Riemann's zeta function

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1. EpiMathSep 2019
   If you've got enough math background to follow it, Harold Edwards' Riemann's Zeta Function is a gem of a book and available inexpensively from Dover. There is an English translation of Riemann's paper at the end of the book. I spent a worthwhile few weeks of spare time working my way through the paper ( and a lot of pencil & paper to work "between" the steps in the paper -- math is not a spectator sport ) with a longish diversion diving into the gamma function along the way.
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2. jordighJan 2017
   Hm, except for using Gamma(n) = (n-1)! instead of Pi(n) = n!, this presentation seems to follow the notation of this book:

   https://books.google.ca/books/about/Riemann_s_Zeta_Function....

   This is most obvious in the naming of the J function near the bottom, which I have not seen anywhere else. Riemann originally called that function a generic f.

   At any rate, Edwards' book is great because it develops the theory from a historical viewpoint. It begins with a very well-annotated exposition of Riemann's original paper and the rest of the book goes on to explain other mathematicians' successive results in filling in all of the gaps that Riemann left in his paper. I recommend this book to any serious student of zeta.

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3. jorgenveisdalJan 2017
   Books used:

   1. Men of Mathematics (E.T. Bell, 1937)

   2. The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike (Peter Borwein et.al, 2007)

   3. Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (John Derbyshire, 2004)

   4. Unknown Quantity: A Real and Imaginary History of Algebra (John Derbyshire, 2007)

   5. Journey through Genius: The Great Theorems of Mathematics (William Dunham, 1991)

   6. Riemann’s Zeta Function (H. M. Edwards, 1971)

   7. Gamma: Exploring Euler’s Constant (Julian Havil, 2009)

   8. The Man Who Loved Only Numbers (Paul Hoffman, 1999)

   9. The Riemann Zeta-Function: Theory and Applications. (Alexander Ivic, 2004)

   10. e: The Story of a Number (Eli Maor, 2009)

   11. An Imaginary Tale: The Story of √-1 (Paul J. Nahin; 1998)

   12. Dr. Euler’s Fabulous Formula (Paul J. Nahin; 2006)

   13. Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers (Dan Rockmore, 2006)

   14. Infinity and the Mind (Rudy Rucker, 2004)

   15. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Karl Sabbagh, 2004)

   16. The Prime Numbers and Their Distribution. (Tenenbaum and Mendès, 2000)

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4. Xmd5aJul 2026
   The link is between numbers and words. Both can be decomposed into primitive components: primes and lyndon words (combinatorics). The Zipf distribution takes this shape when the vocabulary is infinite:

       P(r)=r^-s / Zeta(s)  with s > 1
   

   Replace r^-s with (r + a)^-s, i.e. move onto Mandelbrot's rank-shifted model of Zipf's law and Riemann's Zeta function becomes Hurwitz Zeta function (ZetaHurwitz(s,a+1) with the ranks starting at 1).

   Zipf law != Zipf distribution, still this is intriguing.

   Integers and Prime Numbers: Deriving Zipf's Law (https://arxiv.org/abs/2403.12773)

   > In a prime number decomposition of integers in a given set, the occurrence frequencies of prime numbers are shown to satisfy a general forms of Zipf's law.

   The same holds for Lyndon words decomposition of random strings to a certain extent.

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