Extending the Zeta Function

Complex Analysis · Axiom Academy

LESSON Extending the Zeta Function The series only converges for — yet one analytic function reaches the whole plane. Here is how, and where its zeros hide. 1. The Series Has a Wall — the Function Does Not The defining sum converges absolutely only to the right of . Compare its terms to a p -series: has the same size as , which converges exactly when . The alternating cousin of — the Dirichlet eta function — converges on a larger region, all the way to (conditionally on the strip ). Splitting off the even terms ties it straight back to . Solve for — valid wherever converges Because is finite for every and the factor 1-2^ 1-s vanishes only at isolated points, this single formula extends to the whole half-plane — with one exception, the point s=1 . At s=1 the original series is the divergent harmonic series, and the continued has a simple pole with residue 1 . It is the only singularity anywhere in the plane. 3. The Functional Equation Reflects Across Riemann's functional equation links the value at s to the value at its mirror image 1-s across the line . Once is known on the right half-plane, this equation defines it on the left — completing the continuation to all of . The right-hand side carries the factor . That sine is zero at every even integer, and on the negative axis it forces to vanish: , from the sine factor. "Trivial" because we can see exactly why they occur. Between them, takes clean rational values: , , . 4. The Critical Strip and the Riemann Hypothesis

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