The Cauchy Integral Formula

J. J. P. Veerman ()
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J. J. P. Veerman: Portland State University

Chapter Chapter 12 in Numbers from all Angles, 2026, pp 245-267 from Springer

Abstract: Abstract Again, we need to venture very far, apparently, from number theory to make progress. In the mid-nineteenth century, the main insight in number theory came from Riemann, who realized that the distribution of the primes was intimately connected to the properties of the (analytic continuation of the) Riemann zeta function to the complex plane. In this chapter, we develop the necessary complex analysis tools—essentially the Cauchy integral formula—to study the convergence of a certain improper integral (Theorem 12.18), which is the key to the proof of the prime number theorem in the next chapter (Theorem 13.16 ). For more detailed introductions to complex analysis, we refer to [6, 52, 84].

Date: 2026
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DOI: 10.1007/978-3-032-10000-9_12

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