diverges. So the region of convergence is always a right half-plane, bounded on the left by the vertical line Re s = sigma_0. The word 'abscissa' just means the horizontal coordinate of that boundary line.
This is the exact analogue of the radius of convergence for power series — but the geometry changed from a disk to a half-plane. For a power series sum a_n z^n the boundary is a circle |z| = R; for a Dirichlet series the boundary is a vertical line Re s = sigma_0. Just as the radius could be 0 (converges nowhere) or infinity (converges everywhere), the abscissa can be +infinity (the series never converges), -infinity (it converges on the whole plane), or any finite value in between. On the boundary line itself, anything can happen — convergence, divergence, or a mix — and no general rule settles it. There is a companion threshold, the abscissa of ABSOLUTE convergence sigma_a, with sigma_0 <= sigma_a <= sigma_0 + 1; the gap of width up to 1 is a genuinely Dirichlet-series phenomenon with no power-series counterpart.
Why does it matter? The abscissa tells you the natural domain on which the series itself makes sense before any analytic continuation. For zeta the abscissa is 1 (the series converges for Re s > 1), and the whole drama of zeta — the pole at 1, the zeros in the critical strip, the functional equation — happens by continuing PAST that abscissa into a region the series cannot reach. A common confusion: the abscissa of convergence is about the SERIES, not about the function. zeta(s) is defined and interesting far to the left of Re s = 1, even though its defining series diverges there; the abscissa only marks where the literal sum stops working.
For zeta, sum 1/n^s, the abscissa of convergence is sigma_0 = 1: it converges for Re s > 1 and diverges at s = 1 (the harmonic series). For the alternating eta series sum (-1)^(n-1)/n^s the abscissa is 0, since cancellation buys an extra unit of convergence. For a series with bounded partial sums of coefficients, like sum chi(n)/n^s for a non-principal character, the abscissa can be as far left as 0.
The boundary of convergence is a vertical line Re s = sigma_0, the half-plane analogue of a power series' circle of radius R.
Convergence of the series past the abscissa is not the same as the function being undefined past it. Analytic continuation can extend the function far beyond Re s = sigma_0 even though the literal series diverges there — zeta is the headline example.