The zeta function, L-functions, and irreducible polynomials
نویسنده
چکیده
= 1 1− q1−s , where we have convergence for all s with <(s) > 1. This automatically gives us analytic continuation of ζ to all of C. We note the following simple observations: 1. The Riemann hypothesis: All zeroes of ζ lie on the line <(s) = 1/2, simply because there are no zeroes! 2. ζ has poles at s = 1 + 2πin log q . 3. Locally around s = 1, ζ has the Laurent series expansion: 1 (s− 1) log q + a0 + a1(s− 1) + a2(s− 1) + . . . ,
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