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Abstract
This paper traces the path from Euler’s series 1 + 2−s + 3−s + ··· of 1737 to Riemann’s zeta function of 1859, and it does so at the finest grain: every letter is named the first time it appears, every displayed equation is followed by the single operation that turns it into the next, every technical word is defined before it is used, and every person named is placed in time relative to Riemann’s paper. Euler had the series and its product over the primes for real exponents greater than 1, the values of the series at the even integers, and in 1749 a conjectured relation between the series at s and the series at 1 − s. That relation could not be a theorem, because for s > 1 the series at 1 − s is a sum of growing numbers that settles to no value. Riemann gave the symbol ζ(s) a definite meaning at every complex number s other than 1 and proved the relation. His procedure consisted of three operations on one integral. First, he rewrote each term n−s as an integral of Euler’s Gamma type, which moved the variable s out of the series and into the single factor xs−1 of an integrand. Second, he replaced the path of integration from 0 to ∞ by a loop that comes in from +∞, circles the origin once, and returns to +∞; this turned the integral into a function with a derivative at every complex s, because the loop never meets the one point, x = 0, at which convergence had depended on s. Third, he pushed that loop outward onto the points x = ±2πin where 1/(ex − 1) becomes infinite, and the residues at those points produced the functional equation ζ(1 − s) = 2(2π)−s cos(πs/2) Γ(s) ζ(s). He then gave a second derivation from Jacobi’s transformation of the theta series, which yields the symmetric form π−s/2 Γ(s/2) ζ(s) = π−(1−s)/2 Γ((1 − s)/2) ζ(1 − s) and the function ξ(t) whose zeros are the subject of his hypothesis. Every construction that lives on a line or in a plane is drawn, with each symbol labelled. A section on Riemann’s education and library records which books, lectures and journals supplied him with each tool, a section explains at what single point the primes enter the calculus and by what device they come out, and a closing section settles which functions can select the primes (an elementary one, a polynomial whose positive values are the primes, and one built from ζ), which cannot (a polynomial all of whose values are prime), and states the open problem of the least cost of such a function.
Contents
- 1. Introduction: what is answered, and how the paper is written p. 1
- 2. What Riemann had in his hands: his education, his books and his journals p. 3
- 3. The people, in order of time p. 4
- 4. The terms used, each defined in checkable words p. 7
- 5. Euler's series and the product over the primes (1737), every step written out p. 12
- The series, its terms, and what it means for it to have a value p. 12
- When s>1: the partial sums are bounded, and the bound shows the pole p. 12
- When s<=1: the partial sums exceed every bound, so the series has no value p. 14
- The product over the primes, proved with finite sums p. 16
- The logarithm of the product, and where the sum of the reciprocals of the primes diverges p. 18
- Euler's values at the even whole numbers p. 20
- 6. Where the primes enter the calculus, and where they come out p. 20
- 7. Euler's 1749 relation between the series at s and at 1-s, and why it was not yet a theorem p. 22
- 8. What Riemann brought: the complex exponent, and three facts about functions of a complex variable p. 26
- 9. The Gamma function, and the two identities of Pi that Riemann calls known p. 28
- Definition and convergence p. 28
- The recursion, and the values at whole numbers p. 29
- The integral is analytic in s p. 29
- Continuation of Gamma to the whole plane; its poles; its residues p. 29
- The Beta integral and its relation to Gamma p. 29
- Euler's reflection formula, proved by a loop integral p. 30
- Legendre's duplication formula, proved by the Beta integral p. 31
- 10. Step one: the Gamma integral replaces the series p. 32
- 11. Step two: the loop integral around the origin gives the continuation to the whole plane p. 34
- 12. Step three: the pole at s=1, the trivial zeros, and the values at the negative whole numbers, read off the loop integral p. 37
- The loop integral at a whole number p. 37
- The whole numbers m>=2: no pole, and the value of the series is recovered p. 38
- The point s=1: a simple pole with residue exactly 1 p. 38
- The negative whole numbers: the values Euler attached to divergent series p. 38
- The trivial zeros p. 39
- Consistency with Euler's values at the positive even numbers p. 39
- 13. Step four: moving the contour onto the poles x = +-2 pi i n gives the functional equation p. 39
- 14. Step five: the theta-series route (Jacobi) and the second derivation of the continuation p. 43
- Two preliminary facts p. 43
- The half-argument Gamma integral, term by term p. 43
- Summing over n p. 44
- Jacobi's transformation, proved by Poisson's summation formula p. 44
- Splitting the integral at x=1 and transforming the piece near 0 p. 46
- The right side of (107) is defined for every s p. 47
- What the second form shows at a glance p. 47
- 15. Step six: the function xi(t), its two integral representations, and the zeros p. 47
- 16. What Riemann did next with the continued function, and what was proved later p. 50
- The product over the zeros: what “determined by its zeros” means p. 50
- The logarithm of zeta(s) for complex s, defined so that it is single-valued p. 51
- The prime powers, and the step function J(x) p. 52
- One term as an integral, and the whole series as one integral p. 53
- The inversion: Fourier's theorem in the form Riemann needs, proved p. 54
- From the inversion integral to the explicit formula: what Riemann did, and what is proved p. 56
- Recovering pi(x) from J(x): Möbius inversion, proved p. 57
- What the formula says p. 57
- 17. Functions that select the primes: what exists, what cannot, and the open problem of their cost p. 58
- 18. The sequence of operations in one table p. 61
