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I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function.

The following are excluded:

  • Books by mathematical cranks (especially books by amateurs who claim to prove or disprove RH in their book)

  • Books about prime numbers or analytic number theory in general that include some material about the Riemann Hypothesis or Riemann's Zeta Function

  • Books that consist of collections of mathematical tables

  • Books that are paper-length (say, under 50 pages)

  • Doctoral dissertations (published books based upon doctoral dissertations are, of course, included)

Marko Amnell
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    I wonder if it would fit protocols better to post this as an _answer_ after posting a short _question_ that it answers. – Michael Hardy Feb 06 '13 at 17:33
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    That will be a long list... consider writing it up as a BIBTeX bibliography. – vonbrand Feb 06 '13 at 17:34
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    @vonbrand There are probably a few books missing, but I doubt more than 5-10 at most. I have been collecting books about this topic for years and own copies of all the books on my list except for the two by Laurincikas as I cannot find reasonably priced copies of them. One book I could have included but chose not to is _Infirmation de l'hypothèse de Riemann_ by Henri Berliocchi, who is a respected French economist but apparently claims to disprove RH in the book. – Marko Amnell Feb 06 '13 at 18:46
  • @MarkoAmnell: I am making this [Community Wiki](http://math.stackexchange.com/privileges/community-wiki). If you have some reason that this question should not be CW, flag this question for moderator attention. – robjohn Feb 06 '13 at 19:41
  • [This bibliography list](http://math.fullerton.edu/mathews/c2003/RiemannHypothesisBib/Links/RiemannHypothesisBib_lnk_3.html) may help, albeit may contain overlaps. – Sniper Clown Feb 09 '13 at 07:45
  • @Mahmud: Thanks, but that bibliography seems to contain only one book, _The Riemann Hypothesis and Hilbert's Tenth Problem_ by S. Chowla, which is already on my list. The rest are articles. – Marko Amnell Feb 09 '13 at 17:11
  • I added two books by Michel Lapidus and Machiel van Frakenhuysen which develop the same ideas as Lapidus's later book _In Search of the Riemann Zeros_, which was already on my list. The blurb for the 2006 book says, "The Riemann hypothesis is given a natural geometric reformulation in context of vibrating fractal strings..." – Marko Amnell Mar 06 '13 at 07:12
  • I added Robert Spira's _History of Zeta Functions_ to the list. I am not sure of the exact contents but it certainly includes a great deal of material on Riemann's zeta function. The contents are described as follows: "Covers the range of years from Euler's papers in the 1700's to references appearing in Mathematical Reviews through 1991." – Marko Amnell Apr 01 '13 at 09:26
  • I added Aleksandar Ivic's _The Theory of Hardy's Z-Function_ to the list, as the Z-function is part of the theory of Riemann's Zeta Function. – Marko Amnell Nov 08 '13 at 08:38
  • I added Ivic's _Topics in recent zeta function theory_ (1983) to the list. – Marko Amnell Nov 30 '13 at 11:29
  • I added Emilio Elizalde's _Ten Physical Applications of Spectral Zeta Functions_ to the list. – Marko Amnell Jan 02 '14 at 06:00
  • I added two new books to the list, which will both be published in June, 2014. – Marko Amnell May 25 '14 at 18:02
  • I added Machiel van Frankenhuijsen's _The Riemann Hypothesis for Function Fields_ to the list. The author says the book is a "description of how non-commutative geometry could provide a means to attack the Riemann Hypothesis" so it might help in understanding Alain Connes's related ideas. – Marko Amnell Jun 25 '14 at 14:17
  • I added Henryk Iwaniec's _Lectures on the Riemann Zeta Function_ to the list, which will be published by the AMS on October 30, 2014. – Marko Amnell Sep 13 '14 at 22:46
  • I added _The Bloch-Kato Conjecture for the Riemann Zeta Function_ to the list, which according to the blurb is an "account of a significant body of recent work that resolves some long-standing mysteries concerning special values of the Riemann zeta function." – Marko Amnell Nov 27 '14 at 18:07
  • I added Katz and Sarnak's _Random Matrices, Frobenius Eigenvalues, and Monodromy_ to the list, which, as the Publisher's blurb says, focuses "on the Montgomery-Odlyzko law, the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups." – Marko Amnell Feb 28 '15 at 13:14
  • I added _New Directions in Value-distribution Theory of Zeta and L-functions_ to the list. From the Preface: "Most of the papers deal with zeta- and L-functions which are very powerful tools in Number Theory that encode interesting information about the underlying arithmetical objects in their value-distribution. The famous yet unproved Riemann hypothesis is a prototypical example... One of the main topics of contributions in these proceedings is the spectacular universality theorem of Voronin from 1975..." – Marko Amnell Mar 07 '15 at 14:21
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    I added István Sándor Gál's _Lectures on algebraic and analytic number theory; with special emphasis on the theory of the Zeta functions of number fields and function fields_ to the list. The contents are described as follows: "Lectures given at Yale University and repeated at the University of Minnesota ... 1959-60 and 1960-61, respectively." – Marko Amnell Apr 18 '15 at 11:09
  • As suggested by Michael Hardy, I have edited this question so that my own list of books appears as a separate answer. – Marko Amnell Apr 19 '15 at 11:37

2 Answers2

15

Some of these are paper-length, not book-length, but they come up when I search Math Reviews for books, and who am I to argue with Math Reviews?

  • MR2934277 Reviewed van der Veen, Roland; van de Craats, Jan De Riemann-hypothese. (Dutch) [The Riemann hypothesis] Een miljoenenprobleem. [A million dollar problem] Epsilon Uitgaven, Utrecht, 2011. vi+102 pp. ISBN: 978-90-5041-126-4

  • MR2198605 Reviewed Jandu, Daljit S. The Riemann hypothesis and prime number theorem. Comprehensive reference, guide and solution manual. Infinite Bandwidth Publishing, North Hollywood, CA, 2006. 188 pp. ISBN: 0-9771399-0-5 11M26 (11N05) [From the publisher's description: "The author adopts the real analysis and technical basis to guide and solve the problem based on high school mathematics.''] [This one may not pass the "crank" test...]

  • MR1332493 Reviewed Ramachandra, K. On the mean-value and omega-theorems for the Riemann zeta-function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 85. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1995. xiv+169 pp. ISBN: 3-540-58437-4

  • MR1230387 Reviewed Ivić, A. Lectures on mean values of the Riemann zeta function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 82. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1991. viii+363 pp. ISBN: 3-540-54748-7

  • MR0747304 Reviewed van de Lune, J. Some observations concerning the zero-curves of the real and imaginary parts of Riemann's zeta function. Afdeling Zuivere Wiskunde [Department of Pure Mathematics], 201. Mathematisch Centrum, Amsterdam, 1983. i+25 pp.

  • MR0683287 Reviewed Klemmt, Heinz-Jürgen Asymptotische Entwicklungen für kanonische Weierstraßprodukte und Riemanns Überlegungen zur Nullstellenanzahl der Zetafunktion. (German) [Asymptotic expansions for canonical Weierstrass products and Riemann's reflections on the number of zeros of the zeta function] Nachrichten der Akademie der Wissenschaften in Göttingen II: Mathematisch-Physikalische Klasse 1982 [Reports of the Göttingen Academy of Sciences II: Mathematics-Physics Section 1982], 4. Akademie der Wissenschaften in Göttingen, Göttingen, 1982. 24 pp.

  • MR0637204 Reviewed van de Lune, J.; te Riele, H. J. J.; Winter, D. T. Rigorous high speed separation of zeros of Riemann's zeta function. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics], 113. Mathematisch Centrum, Amsterdam, 1981. ii+35 pp. (loose errata).

  • MR0541033 Reviewed te Riele, H. J. J. Tables of the first 15000 zeros of the Riemann zeta function to 28 significant digits, and related quantities. Afdeling Numerieke Wiskunde [Department of Numerical Mathematics], 67. Mathematisch Centrum, Amsterdam, 1979. 155 pp. (not consecutively paged).

  • MR0565985 Reviewed van de Lune, J. On a formula of van der pol and a problem concerning the ordinates of the non-trivial zeros of Riemann's zeta function. Mathematisch Centrum, Afdeling Zuivere Wiskunde, ZW 16/73. Mathematisch Centrum, Amsterdam, 1973. iii+21 pp.

  • MR0359258 Reviewed \cyr Voĭtovich, N. N.; \cyr Nefedov, E. I.; \cyr Fialkovskiĭ, A. T. \cyr Pyatiznachnye tablitsy obobshchennoĭ dzeta-funktsii Rimana ot kompleksnogo argumenta. (Russian) [Five-place tables of the generalized Riemann zeta-function of a complex argument] With an English preface. Izdat. ``Nauka'', Moscow, 1970. 191 pp.

  • MR0266875 Reviewed Gavrilov, N. I. \cyr Problema Rimana o raspredelenii korneĭdzetafunktsii. (Russian) [The Riemann problem on the distribution of the roots of the zeta function ] Izdat. Lʹvov. Univ., Lvov, 1970 1970 172 pp.

  • MR0117905 Reviewed Haselgrove, C. B.; Miller, J. C. P. Tables of the Riemann zeta function. Royal Society Mathematical Tables, Vol. 6 Cambridge University Press, New York 1960 xxiii+80 pp.

Ivo Terek
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Gerry Myerson
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  • Thanks. The two books that stand out are the ones by Ramachandra and Ivic. The rest seem to be paper-length, collections of tables or in languages I cannot read. Ivic's book seems to be out of print. While looking for copies on Amazon, I stumbled on another book: _Ramanujan Lecture Notes Series, Vol. 2: The Riemann zeta function and related themes: Proceedings of the international conference held at the National Institute of Advanced Studies, Bangalore, December 2003_. If one includes conference proceedings, there are probably more like that one. – Marko Amnell Feb 07 '13 at 05:37
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    @MarkoAmnell: In the meantime, Ivić's book has become available on Kindle: https://www.amazon.com/Riemann-Zeta-Function-Theory-Applications-Mathematics/dp/0486428133. – joriki Apr 16 '20 at 16:04
14

These are all the books I am aware of that meet the criteria I set:

This list is available as a BibTeX bibliography file which can be downloaded from: http://drive.google.com/file/d/1HENOMh-Va368-vpKsI58mHVaGlm13ZaF/view?usp=sharing

  • Bernoulli Numbers and Zeta Functions, by Tsuneo Arakawa, Tomoyoshi Ibukiyama, Masanobu Kaneko, and Don B. Zagier, Springer (June 30, 2014), 274 pp.

  • The Riemann Hypothesis and the Distribution of Prime Numbers, by Naji Arwashan, Nova Science Pub Inc (April 15, 2021), 219 pp.

  • The Riemann Hypothesis - A Twenty-three centuries-long journey in search of the secret of prime numbers, Vol. 1, by Jose Luis Perez Baeza, Parerga Foundation (Calle Major de Sarrià 232 PB, Barcelona 08017 ES), January 1, 2020, ISBN 978-8409257478, 493 pp.

  • Ramanujan Lecture Notes Series, Vol. 2: The Riemann zeta function and related themes (Proceedings of the international conference held at the National Institute of Advanced Studies, Bangalore, December 2003), R. Balasubramanian, K. Srinivas (Eds.), 206 pp.

  • The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, Peter Borwein, Stephen Choi, Brendan Rooney, Andrea Weirathmueller (Eds.), Springer, 2008

  • Equivalents of the Riemann Hypothesis, by Kevin Broughan, 2 volumes [Vol. 1: Arithmetic Equivalents, 400 pages; Vol. 2: Analytic Equivalents, 350 pages], Cambridge University Press (January 31, 2018)

  • Lectures on the Riemann zeta-function, by K. Chandrasekharan, Tata Institute of Fundamental Research, 1953, 148 pp.

  • The Riemann Hypothesis and Hilbert's Tenth Problem, by S. Chowla, Gordon and Breach, Science Publishers, Ltd., 1965

  • The Bloch-Kato Conjecture for the Riemann Zeta Function, John Coates, A. Raghuram, Anupam Saikia, R. Sujatha (Eds.), London Mathematical Society Lecture Note Series (Book 418), Cambridge University Press (April 30, 2015), 320 pp.

  • Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, by John Derbyshire, Joseph Henry Press, 2003

  • Reassessing Riemann's Paper: On the Number of Primes Less Than a Given Magnitude, by Walter Dittrich, Springer (August 1, 2018), 65 pp.

  • The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, by Marcus du Sautoy, HarperCollins, 2003

  • Riemann's Zeta Function, by Harold M. Edwards, Academic Press, 1974

  • Elizalde, Emilio, Ten Physical Applications of Spectral Zeta Functions, Lecture Notes in Physics 855, Springer, Berlin, 2012 (2nd edition), 290 pages

  • Elizalde, Emilio, Sergei D. Odintsov, August Romeo, A.A. Bytsenko, and S. Zerbini, Zeta Regularization Techniques with Applications, World Scientific Publishing Company (1994), 336 pp.

  • Gál, István Sándor, Lectures on algebraic and analytic number theory; with special emphasis on the theory of the Zeta functions of number fields and function fields, Jones Letter Service, Minneapolis, 1961, 453 pp.

  • Gavrilov, N. I. Problema Rimana o raspredelenii korneidzetafunktsii. (Russian) [The Riemann problem on the distribution of the roots of the zeta function] Izdat. L'vov. Univ., Lvov, 1970 172 pp.

  • Simply Riemann (Great Lives), by Jeremy Gray, Simply Charly (March 20, 2020), 167 pp.

  • The Mysteries of the Real Prime, by M.J. Shai Haran, London Mathematical Society (December 6, 2001), 256 pp.

  • The Riemann hypothesis in algebraic function fields over a finite constants field, by Helmut Hasse, Dept. of Mathematics, Pennsylvania State University, 1968, 235 pp. [Verbatim reproduction of lectures given at Pennsylvania State University, Spring term, 1968]

  • Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality, by Hafedh Herichi, World Scientific Pub Co Inc (July 31, 2019), 400 pp.

  • Ivic, A. Lectures on mean values of the Riemann zeta function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 82. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1991. viii+363 pp. ISBN: 3-540-54748-7

  • The Riemann Zeta-Function: Theory and Applications, by Aleksandar Ivic, John Wiley & Sons, Inc., 1985

  • Ivic, A. The Theory of Hardy's Z-function. Cambridge Tracts in Mathematics 196. Cambridge: Cambridge University Press. ISBN 978-1-107-02883-8, 264 pages, 2012

  • Ivic, A. Topics in recent zeta function theory. Publ. Math. d'Orsay, Université de Paris-Sud, Dép. de Mathématique, 1983, 272 pages

  • Lectures on the Riemann Zeta Function, by H. Iwaniec, American Mathematical Society (October 30, 2014), 119 pp.

  • Contributions to the Theory of Zeta-Functions: The Modular Relation Supremacy, by Shigeru Kanemitsu and Haruo Tsukada, World Scientific Publishing Company (June 30, 2014), 280 pp.

  • The Riemann Zeta-Function, by Anatoly A. Karatsuba and S. M. Voronin, Walter de Gruyter & Co., 1992

  • Random Matrices, Frobenius Eigenvalues, and Monodromy, by Nicholas M. Katz and Peter Sarnak, American Mathematical Society (November 24, 1998), 419 pp.

  • Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions, by Michel L. Lapidus and Machiel van Frankenhuysen, Birkhäuser, 1999

  • Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, by Michel L. Lapidus and Machiel van Frankenhuysen, Springer, 2006

  • In Search of the Riemann Zeros: Strings, Fractal Membranes, and Noncommutative Spacetimes, by Michel L. Lapidus, American Mathematical Society, 2008

  • Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions, by Michel L. Lapidus, Goran Radunović and Darko Žubrinić, Springer (February 1, 2017), 704 pp.

  • Limit Theorems for the Riemann Zeta-Function, by Antanas Laurincikas, Kluwer Academic Publishers, 1996

  • The Lerch zeta-function, by Antanas Laurincikas and Ramunas Garunkstis, Kluwer Academic Publishers, 2002

  • Recent Progress on Topics of Ramanujan Sums and Cotangent Sums Associated with the Riemann Hypothesis, by Helmut Maier, Laszlo Toth and Michael Th. Rassias, World Scientific Publishing Co Pte Ltd (March 10, 2022), 180 pp.

  • Prime Numbers and the Riemann Hypothesis, by Barry Mazur and William Stein, Cambridge University Press (October 31, 2015), 150 pp.

  • Exploring the Riemann Zeta Function: 190 years from Riemann's Birth, Hugh Montgomery, Ashkan Nikeghbali, Michael Th. Rassias (Eds.), Springer (September 9, 2017), 272 pp.

  • Spectral Theory of the Riemann Zeta-Function, by Yoichi Motohashi, Cambridge University Press, 1997

  • A Study of Bernhard Riemann's 1859 Paper, by Terrence P. Murphy, Paramount Ridge Press (September 18, 2020), 182 pp.

  • In Pursuit of Zeta-3: The World's Most Mysterious Unsolved Math Problem, by Paul J. Nahin, Princeton University Press (October 19, 2021), 344 pp.

  • An Introduction to the Theory of the Riemann Zeta-Function, by S. J. Patterson, Cambridge University Press, 1988

  • Ramachandra, K. On the mean-value and omega-theorems for the Riemann zeta-function. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 85. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1995. xiv+169 pp. ISBN: 3-540-58437-4

  • The Theory of the Hurwitz Zeta Function of the Second Variable, by Vivek V. Rane, Alpha Science International Ltd (December 31, 2015), 300 pp.

  • Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers, by Dan Rockmore, Random House, Inc., 2005

  • The Riemann Hypothesis in Characteristic p in Historical Perspective, by Peter Roquette, Springer (September 30, 2018), 300 pp.

  • The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, by Karl Sabbagh, Farrar, Straus, and Giroux, 2002

  • History of Zeta Functions, by Robert Spira, 3 volumes, Quartz Press (392 Taylor Street, Ashland OR 97520-3058), 1218 pages, 1999, ISBN 0-911455-10-8

  • Seminar on the Riemann Zeta Function 1965-1966, by Robert Spira, Mimeographed typescript, University of Tennessee, Knoxville, 57 pages

  • Zeta and q-Zeta Functions and Associated Series and Integrals, by H. M. Srivastava and Junesang Choi, Elsevier Inc., 2012

  • New Directions in Value-distribution Theory of Zeta and L-functions: Wurzburg Conference, October 6-10, 2008 (Berichte aus der Mathematik), Rasa Steuding, Jörn Steuding (Eds.), Shaker Verlag GmbH, Germany (December 31, 2009), 346 pp.

  • Bohr-Jessen Limit Theorem, Revisited, by Satoshi Takanobu, Mathematical Society of Japan Memoirs (Book 31), Mathematical Society of Japan (July, 2013), 216 pp.

  • Zeta and eta functions: A new hypothesis, by Ashwani Kumar Thukral, CreateSpace Independent Publishing Platform (December 17, 2015), 56 pp.

  • The Theory of the Riemann Zeta-Function, by E. C. Titchmarsh, D. R. Heath-Brown (Ed.), Second edition, Oxford University Press, 1986

  • Pseudodifferential Methods in Number Theory, by André Unterberger, Birkhäuser (July 24, 2018), 180 pages

  • Van der Veen, Roland; van de Craats, Jan De Riemann-hypothese. (Dutch) [The Riemann hypothesis] Een miljoenenprobleem. [A million dollar problem] Epsilon Uitgaven, Utrecht, 2011. vi+102 pp. ISBN: 978-90-5041-126-4

  • The Riemann Hypothesis, by Roland van der Veen and Jan van de Craats, The Mathematical Association of America (January 6, 2016), 154 pp.

  • Van Frankenhuijsen, Machiel, The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators, London Mathematical Society Student Texts (Book 80), Cambridge University Press (January 9, 2014), 162 pp.

  • Zeta Functions over Zeros of Zeta Functions, by André Voros, Springer-Verlag, 2010

  • Zeta Functions of Reductive Groups and Their Zeros, by Lin Weng, World Scientific Publishing Co Pte Ltd (May 19, 2018), 550 pp.

Marko Amnell
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  • you should classify them, putting the Titchmarsh in good place (but say that there is nothing in it about Dirichlet characters, number fields, nor automorphic forms) and maybe adding a book on number fields, and another on automorphic forms ? – reuns May 27 '16 at 00:54
  • @user1952009: That would be a different (and perhaps worthwhile) project. If I started to add books about various topics in number theory (or related fields) which aren't devoted entirely (or mainly) to the Riemann Hypothesis or the Riemann zeta function, the list would quickly grow to be much longer, and it would be very hard to decide what books to include, and what to exclude. Similarly, classifying the books in some way, or adding descriptions of their contents, is yet another different project. My question (and the resulting list) is defined more narrowly. Even now, decisions about... – Marko Amnell May 29 '16 at 03:37
  • (continued): whether to include a particular book can be open to debate. For example, I excluded Marcus du Sautoy's _Zeta Functions of Groups and Rings_ because it doesn't seem to me to be directly relevant to RH but I freely admit I could turn out to be wrong about that. But I included _Random Matrices, Frobenius Eigenvalues, and Monodromy_ by Nicholas Katz and Peter Sarnak because Sarnak himself says he thinks the ideas in that book will be crucial to finding a proof of RH. See e.g. http://math.stackexchange.com/questions/327693/a-cohomological-statement-equivalent-to-the-riemann-hypothesis – Marko Amnell May 29 '16 at 03:47
  • in my opinion it is not "various topics in number theory" but only some of the main aspects of the problem [www.claymath.org/sites/default/files/official_problem_description.pdf](http://www.claymath.org/sites/default/files/official_problem_description.pdf). and do you have pdf copies of all those books ? – reuns May 29 '16 at 14:06
  • Fair enough, but how would you decide which books about number fields, or automorphic forms, to include? All of them? I actually own printed copies of all the books on my list except for five, and I will hopefully acquire one more in a week or two if Vivek Rane's The Theory of the Hurwitz Zeta Function of the Second Variable is finally published on May 31 after several delays. – Marko Amnell May 29 '16 at 17:14
  • do you think making a copy for us of all those .pdf books ? if you are afraid I can host them on my ftp : send me the .zip to acx01c@gmail.com . and see this great list of books : http://nozdr.ru/biblio/kolxo3/m/mt only and http://nozdr.ru/biblio/kolxo3/m/mc a few on $\zeta(s)$ but more than 700 number theory books ! – reuns May 29 '16 at 17:39
  • @Neves: Thank you for your interest in my question. I reverted your addition of Jean-Marie De Koninck's book _The Life of Primes in 37 Episodes_ to my answer because the book does not specifically deal with the Riemann Hypothesis or the Riemann zeta function. Of its 37 "episodes" or chapters, only three deal with RH or the Riemann zeta function. Please note that in my question I said that "Books about analytic number theory in general that include some material about the Riemann Hypothesis or Riemann's Zeta Function" are excluded. I meant to also exclude books about prime numbers in general. – Marko Amnell May 01 '22 at 01:29