Reimann Hypothesis Forum

Euler Product Convergence on Re(s) = 1: Common Misconceptions

Students often assume the Euler product converges on Re(s) = 1, s ≠ 1 simply because ζ(s) has an analytic continuation there. In fact, convergence of the product on the 1-line is equivalent in depth to the Prime Number Theorem.

Right—taking logs reduces it to the convergence of ∑ p^{-1-it}, which via partial summation relies on π(x) = li(x) + o(x / log x) or a quantitative zero-free region.