Analytic Continuation

The Riemann Hypothesis is one of 7 Millennium Problems ($1M). It is only stated via analytic continuation: the zeta series converges only for Re(s)>1, but zeros lie in the critical strip.

• **Number theory:** zeta function via continuation is linked to prime distribution. RSA encryption rests on the hardness of factoring.
• **CAS (Mathematica, Maple):** Risch algorithm integrates sqrt(P(x)) via Riemann surfaces of algebraic curves.
• **Quantum physics:** S-matrix amplitudes in field theory defined via analytic continuation. Poles are particle resonances.

Analytic Continuation

The Riemann Hypothesis about the zeros of zeta(s) -- one of the 7 Millennium Prize Problems ($1M reward) -- can only be stated via analytic continuation: the series sum n^{-s} converges only for Re(s)>1, but zeta(s) extends to the entire complex plane.

Riemann Surfaces

Wolfram Mathematica, when computing integrals of the form integral sqrt(P(x)) dx, automatically works with the Riemann surface of the algebraic curve y^2=P(x). This is the Risch algorithm (1969), the foundation of all CAS systems.

Key ideas

• **Identity principle:** continuation is unique. If f1=f2 on an open set, they agree everywhere in a connected domain.
• **Branches and cuts:** Log z, sqrt(z) are multivalued. Cut C\(-inf,0] selects the principal branch. Riemann surface eliminates multivaluedness.
• **Zeta function:** zeta(s)=sum n^{-s} for Re(s)>1 extends to C\{1}. Zeros in the critical strip -- the Riemann problem.