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01 Euler and Ζ(S) 1. Euler and the Basel Problem

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01 Euler and Ζ(S) 1. Euler and the Basel Problem (September 14, 2015) 01 Euler and ζ(s) Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2015-16/01 Euler and zeta.pdf] 1. Euler and zeta 2. Euler product expansion of sin x 3. Quantitative infinitude of primes 4. Proof of Euler product 5. Appendix: product expansion of sin x Infinite sums that telescope can be understood in simpler terms. The archetype is 1 1 1 1 1 1 1 1 1 + + + ::: = − + − + − + ::: = 1 1 · 2 2 · 3 3 · 4 1 2 2 3 3 4 Other subtler examples analyze-able via trigonometric functions give more interesting answers, but these are still elementary, the archetype being [1] 1 1 1 x x3 x5 Z x dt π − − + ::: = − + − ::: = = arctan 1 = 2 1 3 5 1 3 5 x=1 0 1 + t x=1 4 As late as 1735, no one knew how to express in simpler terms 1 1 1 1 1 + + + + + ::: 12 22 32 42 52 This was the Basel problem, after the town in Switzerland home to the Bernoullis, collectively a dominant force in European mathematics at the time. L. Euler solved the problem, winning him useful notoriety at an early age, but more notably introducing larger ideas, as follows. 1. Euler and the Basel Problem Given non-zero numbers a1; : : : ; an, a polynomial with constant term 1 having these numbers as roots is x x x 1 1 1 1 1 xn 1 − 1 − ::: 1 − = 1 − ( ::: + )x + ( + + + :::)x2 + ::: + (−1)n a1 a2 an a1 an a1a2 a1a3 a2a3 a1 : : : an sin πx Imagine that πx has an analogous product expansion, in terms of its zeros at ±1; ±2; ±3;:::, up to a normalizing constant needing determination: using the power series expansion of sin x, (πx)3 1 (πx) − + ::: sin πx Y x2 X 1 3! = = C · 1 − = C · x − x3 + ::: πx πx n2 n2 n=1 n Assuming this works, equating constant terms gives C = 1, and equating coefficients of x2 gives π2 X 1 = 6 n2 P 1 Slightly messier manipulations yield all values n2k . Euler did not manage to prove that the sine function had such a product expansion for some years. Nevertheless, just this heuristic, without the eventual proof, was what won him considerable notoriety, in part because it suggested an underlying mechanism. Further, everyone believed the heuristic because, the numerical plausibility was easy to check, once observed. [1] There is some delicacy about convergence, but this is secondary. 1 Paul Garrett: 01 Euler and ζ(s) (September 14, 2015) 2. Euler product expansion of ζ(s) In effect, the Basel problem was about explaining the values ζ(2); ζ(4);:::, of the simplest zeta function 1 X 1 ζ(s) = (real s?, complex s?) ns n=1 Euler made another striking observation about ζ(s): it has an Euler product factorization/expansion, coming from unique factorization in Z: X 1 1 1 1 1 1 1 1 1 1 1 1 1 = + + + + + + + + + + + + :::::: ns 1s 2s 3s 4s 5s 6s 7s 8s 9s 10s 11s 12s n 1 1 1 1 1 1 1 1 1 1 1 1 = + + + + + + + + + + + + ::: 1s 2s 3s 22s 5s 2s · 3s 7s 23s 32s 2s · 5s 11s 22s · 3s 1 1 1 1 1 1 1 1 1 Y 1 = 1 + + + + ::: 1 + + + + ::: 1 + + + + ::: ::: = 2s 22s 23s 3s 32s 33s 5s 52s 53s 1 primes p 1 − ps from factoring uniquely into prime powers and massively regrouping in terms of geometric series for each prime: 1 1 1 1 1 + + + + ::: = ps p2s p3s 1 1 − ps In short, 1 X 1 Y 1 ζ(s) = = ns 1 n=1 primes p 1 − ps That is, we have an expression involving just primes equated to an expression not overtly involving primes, suggesting the relevance of the zeta function to prime numbers. Whether or not we care greatly about prime numbers, the connection between primes and behavior of ζ(s) is striking. 3. Quantitative infinitude of primes The ancient Greek mathematicians proved the infinitude of primes qualitatively, but experimentation quickly suggests that the estimates on asymptotic density arising from that classical argument are laughably bad. The first serious qualitative result about primes was Euler's, using the Euler product expansion of ζ(s), proving X 1 = 1 p all primes p R 1 −s + Comparison with 1 x dx for real s > 1 proves that ζ(s) ! +1 as s ! 1 : 1 Z 1 = x−s dx ≤ ζ(s) (for real s > 1) s − 1 1 On the other hand, from − log(1 − x) = x + x2=2 + x3=3 + :::, X 1 X 1 1 1 log ζ(s) = − log 1 − = + + + ::: ps ps 2p2s 3p3s p p 2 Paul Garrett: 01 Euler and ζ(s) (September 14, 2015) The terms after the initial 1=ps do not contribute to any blow-up as s ! 1+, by crudely estimating primes by positive integers: X X 1 X X 1 X 1 Z 1 dx ≤ ≤ `p`s `n`s ` x`s p `≥2 n≥2 `≥2 `≥2 1 X 1 1 X 1 1 = · ≤ · < 1 (uniformly for s ≥ 1) ` `s − 1 ` ` − 1 `≥2 `≥2 Thus, for some finite constant C, 1 X 1 log < log ζ(s) ≤ C + (for s > 1) s − 1 ps p + [2] P Thus, the blow-up of log ζ(s) as s ! 1 forces the divergence of p 1=p. P 1 This argument by itself did not attempt quantify how fast p<x p blows up as a function of x ! 1, although a similar idea can make progress there. A related question of at least whimsical interest is about the prime-counting function [3] X π(x) = 1 p0 <x That is, the weights in the counting are not 1=ps, not 1=p, but just 1. About 160 years after Euler's first work on zeta, in 1896 Hadamard and de la Vall´ee-Poussin (independently) proved x π(x) ∼ (as x ! +1) log x meaning that π(x)=x= log x ! 1 as x ! +1. The question of refining the error term in this asymptotic is very much open currently. Riemann's 1859 paper showed that the connection between prime numbers and the zeta function is far deeper than the above discussion suggests. 4. Proof of Euler product Although the heuristic may be clear, we could be concerned about 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 " − 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. [4] Thus, although an " − δ discussion is alien to Euler's viewpoint, it is more familiar to contemporary readers, and we conduct the discussion in such terms. [2] P −s P + For real s > 1, certainly p p ≤ p 1=p, so the blow-up of the left-hand side as s ! 1 implies divergence of the right. [3] Slightly unfortunate notation π(x), but it is traditional. [4] 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, 3 Paul Garrett: 01 Euler and ζ(s) (September 14, 2015) [4.1] 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 pm < T in a unique manner, Y X 1 X 1 X 1 − < pms ns jnsj prime p<T m : pm<T n<T n≥T since the finitely-many leftovers from the product produce integers n ≥ T at most once each. The latter sum goes to 0 as T ! 1, for fixed Re(s) > 1, by comparison with an integral. P s The finite sums n<T 1=n are the usual partial sums of the infinite sum, and (simple) convergence is the P s assertion that this sequence converges to n 1=n . Thus, this already proves that Y X 1 X 1 lim = T !1 pms ns prime p<T m : pm<T n≥1 [4.2] Limits of varying products In contrast, the auxiliary question about infinite products is more complicated. We have products whose factors themselves vary: we would like to prove that because 1 1 1 1 1 + + + ::: + −! (for Re(s) > 1) ps p2s pms 1 1 − ps the limit of changing products converges: Y X 1 Y 1 −! (for Re(s) > 1) pms 1 p<T m:pm<T p 1 − ps where the infinite product on the right is the limit of its finite partial products. That the individual factors on the left approach the individual factors on the right is not necessary (for fixed s), and in any case is not sufficient. Taking logarithms is convenient, since error estimates on sums is easier than error estimates on products. That is, we claim that Y X 1 Y 1 log −! log (for Re(s) > 1) pms 1 p<T m:pm<T p 1 − ps The infinite product on the right is easily verified to converge to a non-zero limit, so continuity of log away from 0 allows us to move the logarithm inside the infinite product.
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