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A Primer on the Riemann Hypothesis

• First Online: 15 September 2023

Cite this chapter

• Michael E. N. Tschaffon 21 ,
• Iva Tkáčová 22 ,
• Helmut Maier 23 &
• Wolfgang P. Schleich 24 , 25  

Part of the book series: Lecture Notes in Physics ((LNP,volume 1000))

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We provide an introduction for physicists into the Riemann Hypothesis. For this purpose, we first introduce, and then compare and contrast the Riemann function and the Dirichlet L-functions, with the Titchmarsh counterexample. Whereas the first two classes of functions are expected to satisfy the Riemann Hypothesis, the Titchmarsh counterexample is known to violate it. Throughout our article we employ elementary mathematical techniques known to every physicist. Needless to say, we do not verify the Riemann Hypothesis but suggest heuristic arguments in favor of it. We also build a bridge to quantum mechanics by interpreting the Dirichlet series central to this field as a superposition of probability amplitudes leading us to an unusual potential with a logarithmic energy spectrum opening the possibility of factoring numbers.

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Acknowledgements

We thank P. C. Abbott, M. B. Kim, H. L. Montgomery, J. W. Neuberger, M. Zimmermann for many fruitful discussions, E. P. Glasbrenner for technical assistance and J. Pohl for her help with the scans of Riemann’s article. Moreover, we are most grateful to the Niedersächsische Staats- und Universitätsbibliothek and, in particular, R. B. Röper for allowing us to present excerpts of Riemann’s original manuscript. W. P. S. is also grateful to Texas A & M University for a Faculty Fellowship at the Hagler Institute for Advanced Study at Texas A& M University and to Texas A& M AgriLife for the support of this work. The research of the IQST is financially supported by the Ministry of Science, Research and Arts Baden-Württemberg. I. B. is grateful for the financial support from the Technical University of Ostrava, Grant No. SP2018/44.

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Institut für Quantenphysik and Center for Integrated Quantum Science and Technology (IQST), Universität Ulm, Ulm, Germany

Michael E. N. Tschaffon

Department of Physics, Faculty of Electrical Engineering and Computer Science, VSB-Technical University of Ostrava, Ostrava, Poruba, Czech Republic

Iva Tkáčová

Institut für Reine Mathematik, Universität Ulm, Ulm, Germany

Helmut Maier

Wolfgang P. Schleich

Hagler Institute for Advanced Study, Institute for Quantum Science and Engineering (IQSE), and Texas A&M AgriLife Research, Texas A&M University, College Station, TX, USA

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Correspondence to Michael E. N. Tschaffon .

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Department of Physics “E.R.Caianiello”, University of Salerno, Fisciano, Italy

Roberta Citro

Quantum Optics Theory, Institute of Photonic Sciences, Castelldefels (Barcelona), Spain

Maciej Lewenstein

Theory Department, Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany

Angel Rubio

Institute of Quantum Physics, University of Ulm, Ulm, Germany

Physics Department, University of Michigan–Ann Arbor, Ann Arbor, MI, USA

James D. Wells

CSPAR & SPA, University of Alabama in Huntsville, Huntsville, AL, USA

Gary P. Zank

Appendix 1: Functional Equation of Jacobi Theta Function

In this Appendix we briefly rederive the functional equation, Eq. ( 26 ), of the Jacobi theta function

defined by Eq. ( 21 ), in order to bring out most clearly the origin of the difference to the corresponding relation, Eq. ( 155 ), for the generalized Jacobi theta function given by Eq. ( 154 ).

Since the summation index n appears in \(\omega \) as a square, we can immediately include negative values of n together with a factor of \(1/2\) . However, the term corresponding to \(n=0\) is unity. Thus, we have to subtract this contribution, and arrive at the representation

of \(\omega \) .

Next we employ the Poisson summation formula

where \(b=b(\nu )\) is a continuous extension of the discrete coefficients \(b_n\) . We emphasize that the specific form of the extension is of no consequence as long as \(b(n)\equiv b_n\) .

Indeed, this fact is a result of the identity

which is at the very heart of the Poisson summation formula.

For the choice

and with the help of the integral relation

we arrive at the identity

which leads us with the representation, Eq. ( 195 ), of \(\omega \) to

that is, the familiar functional equation

We emphasize that the second term in Eq. ( 202 ), which is independent of \(\omega \) , is a consequence of the absence of the term \(n=0\) in the definition, Eq. ( 194 ), of the Jacobi theta function \(\omega \) leading to the subtraction of unity in Eq. ( 195 ).

Appendix 2: Equivalent Condition for Non-trivial Zeros

In Sect. 2.4 we have derived the condition, Eq. ( 39 ) , for a zero on the critical line given by the identity of an integral

containing an oscillatory integrand with a Lorentzian. We devote this appendix to the derivation of an alternative but equivalent formulation by casting the integral into a different form.

In particular, we analyze the asymptotic behavior of J . For this purpose, we first derive an exact alternative expression for J , and then consider the limit \(\tau \rightarrow \infty \) .

We start by introducing in J the new integration variable \(y\equiv (1/2)\ln x\) , or \(x=\exp (2y)\) , which with the identity \(\mathrm {d}y=\mathrm {d}x/(2x)\) leads us to the form

Next, we obtain by integration by parts the equivalent representation

where we have used the fact that the boundary terms vanish, and find performing the differentiation

Finally, we introduce the new integration variable \(\theta \equiv \tau y\) , and arrive at the expression

When we recall from Eq. ( 21 ) the definition

of \(\omega \) , the differentiation yields the explicit formula

which decays for increasing \(\tau \) as \(1/ \tau ^2\) which is identical to the decrease of the Lorentzian on the right-hand side of Eq. ( 39 ).

When we now compare this representation of J to the initial one, Eq. ( 203 ), we find, apart from the decay with \(1/\tau ^2\) , three distinct features: (i) The oscillatory function in the integral is independent of \(\tau \) . (ii) We have a double-exponential decay, but (iii) this rapid decay only sets in for integration variables \(\theta > \tau /2\) .

Hence, the expression, Eq. ( 209 ), for J shows that the main contribution to this integral results from the decays characterized by the summation index n that are fast compared to the oscillation period, in complete agreement with the discussion of Sect. 2.4 .

Appendix 3: Exponential Product

In this appendix we cast the product

where q is an integer and \(\kappa =0,1\) , into an exponential form, which allows us to derive in a straight-forward way an asymptotic limit of \(\lambda \) . This term is not only a factor in the definition, Eq. ( 151 ), of the Dirichlet L-function \(\Lambda \) but also appears in the Riemann function \(\xi \) , defined by Eq. ( 40 ) for the special case \(q=1\) and \(\kappa =0\) .

We first derive a general expression for \(\lambda \) and then perform the asymptotic limit. Finally, we address the special case of the Riemann function.

1.1 General Expression

We start from the definition, Eq. ( 210 ), of \(\lambda \) in the exponential form

and recall the representation [ 17 ]

of the logarithm of the Gamma function with the remainder

which leads us for the argument \((s+\kappa )/2\) to the expression

Here we have introduced the abbreviation

and \(\bar {R}\) is defined by Eq. ( 213 ).

1.2 Asymptotic Limit

Next we consider the limit \(s \rightarrow \infty \) . In this case we find with the help of the expansion

the relation

Moreover, the generating function [ 1 ]

of the Bernoulli numbers \(B_n\) with

allows us to expand the remainder \(\bar {R}\) given by Eq. ( 213 ) into inverse powers of s with the leading contribution

Hence, we arrive in the lowest order of \(1/s\) at the asymptotic expression

for \(\mathcal {R}\) , which apart from the constant term decays as \(1/s\) .

1.3 Special Case: Riemann Function

Finally, we discuss the example \(q=1\) and \(\kappa =0\) corresponding to the Riemann function \(\xi \) . In this case, Eq. ( 214 ) reduces to

where \(\mathcal {R}\) given by Eq. ( 215 ) simplifies to

With the help of the asymptotic limit Eq. ( 221 ), we find the expression

for the remainder, which together with Eq. ( 224 ) is consistent with Eq. ( 222 ) for \(\kappa =0\) .

Appendix 4: Lines of Constant Height and Constant Phase

In Sect. 3.6.3 we have derived an asymptotic expression for \(\xi \) which arises solely from the product \((\wp /2)\pi ^{-s/2}\Gamma \left (s/2\right )\) since we have made the approximation \(\zeta =1\) . This formula is in terms of the complex variable s .

In this appendix, we first cast this expression in terms of amplitude and phase, and then consider two special cases. We conclude by verifying these approximations using the Cauchy-Riemann differential equations in amplitude and phase discussed in the Chapter “Insights into Complex Functions” of this volume.

1.1 Decomposition in Amplitude and Phase

We start by decomposing the approximation

of \(\alpha _{\xi }\) into its real and imaginary parts \(\Sigma \) and \(\Phi \) , that is,

determining the absolute value

and the phase \(\Phi \) of \(\xi \) . Here, we first derive general expressions for \(\Sigma \) and \(\Phi \) , and then consider the two extreme limits \(1<\sigma \ll \tau \) and \(0\leq \tau \ll \sigma \) .

With the identity

leading us to

we find the explicit expressions

We emphasize that the decomposition, Eq. ( 227 ), of the approximation, Eq. ( 226 ), of \(\alpha _{\xi }\) into \(\Sigma \) and \(\Phi \) given by Eqs. ( 231 ) and ( 232 ) is exact.

1.2 Special Limits

In order to gain insight into the dependence of \(\Sigma \) and \(\Phi \) on \(\sigma \) and \(\tau \) , we now consider two extreme limits, and derive approximate but analytic expressions for \(\Sigma \) and \(\Phi \) .

1.2.1 At the Center of the Complex Plane

We start with the case of \(1<\sigma \ll \tau \) , and arrive with the help of the approximations

at the expressions

In this limit, the lines \(\tau _{\Phi } = \tau _{\Phi }(\sigma )\) of constant phase \(\Phi \) satisfy an interesting symmetry relation. Indeed, for a given value \(\tau \) the lines of constant phase corresponding to \(\Phi \) and \(\Phi +\pi \) are separated in their values of \(\sigma \) and \(\sigma ^{\prime }\) by 4. This property follows from Eq. ( 237 ) which for \(\Phi +\pi \) reads

and a comparison with Eq. ( 237 ) immediately yields the connection

and thus the identity

Next, we derive an explicit expression for \(\tau _{\Phi }\) . Unfortunately, Eq. ( 237 ) is a transcendental equation, that is, the Lambert equation, and cannot be solved directly. However, in the limit of large \(\tau \) we obtain an approximate expression for \(\tau _{\Phi }\) by iteration.

For this purpose, we first cast Eq. ( 237 ) into the form

and replace \(\tau _{\Phi }\) in the slowly varying logarithm by the right-hand side leading us to the formula

We conclude by noting that this expression obviously also satisfies the periodicity property, Eq. ( 240 ).

1.2.2 At the Right Edge of the Complex Plane

Next we consider the case \(0\leq \tau \ll \sigma \) and arrive with

In this limit \(\Sigma \) , and thus \(|\xi |\) grows exponentially with \(\sigma \ln (\sigma /2\pi )\) for increasing \(\sigma \) , but decays like a Gaussian due to the term \(-\tau ^2/(4\sigma )\) for \(\tau \) increasing from zero. Moreover, \(\Phi \) increases linearly with \(\tau \) with a rate mainly given by \(\ln (\sigma /2\pi )\) .

According to Eq. ( 247 ), in this asymptotic limit the lines \(\tau _{\Phi } = \tau _{\Phi }(\sigma )\) of constant phase \(\Phi \) read

and decay with a rate, that is, inversely proportional to \(\ln (\sigma /(2\pi ))\) . In particular, \(\tau = 0\) , that is, the real axis, corresponds to the phase \(\Phi = 0\) , and all phase lines approach it as \(\sigma \rightarrow \infty \) .

In the last step of Eq. ( 248 ), we have neglected the correction term \(3/\sigma \) since in obtaining the general expression, Eq. ( 226 ), of \(\alpha _{\xi }\) we have already neglected terms of the order \(1/s\) .

1.3 Cauchy-Riemann Differential Equations

Next we probe the consistency of our approximations, Eqs. ( 236 ) and ( 246 ), as well as Eqs. ( 237 ) and ( 247 ) for the amplitude \(\Sigma \) as well as the phase \(\Phi \) of the exponential representation

of \(\xi \) by applying the Cauchy-Riemann differential equations

discussed in the Chapter “Insights into Complex Functions” of this volume.

We start with the asymptotic expressions

obtained in the previous section for \(1<\sigma \ll \tau \) .

Indeed, by direct differentiation we find

in complete agreement with the Cauchy-Riemann differential equation, Eq. ( 250 ).

Moreover, we also obtain the result

which is consistent with the expression

that is, with Eq. ( 256 ) when we neglect the terms proportional to \(1/\tau \) . Indeed, in the derivation of the expression, Eq. ( 253 ) for \(\Phi \) we have already neglected contributions of this order.

Next, we address the approximations

valid for \(0\leq \tau \ll \sigma \) , and find immediately

where in the last step we have neglected the term \((\tau /\sigma )^2\ll 1\) . Hence, we have verified the Cauchy-Riemann differential equation, Eq. ( 250 ).

Moreover, we obtain from Eq. ( 258 ) the relation

and Eq. ( 259 ) yields

Hence, we satisfy the Cauchy-Riemann differential equation, Eq. ( 251 ), in the approximation \(\sigma \ll 1\) .

Appendix 5: Special Examples of Normalized Gauss Sums

In Sect. 5 we have shown that the normalized Gauss sum

is given by a phase factor whose phase \(\beta \) is determined by the Dirichlet character \(\chi \) . We now illustrate this result by evaluating the normalized Gauss sum of three different characters.

For this purpose, we first consider the normalized Gauss sums associated with the Dirichlet characters forming the Titchmarsh counterexample. In these examples, the phase is non-vanishing. We then calculate the normalized Gauss sum of a character where the phase does vanish.

1.1 Non-vanishing Phase

We start by evaluating the Gauss sums \(G_1\) and \(G_2\) corresponding to the characters \(\chi _1\) and \(\chi _2\) defined \(\mathrm {mod}\;5\) with the values

and demonstrate that they are phase factors, in complete accordance with Eq. ( 143 ). Since \(\chi _1\) and \(\chi _2\) are the complex conjugate of each other, the corresponding phases must satisfy the symmetry relation, Eq. ( 144 ).

We start by noting that in both characters \(q=5\) . Thus, we find with \(\chi _j(-1) = \chi _j(4-5) = \chi _j(4)\) for \(j=1,2\) , and the definitions Eqs. ( 264 ) and ( 265 ) of \(\chi _1\) and \(\chi _2\) , the result \(\chi _1(-1)=\chi _2(-1)=-1\) which yields with Eq. ( 127 ) the parameter

Hence, the Gauss sum \(G_1\) , corresponding to the character \(\chi _1\) reads

and takes with the help of the definition, Eq. ( 264 ), of \(\chi _1\) , the form

where in the last step we have taken advantage of the \(2\pi \) -periodicity of the Fourier factors.

In terms of trigonometric functions, we find

and the values

finally yield the expression

for the Gauss sum \(G_1\) as a phase factor where

We conclude by briefly addressing the Gauss sum

associated with

Indeed, this symmetry enforces that in \(G_2\) the second and the third term in Eq. ( 267 ) change their signs, but everything else remains the same.

As a result, we arrive at a negative rather than a positive phase, that is,

We emphasize that the sign change in the phase due to the transition from \(G_1\) to \(G_2,\) Eqs. ( 269 ) and ( 272 ), can also be viewed as a consequence of the connection, Eq. ( 271 ), between \(\chi _1\) and \(\chi _2\) , and the symmetry relation, Eq. ( 144 ).

1.2 Vanishing Phase

We conclude by discussing an example of a character \(\chi \) where the phase of the corresponding normalized Gauss sum vanishes. For this purpose, we consider the real-valued character

with an integer k .

Since \(\chi \) is \(\mathrm {mod} \; 4\) we find \(q=4\) and with \(\chi (-1) = \chi (3-4)=\chi (3) = -1\) following from the definition, Eq. ( 273 ), of \(\chi \) , we obtain with the help of Eq. ( 127 ) the value \(\kappa =1.\)

Hence, the normalized Gauss sum, Eq. ( 263 ), takes the form

which with the definition, Eq. ( 273 ), of the character \(\chi \) reads

For the character \(\chi \) , given by Eq. ( 273 ), we obtain indeed a vanishing phase \(\beta \) for G .

Appendix 6: Functional Equation of Generalized Jacobi Theta Function

Since the functional equation

of the generalized Jacobi theta function plays a crucial role in the analytic continuation of the Dirichlet L-functions, as discussed in Sect. 6 , we devote the present Appendix to a brief summary of the derivation. This analysis also brings out most clearly the origins of the Gauss sum G and the character in its complex conjugate form.

Here we proceed in two steps: (i) We first show that due to the appearance of the character \(\chi \) and the term \(n^\kappa \) in the definition of \(\omega \) we can express the summation in \(\omega \) from unity to infinity, into one from minus infinity to plus infinity by merely introducing a factor \(1/2\) . This additional symmetry is the deeper reason for the simplicity of the functional equation, Eq. ( 275 ), of the Jacobi theta function for \(\Lambda ,\) compared to the one for \(\xi \) . (ii) Next we take advantage of the periodicity of the character together with the Poisson summation formula, and arrive at the functional equation, Eq. ( 275 ), of \(\omega \) corresponding to \(\Lambda .\)

We conclude by briefly comparing the derivation of the functional equation of the Jacobi theta function \(\omega \) corresponding to \(\Lambda \) to that for \(\xi \) . In particular, we show that the origin of the quadratic polynomial \(s(s-1)\) and the off-set \(1/2\) in the analytic continuation, Eq. ( 43 ), of \(\xi \) is the fact that the term \(n=0\) in \(\omega \) corresponding to \(\xi \) is nonzero.

1.1 Extension of Summation

We start by verifying the representation

For this purpose we consider the sum

and decompose S into a sum over all positive, and one over all negative integers including zero, that is,

where in the sum over negative n we have replaced \(-n\) by \(n.\)

When we recall the connection, Eq. ( 128 ), between \(\chi (-n)\) and \(\chi (n)\) together with \(\chi (0) = 0\) , Eq. ( 119 ), we find the expression

The values, Eq.( 127 ), of \(\kappa \) together with the relation \(\chi (-1) = \pm 1\) yield the identity

and the definition, Eq. ( 277 ), of \(\omega \) the desired representation, Eq. ( 276 ).

1.2 Emergence of Gauss Sum

Since the character \(\chi \) satisfies the periodicity condition, Eq. ( 116 ), it is useful to apply the summation formula

valid for any appropriately converging sum of coefficients \(d(n)\) , to S in the form

where we have introduced the abbreviation

Indeed, with the help of Eq. ( 116 ) we immediately find the representation

of S which with the Poisson summation formula

takes the form

Finally, the new integration variable

yields the expression

where we have interchanged the two summations, and have recalled the definition, Eq. ( 130 ), of the Gauss sum \(\tilde {G}(\chi ,m).\)

Next we use the reduction formula, Eq. ( 129 ), which allows us to factor \(\tilde {G}(\chi ,1)\) out of the sum over m and to find

With the definition of h , Eq. ( 282 ), and the integral formula

we finally arrive at

that is, the functional equation, Eq. ( 275 ), of \(\omega \) when we use the relation, Eq. ( 276 ). Moreover, we have recalled the definition, Eq. ( 142 ), of the Gauss sum.

1.3 Comparison to Jacobi Theta Function of \(\xi \)

We conclude by briefly comparing and contrasting this derivation of the functional equation of the Jacobi theta function of \(\Lambda \) to the corresponding one for the Jacobi theta function of \(\xi \) , defined by Eq. ( 21 ) and discussed in Appendix 1.

In contrast to the generalized Jacobi theta function, Eq. ( 154 ), where the term \(n=0\) vanishes due to \(\chi (0)=0\) , Eq. ( 119 ), the contribution in the one for \(\xi \) is unity. This difference is important since the term in Eq. ( 202 ) independent of \(\omega \) is a consequence of the subtraction of the term \(n=0\) in Eq. ( 195 ). As a result, in the corresponding functional equation, Eq. ( 275 ), for the generalized Jacobi theta function only the contribution proportional to \(\omega (1/x)\) enters.

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Tschaffon, M.E.N., Tkáčová, I., Maier, H., Schleich, W.P. (2023). A Primer on the Riemann Hypothesis. In: Citro, R., Lewenstein, M., Rubio, A., Schleich, W.P., Wells, J.D., Zank, G.P. (eds) Sketches of Physics. Lecture Notes in Physics, vol 1000. Springer, Cham. https://doi.org/10.1007/978-3-031-32469-7_7

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The Riemann hypothesis concerns the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ...Ubiquitous and fundamental in mathematics as they are, it is important and interesting to know as much as possible about these numbers. Simple questions would be: how are the prime numbers distributed among the positive integers? What is the number of prime numbers of 100 digits? Of 1,000 digits? These questions were the starting point of a groundbreaking paper by Bernhard Riemann written in 1859. As an aside in his article, Riemann formulated his now famous hypothesis that so far no one has come close to proving: All nontrivial zeroes of the zeta function lie on the critical line. Hidden behind this at first mysterious phrase lies a whole mathematical universe of prime numbers, infinite sequences, infinite products, and complex functions. The present book is a first exploration of this fascinating, unknown world. It originated from an online course for mathematically talented secondary school students organized by the authors of this book at the University of Amsterdam. Its aim was to bring the students into contact with challenging university level mathematics and show them what the Riemann Hypothesis is all about and why it is such an important problem in mathematics.

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Frontmatter pp i-iii

Dedication pp iv-iv, anneli lax new mathematical library pp v-vi, contents pp vii-viii, preface pp ix-xii, 1 - prime numbers pp 1-20, 2 - the zeta function pp 21-40, 3 - the riemann hypothesis pp 41-58, 4 - primes and the riemann hypothesis pp 59-86, appendix a - why big primes are useful pp 87-90, appendix b - computer support pp 91-98, appendix c - further reading and internet surfing pp 99-100, appendix d - solutions to the exercises pp 101-142, index pp 143-144, altmetric attention score, full text views.

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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 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)

• number-theory
• reference-request
• analytic-number-theory
• riemann-zeta
• riemann-hypothesis
• 7 $\begingroup$ I wonder if it would fit protocols better to post this as an answer after posting a short question that it answers. $\endgroup$ –  Michael Hardy Commented Feb 6, 2013 at 17:33
• 4 $\begingroup$ That will be a long list... consider writing it up as a BIBTeX bibliography. $\endgroup$ –  vonbrand Commented Feb 6, 2013 at 17:34
• 1 $\begingroup$ @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. $\endgroup$ –  Marko Amnell Commented Feb 6, 2013 at 18:46
• $\begingroup$ @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. $\endgroup$ –  robjohn ♦ Commented Feb 6, 2013 at 19:41
• 1 $\begingroup$ 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." $\endgroup$ –  Marko Amnell Commented Apr 18, 2015 at 11:09

2 Answers 2

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.

• $\begingroup$ 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. $\endgroup$ –  Marko Amnell Commented Feb 7, 2013 at 5:37
• 1 $\begingroup$ @MarkoAmnell: In the meantime, Ivić's book has become available on Kindle: amazon.com/… . $\endgroup$ –  joriki Commented Apr 16, 2020 at 16:04

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, 3 volumes [Vol. 1: Arithmetic Equivalents , 400 pages; Vol. 2: Analytic Equivalents , 350 pages], Cambridge University Press (January 31, 2018); Vol. 3: Further Steps towards Resolving the Riemann Hypothesis , 704 pages (September 30, 2023)

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.

• $\begingroup$ 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 ? $\endgroup$ –  reuns Commented May 27, 2016 at 0:54
• $\begingroup$ @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... $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 3:37
• $\begingroup$ (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/… $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 3:47
• $\begingroup$ 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 ? $\endgroup$ –  reuns Commented May 29, 2016 at 14:06
• $\begingroup$ 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. $\endgroup$ –  Marko Amnell Commented May 29, 2016 at 17:14

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Riemann_Hypothesis_Proof.pdf

Easy proof using laplace transform and fractional part function

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the riemann hypothesis and hilberts tenth problem

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