Abstract
The hero of the book is Riemann, and in this concluding chapter the discussion revolves around his greatest contribution to mathematics: the still unresolved Riemann hypothesis concerning the zeros of the zeta function. In the flow of the discussion a number of interesting integrals are encountered as the functional equation of the zeta function is derived. The book ends with the exact evaluation of ζ′(0), using results from earlier in the book.
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Notes
- 1.
To derive (9.1.2) is not difficult, just ‘devilishly’ clever; you can find a proof in any good book on number theory. Or see my book, An Imaginary Tale: the story of \( \sqrt{-1} \), Princeton 2016, pp. 150–152.
- 2.
A slight variation is the modern \( \mathrm{li}\left(\mathrm{x}\right)={\int}_2^{\mathrm{x}}\frac{\mathrm{du}}{\ln \left(\mathrm{u}\right)}+1.045\dots \) which, for x = 1,000 (for example), MATLAB computes integral(@(x)1./log(x),2,1000)+1.045 = 177.6.
- 3.
The English mathematician A. E. Ingham (1900–1967) opens his book The Distribution of the Primes (Cambridge University Press 1932) with the comment “A problem at the very threshold of mathematics [my emphasis] is the question of the distribution of the primes among the integers.”
- 4.
The methods used to compute the non-trivial zeros are far from ‘obvious’ and beyond the level of this book. If you are interested in looking further into how such computations are done, I can recommend the following four books: (1) H. M. Edwards, Riemann’s Zeta Function, Academic Press 1974; (2) E. C. Titchmarsh, The Theory of the Riemann Zeta-function (2nd edition, revised by D. R. Heath-Brown), Oxford Science Publications 1986; (3) Aleksandar Ivić, The Riemann Zeta-Function, John Wiley & Sons 1985; and (4) The Riemann Hypothesis (Peter Borwein et al., editors), Springer 2008.
- 5.
Because of the symmetry properties of the complex zero locations, one only has to consider the case of t > 0. The value of t for a zero is called the height of the zero, and the zeros are ordered by increasing t (the first six zeros are shown in Fig. 9.1.2, where a zero occurs each place \( \left|\upzeta \left(\frac{1}{2}+i\ \mathrm{t}\right)\right| \) touches the vertical t-axis). The heights of the first six zeros are 14.134725, 21.022040, 25.010856, 30.424878, 32.935057, and 37.586176. In addition to the first 1013 zeros, billions more zeros at heights as large as 1024 and beyond have also all been confirmed to be on the critical line.
- 6.
From Littlewood’s essay “The Riemann Hypothesis,” in The Scientist Speculates: An Anthology of Partly-Baked Ideas (I. J. Good, editor), Basic Books 1962.
- 7.
In my book Dr. Euler’s Fabulous Formula, Princeton 2017, pp. 246–253, you’ll find a derivation of the identity \( {\sum}_{\mathrm{k}=-\infty}^{\infty }{\mathrm{e}}^{-{\upalpha \mathrm{k}}^2}=\sqrt{\frac{\uppi}{\upalpha}}{\sum}_{\mathrm{n}=-\infty}^{\infty }{\mathrm{e}}^{-{\uppi}^2\ {\mathrm{n}}^2/\upalpha} \). If you write α = πx then (9.2.5) immediately results. The derivation in Dr. Euler combines Fourier theory with what mathematicians call Poisson summation , all of which might sound impressively exotic. In fact, it is all at the level of nothing more than the end of freshman calculus. If you can read this book then you can easily follow the derivation of (9.2.5) in Dr. Euler.
- 8.
We know we can do this because, as shown in the previous section, \( \upzeta \left(\frac{1}{2}\right)=-1.4\dots \ne 0 \).
- 9.
You can read more about Ramanujan’s amazing life in the biography by Robert Kanigel, The Man Who Knew Infinity: a life of the genius Ramanujan, Charles Scribner’s Sons 1991.
- 10.
I use the word obviously because, over the entire interval of integration, the integrand is finite and goes to zero very fast as x → ∞. Indeed, the integrand vanishes even faster than exponentially as x → ∞, which you can show by using (9.2.4) to write \( \uppsi \left(\mathrm{x}\right)={\sum}_{\mathrm{n}=1}^{\infty }{\mathrm{e}}^{-{\mathrm{n}}^2\uppi \mathrm{x}}={\mathrm{e}}^{-\uppi \mathrm{x}}+{\mathrm{e}}^{-4\uppi \mathrm{x}}+{\mathrm{e}}^{-9\ \uppi \mathrm{x}}+\dots <{\mathrm{e}}^{-\uppi \mathrm{x}}+{\mathrm{e}}^{-2\uppi \mathrm{x}}+{\mathrm{e}}^{-3\ \uppi \mathrm{x}}+\dots \) , a geometric series easily summed to give \( \uppsi \left(\mathrm{x}\right)<\frac{1}{{\mathrm{e}}^{\uppi \mathrm{x}}-1},\mathrm{x}>0 \), which behaves like e−πx for x ‘large.’ With s = 1 the integrand behaves (for x ‘large’) like \( \frac{\uppsi \left(\mathrm{x}\right)}{{\mathrm{x}}^{3/2}+{\mathrm{x}}^{1/2}}\approx \frac{{\mathrm{e}}^{-\uppi \mathrm{x}}}{\mathrm{x}\sqrt{\mathrm{x}}} \) for x ‘large.’
- 11.
All the details in the derivation of (9.2.20) can be found on pp. 154–155 of Dr. Euler.
- 12.
See, for example, my An Imaginary Tale: the story of \( \sqrt{-1} \), Princeton 2016, pp. 155–156.
- 13.
When Riemann submitted his doctoral dissertation (Foundations for a General Theory of Functions of a Complex Variable) to Gauss, the great man pronounced it to be “penetrating,” “creative,” “beautiful,” and to be a work that “far exceeds” the standards for such works. To actually be able to teach, however, Riemann had to give a trial lecture to an audience of senior professors (including Gauss), and Gauss asked that it be on the foundations of geometry. That Riemann did in June 1854 and—in the words of Caltech mathematician Eric Temple Bell in his famous book Men of Mathematics—it “revolutionized differential geometry and prepared the way for the geometrized physics of [today].” What Bell was referring to is Einstein’s later description of gravity as a manifestation of curved spacetime, an idea Riemann might have come to himself had he lived. After Riemann’s death the British mathematician William Kingdon Clifford (1845–1879) translated Riemann’s lecture into English and was perhaps the only man then alive who truly appreciated what Riemann had done. Clifford, himself, came within a whisker of the spirit of Einstein’s theory of gravity in a brief, enigmatic note written in 1876 (“On the Space Theory of Matter”), 3 years before Einstein’s birth. Sadly, Clifford too died young of the same disease that had killed Riemann.
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Nahin, P.J. (2020). Epilogue. In: Inside Interesting Integrals. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-43788-6_9
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