The Riemann Zeta Function

Complex Analysis · Axiom Academy

LESSON The Riemann Zeta Function One infinite sum, secretly a product over the primes — extended to the whole complex plane, where its zeros hide the deepest unsolved problem in mathematics. 1. A Sum That Is Secretly a Product For , the zeta function is the convergent sum of n^ -s over every positive integer. But every integer factors uniquely into primes — so each integer can be assembled from prime building blocks. Group the sum by those blocks and it collapses into one factor per prime . …equals a product over the primes p The sum only converges when — to the left of that line the terms blow up. Yet there is exactly one analytic function that agrees with the sum on the right and extends smoothly everywhere else (with a single pole at s=1 ). That extension is what we mean by on the rest of the plane. For the series converges — e.g. . At s=1 , has a simple pole; near it . Continuation gives finite values where the sum diverges: , . It relates and , mirroring values across . 3. The Zeros and the Critical Line The Euler product has no zeros for , and the functional equation forces "trivial" zeros at the negative even integers . Every other zero must lie in the critical strip — and remarkably, the ones found so far all sit on the central line . Trivial zeros: the factor in the functional equation vanishes at — these are well understood. Nontrivial zeros: these lie inside the strip and come in symmetric sets — if is a zero, so are and . The first few sit at

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