# Books about the Riemann Hypothesis

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 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)

• 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 6 '13 at 17:33
• That will be a long list... consider writing it up as a BIBTeX bibliography. – vonbrand Feb 6 '13 at 17:34
• @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 6 '13 at 18:46
• @MarkoAmnell: I am making this Community Wiki. If you have some reason that this question should not be CW, flag this question for moderator attention. – robjohn Feb 6 '13 at 19:41
• This bibliography list may help, albeit may contain overlaps. – Sniper Clown Feb 9 '13 at 7:45

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.

• 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 7 '13 at 5:37

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

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

• 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

• 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

• 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.

• 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 0:54
• @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 3:37
• (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. math.stackexchange.com/questions/327693/… – Marko Amnell May 29 '16 at 3:47
• 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. 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