notes 1 - Euler product of(s Paul Garrett...

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( August 24, 2013 ) Euler product of ζ ( s ) Paul Garrett [email protected] http: / /www.math.umn.edu/ e garrett/ [This document is http://www.math.umn.edu/˜garrett/m/mfms/notes 2013-14/01a Euler product.pdf] Euler’s discovery that X n =1 1 n s = Y p prime 1 1 - p - s (for Re( s ) > 1) using the geometric series expansion 1 1 - p - s = 1 + p - s + ( p 2 ) - s + ( p 3 ) - s + . . . and unique factorization in Z , may or may not seem intuitive, although we should certainly revise our intuitions to make this a fundamental fact of nature. This note’s concern is convergence of the infinite product of these geometric series to the sum expression for ζ ( s ), in Re( s ) > 1. All the worse if the Euler product seems intuitive, it is surprisingly non-trivial to carry out a detailed verification of the Euler factorization. Ideas about convergence were different in 1750 than now. It is not accurate to glibly claim that the Cauchy- Weierstraß ε - δ viewpoint gives the only possible correct notion of convergence , since A. Robinson’s 1966 non-standard analysis offers a rigorous modernization of Leibniz’, Euler’s, and others’ use of infinitesimals and unlimited natural numbers to reach similar goals. [1] Thus, although an ε - δ discussion is alien to Euler’s viewpoint, it is more familiar to contemporary readers, and we conduct the discussion in such terms. [0.1] The main issue One central point is the discrepancy between finite products of finite geometric series involving primes, and finite sums of natural numbers. For example, for T > 1, because every positive integer n < T is a product of prime powers p m < T in a unique manner, Y prime p<T X m : p m <T 1 p ms - X n<T 1 n s < X n T 1 | n s | since the finitely-many leftovers from the product produce integers n T at most once each. The latter sum goes to 0 as T → ∞ , for fixed Re( s ) > 1, by comparison with an integral. The finite sums n<T 1 /n s are the usual partial sums of the infinite sum, and (simple) convergence is the assertion that this sequence converges to n 1 /n s . Thus, this already proves that lim T →∞ Y prime p<T X m : p m <T 1 p ms = X n 1 1 n s [1] A. Robinson’s Non-standard Analysis , North-Holland, 1966 was epoch-making, and really did justify some of the profoundly intuitive ideas in L. Euler’s Introductio in Analysin Infinitorum , Opera Omnia, Tomi Primi, Lausanne, 1748. E. Nelson’s reformulation Internal set theory, a new approach to NSA , Bull. AMS 83 (1977), 1165-1198, significantly improved the usability of these ideas. A. Robert’s Non-standard Analysis
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