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A Comprehensive Course in Number Theory by Alan Baker

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14

On the zeros of the zeta-function

14.1 Introduction

We recall from Section 2.8 that the Riemann zeta-function is defined for any complex number s = σ + it with σ > 1 by the equation

image

The series converges in the region of definition and indeed uniformly for σ > 1 + δ with δ > 0. The function can be continued analytically to the region σ > −1 by the equation

image

where f (x) is the ‘saw-tooth’ function given by

f (x) = [x] − x + image

and xs+1 = exp((s + 1) ...

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