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Extra info for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods

Example text

We follow the exposition of Davenport from [42]. We present the proof in more detail than is typical in this book, since it exemplifies the general form of arguments in this field. Such arguments can be relatively complicated and technical; the reader is forewarned. 9. For N (T ) defined above, we have N (T ) = T T T log − + O(log T ). 2π 2π 2π Proof. 6). Since ξ(s) has the same zeros as ζ(s) in the critical strip, and ξ(s) is an entire function, we can apply the argument principle to ξ(s) instead of ζ(s).

The argument principle in complex analysis gives a very useful tool to count the zeros or the poles of a meromorphic function inside a specified region. For a proof of this well-known result in complex analysis see §79 of [32]. The Argument Principle. Let f be meromorphic in a domain interior to a positively oriented simple closed contour C such that f is analytic and nonzero on C. Then, 1 ∆C arg(f (s)) = Z − P, 2π where Z is the number of zeros, P is the number of poles of f (z) inside C, counting multiplicities, and ∆C arg(f (s)) counts the changes in the argument of f (s) along the contour C.

2π 2π 2π This completes the proof. We now have the theoretical underpinnings for verifying the Riemann hypothesis computationally. We can adapt these results to give us a feasible method of counting the number of zeros of ζ(s) in the critical strip up to any desired height. Combined with efficient methods of computing values of ζ( 12 + it) we can gain heuristic evidence in favor of the Riemann hypothesis and, in part, justify over a century of belief in its truth. For a detailed development of these ideas, see Chapter 3.