proposed (dis)proofs of the Riemann Hypothesis
"*Without doubt it would be desirable to have a rigorous proof of this
proposition; however I have left this research aside for the time being
after some quick unsuccessful attempts, because it appears to be
unnecessary for the immediate goal of my study...*"
Bernhard Riemann, 1859
If you are a university mathematics lecturer who teaches analytic number
theory, you might want to consider setting your students the task of
deconstructing the more serious of these. They may otherwise never
be given any serious attention, which would be a shame. As someone once joked, "It's easier to prove the RH than to get someone to read your proof!"
D. Biswas, "Analytical expression of complex Riemann xi function $\xi(s)$ and proof of Riemann Hypothesis"
[abstract:] "In this paper explicit analytical expression for Riemann xi function $\xi(s)$ is worked out for complex values of $s$. From this expression Riemann Hypothesis is proved. Analytic Expression for non-trivial zeros of Riemann zeta function $\zeta(s)$ is also found. A second proof of Riemann Hypothesis is also given"
J. Feliksiak, "The elementary proof of the Riemann's Hypothesis"
[abstract:] "This research paper aims to explicate the complex issue of the Riemann's Hypothesis and ultimately presents its elementary proof. The method implements one of the binomial coefficients, to demonstrate the maximal prime gaps bound. Maximal prime gaps bound constitutes a comprehensive improvement over the Bertrand's result, and becomes one of the key elements of the theory. Subsequently, implementing the theory of the primorial function and its error bounds, an improved version of the Gauss' offset logarithmic integral is developed. This integral serves as a Supremum bound of the prime counting function $\pi(n)$. Due to its very high precision, it permits to verify the relationship between the prime counting function $\pi(n)$ and the offset logarithmic integral of Carl Gauss. The collective mathematical theory, via the Niels F. Helge von Koch equation, enables to prove the Riemann's Hypothesis conclusively."
Eminent mathematical physicist Sir Michael Atiyah presented a purported "simple proof" of the RH at a conference in Heidelberg on 24th September 2018. It seems that he arrived at this as an unexpected "bonus" when attempting to derive the fine structure constant. Details are still coming in, but it seems that the key ingredient is the Todd function. Here is Atiyah's paper, and here is another one it heavily references. Here is a video of Atiyah's talk. Reactions thus far have been predominantly sceptical – see, e.g., this. Check back here for updates.
J.D. Cook, "The Riemann Hypothesis, the fine structure constant and the Todd function"
R.J. Lipton, "Reading into Atiyah's Proof"
"Setting the stage for societal entropy and a dicey proof of the Riemann Hypothesis": An interview with Henk Diepenmaat (May 2018)
F.I. Moxley III, "Solving the Riemann Hypothesis with Green's function and a Gelfand triplet" (June 2018)
[abstract:] "The Hamiltonian of a quantum mechanical system has an aliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian that maps to the nontrivial zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results using Green's
functions. A Gelfand triplet is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at $\sigma = 1/2$, and a general solution is obtained by performing an invariant similarity transformation."
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis" (2019)
F.I. Moxley III, "Decidability of the Riemann Hypothesis" (September 2018)
[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for
this spectrum to be observable, the Hamiltonian should be Hermitian. Such quantum Riemann zeta
function analogies have led to the Bender–Brody–Müller (BBM) conjecture, which involves a nonHermitian Hamiltonian whose eigenvalues are the nontrivial zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. In this
case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator
is for integrable quantum systems. As such, herein we perform a symmetrization procedure of the
BBM Hamiltonian to obtain a Hermitian Hamiltonian using a similarity transformation, and provide
an analytical expression for the eigenvalues of the results using Green's functions. A Gelfand triplet
is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting
eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian
operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable,
i.e., the Riemann Hypothesis is not decidable. Moreover, a second quantization of the resulting
Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic
continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the
eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists
at $\sigma = 1/2$, and a general solution is obtained by performing an invariant similarity transformation"
F.I. Moxley III, "A Schrödinger equation for solving the Bender–Brody–Muller Conjecture" (AIP Conference Proceedings, 2017)
[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Muller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and discuss the eigenvalues of the results. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, the Hilbert–Pólya conjecture is discussed, and it is heuristically shown that the real part of every nontrivial zero of the Riemann zeta function converges at $\sigma = 1/2$."
F. Galetto, "Riemann's Hypothesis and the Mertens function for the proof of RH" (2018)
[abstract:] "It is well known that 'The Riemann hypothesis is equivalent to the statement that for every $\epsilon > 0$ the function $M(x)/x^{1/2+\epsilon}$ pproaches zero as $x\to\infty$', where $M$ is the Mertens function. Starting from the definitions of $M(x)$ and $\zeta(s)$ we derive an integral equation about $M(x)$: from that we derive that $|M(x)| < x^{1/2+\epsilon}$; the proof of the Riemann hypothesis follows."
F. Galetto, "Riemann's Hypothesis and the Liouville function for the proof of RH" (2018)
[abstract:] "It is well known that 'The Riemann hypothesis is equivalent to the statement that for every $\epsilon > 0$ the function $L(x)/x^{1/2+\epsilon}$ approaches zero as $x\to\infty', where $L$ is the 'cumulative' Liouville function. Starting from the definitions of $L(x)$ and $\zeta(s)$ we derive an integral equation about $L(x)$: from that we derive that $|L(x)|< x^{1/2+\epsilon}$; the proof of the Riemann hypothesis follows."
K. Eswaran, "The final and exhaustive proof of the Riemann Hypothesis from first principles" (May 2018)
[abstract:] "As is well-known, the celebrated Riemann Hypothesis (RH) is the prediction that all the non-trivial zeros of the zeta function $\zeta(s)$ lie on a vertical line in the complex s-plane at Re(s) = 1/2. Very many efforts to prove this statement have been directed to investigating the analytic properties of the zeta function, however all these efforts have not been able to substantially improve on Riemann's initial discovery: that all the non trivial zeros lie in verical strip of unit width whose centre is the critical line. The efforts have been rendered difficult because of a lack of a suitable functional representation (formula) for $\zeta(s)$ (or $1/\zeta(s)$) , which is valid and analytic over all regions of the Argand plane; these difficulties are further complicated by the presence of prime numbers in the very definition of the zeta function and the lack of predictability in the behaviour of prime numbers which makes the analysis intractable. In this paper we make our first headway by looking at the analyticity of the function $F(s) = \zeta(2s)/\zeta(s)$ that has poles in exactly those positions where $\zeta(s)$ has a non trivial zero. Further, the trivial zeros of the zeta function, which occur at the negative even integers, conveniently cancel out in $F(s)$ and do not appear as poles of the latter (however there is an isolated pole of $F(s)$, viz. $s = 1/2$, which is actually a pole of $\zeta(2s)$ but this will not worry us because it is on the critical line). So the task of proving the RH is some what 'simplified' because all we have to show is: All the poles of $F(s)$ occur on the critical line, which then is the main aim of this paper. We then investigate the Dirichlet series that obtains from the function $F(s)$ and employ novel methods of summing the series by casting it as an infinite number of sums over sub-series. In this procedure, which heavily invokes the prime factorization theorem, each sub-series has the property that it oscillates in a predictable fashion, rendering the analytic properties of the function $F(s)$ determinable. With the methods developed in the paper many theorems are proved, for example we prove: that for every integer with an even number of primes in its factorization, there is another integer that has an odd number of primes (multiplicity counted) in its factorization; by this demonstration, and by the proof of several other theorems, a similarity between the factorization sequence involving (Liouville's multiplicative functions) and a sequence of coin tosses is mathematically established. Consequently, by placing this similarity on a firm foundation, one is then empowered to demonstrate, that Littlewood's (1912) sufficiency condition involving Liouville's summatory function, $L(N)$, is satisfied. It is thus proved that the function $F(s)$ is analytic over the two half-planes $\Re(s) > 1/2$ and $\Re(s) < 1/2$, clearly revealing that all the nontrivial zeros of the Riemann zeta function are placed on the critical line $\Re(s) = 1/2$."
R. Raghavan, Answer to Herrington's objection" (2018)
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis updated to include a response to R. Raghavan's Note of 24 November 2018" (2019)
R. Raghavan, response to Herrington's note (January 2019)
Y. Shi, "On the zeros of Riemann $\Xi$ function" (2018)
[abstract:] "In this paper we use complex analysis method and present an elementary and positive proof to the Riemann Hypothesis that all the zeros of the Riemann $\Xi(z)$ function are real. (A) We first construct a family of functions $\{W(n,z)\}_{n\geqslant 2}$ that converges to $\Xi(z)$ uniformly in the critical strip $S_{1/2}:=\{|\Im(z)|< \frac{1}{2}\}$. (B) We then show that $W(n,z)=U(n,z)-V(n,z)$. Using a theorem by Polya, we show that $U(n,z)$ and $V(n,z)$ have only real, simple zeros. (C) We next show that all the zeros of $U(n\geqslant N,z^{1/2})$ and $V(n\geqslant N,z^{1/2})$ are real, positive, simple, and interlacing. Using an entire function equivalent to Hermite–Kakeya Theorem for polynomials we show that $W(n\geqslant N,z^{1/2})$ has only real, positive and simple zeros. Thus $W(n\geqslant N,z)$ have only real and simple zeros. (D) Using a corollary of Hurwitz's theorem in complex analysis we prove that all the zeros of $\Xi(z)$ in the critical strip $S_{1/2}$ are real."
A. A. Logan, "A proof of the Riemann Hypothesis" (February 2018)
[abstract:] "This paper investigates the characteristics of the power series representation of the Riemann xi function. A detailed investigation of the behaviour of the zeros of the real part of the power series and the behaviour of the curve, combined with a substitution of polar coordinates in the power series and in the definition of the critical strip (leading to a critical area), and the relationship with the zeros of the imaginary part of the power series leads to the conclusion that the Riemann xi function only has real zeros."
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2018)
A. Logan, "A Proof of the Riemann Hypothesis version 3.1: Zeros of the Dirichlet eta function" (2018)
[abstract:] "This paper investigates the characteristics of the zeros of the real component of the Riemann zeta function (of $s$) in the critical strip by using the real component of the Dirichlet eta function, which has the same zeros (A necessary condition for a zero of the complete function is a zero of the real component). The derivative of the real component for a fixed imaginary part of s is shown to be always positive for negative or zero values of the real component of the function, meaning that each value of the imaginary part of s produces at most one zero. Combined with the fact that the zeros of the Riemann $\xi$ function are also the zeros of the zeta function and $\xi(s) = \xi(1-s)$, this leads to the conclusion that the Riemann Hypothesis is true."
A. Grytczuk, "Robin's inequality for sum of divisors function and
the Riemann Hypothesis", *Journal of Informatics and Mathematical Sciences* **4**(1) (2012) 15–21
[abstract:] "Let $\sigma(n)$ denote the sum of divisors function. In this paper we give a simple proof of the Robin inequality: $\sigma(n) < e^{\gamma}n\log\log n$, for all positive integers $n \geq 5041$. The Robin inequality implies Riemann Hypothesis."
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2017)
Although the author never explicitly claims to have proven the RH, this purports to prove that all nontrivial zeros of the zeta function have real part 1/2:
F. Qiu, "A necessary condition for the existence of the nontrivial zeros of the Riemann zeta function" (preprint 01/2008)
[abstract:] "Starting from the symmetrical reflection functional equation of the zeta function, we have found that the $\sigma$ values satisfying $\zeta(s) = 0$ must also satisfy $|\zeta(s)| = |\zeta(1 - s)|$. The $\sigma$ values satisfying this requirement are a necessary condition for the existence of the nontrivial zeros. We have shown that $\sigma = 1/2$ is the only numeric solution that satisfies the requirement."
Riemann Hypothesis not proved by Nigerian mathematician Opeyemi Enoch, despite numerous media claims published on 17th November 2015.
Enoch gave a twenty minute presentation, "A Matrix That Generates the Point Spectral of the Riemann Zeta Function" at the "International Conference on Mathematics and Computer Science" earlier in November 2015. Here is the (very confusing, seemingly meaningless) abstract. The conference appears not to have been affiliated with any university or maths/science organisation and the abstract doesn't claim a proof of the RH. All very puzzling!
W. Raab, "Proof of the Riemann Hypothesis"
(2013? – this was seemingly uploaded by Enoch to academia.edu and rather unconvincingly claimed as his own)
F. Galetto has produced a very forceful denunciation of Enoch's publications (2017)
"Fantasy and mathematics by Enoch Opeyemi Oluwole" another forceful response from Galetto to Opeyeme's 2018 research report "The Fourier Laplace approach of proving the Riemann Hypothesis"
P. F. Roggero, "Proof of why all the zeros of the Riemann zeta function are on the critical line" (2016)
P. Braun, "Euler's constant and the Riemann Hypothesis" (2015, updated 2016)
[This claims to show the unprovability of the RH in arithmetic which then leads to a proposed proof in the analytic realm.]
J. Bredakis, "Riemann Hypothesis confirmed by John Bredakis formulas – a brief version" (2015)
J. Kim, "A proof of the Riemann hypothesis using the remainder term of the Dirichlet eta function" (2015)
[abstract:] "The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make sure that the error between remainder term and its approximation is reduced as n goes to infinity. According to the Riemann zeta functional equation, if $\eta(\sigma+it)=0$ then $\eta(1-\sigma-it)=0$. In this case, $n$-th partial sum also can be approximated by expression for $n$. Based on this approximation, we prove the Riemann hypothesis."
K. Braun, "A Kummer function based zeta function theory to prove the Riemann Hypothesis" (2015)
[abstract:] "All nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. A positive answer to this conjecture is proven, enabled by an alternative Kummer function based on zeta function theory."
T. Gomis, *et al.*, "Quadratic theory of the Riemann Hypothesis and connection to operators and random matrix theories" (2015)
[abstract:] "In a bold move, the non-trivial zeros of the Riemann zeta functions are considered as complex zeros of a canonical quadratic equation. The Riemann Hypothesis confirmation and its links to operators and random matrix theories are the direct mathematical implications of this bold view."
L. de Branges, "A proof of the Riemann Hypothesis, I" (11/2014)
P.F. Roggero, "From Riemann Hypothesis to Riemann Thesis" (2014)
F. Galetto, "Riemann Hypothesis proved" (2014)
[abstract:] "We show a proof of the so-called Riemann Hypothesis. We prove the RH using the theory of 'inner product spaces' $I$ and $l^2$ Hilbert spaces, where is
defined the 'functional' $(a,b)$, named scalar [or inner] product of the vectors $a$ and $b$. The proof is so simple that we suspect that there could be an error that
we are unable to find."
"Kazakhstan citizen claims solving Riemann Hypothesis", *Tengri News*, 23 September 2013
"Yessenbek Ushtenov, a mechanical engineer from Saryagash town, Kazakhstan, claims to to have solved one of the seven problems of the millennium, the Riemann hypothesis, Alau-Kazakhstan public-political magazine writes..."
A. Prástaro, "The Riemann hypothesis proved" (05/2013)
[abstract:] "The Riemann hypothesis is proved by quantum-extending the zeta Riemann function to a quantum mapping between
quantum $1$-spheres with quantum algebra $A = \mathbb{C}$, in the sense of A. Prástaro [2009, 2013]. Algebraic topologic
properties of quantum-complex manifolds and suitable bordism groups of morphisms in the category $\mathfrak{Q}_{\mathbb{C}}$ of
quantum-complex manifolds are utilized."
A. Garcés Doz, "A potential elementary proof of the Riemann Hypothesis" (2013)
[abstract:] "This paper presents a possible elementary proof of the Riemann hypothesis. We say possible or potential, you have to be very cautious and skeptical of the potential of the evidence presented, is free of a crucial
error that invalidate the proof. After several months of extensive review, the author, having found no error we have decided to publish it in the hope that someone will find the error. However, it is considered that the method may be useful in some way. This potential proof uses only the rudiments of analysis and arithmetic inequalities. It includes a first part of the reason why we think that the Riemann hypothesis seems to be true."
J. Ghannouchi, "The real primes and the Riemann hypothesis" (2013)
[abstract:] "In this paper, we present the Riemann problem and define the real primes. It
allows to generalize the Riemann hypothesis to the reals. A calculus of integral
solves the problem. We generalize the proof to the integers."
Jeffrey N. Cook, "On exponential decay and the Riemann hypothesis" (2013)
[abstract:] "A Riemann operator is constructed in which sequential elements are removed from a decaying set by means of prime factorization, leading to a form of exponential decay with zero degeneration, referred to as the root of exponential decay. A proportionate operator is then constructed in a similar manner in terms of the non-trivial zeros of the Riemann zeta function, extending proportionately, mapping expectedly always to zero, which imposes a ratio of the primes to said zeta roots. Thirdly, a statistical oscillation function is constructed algebraically into an expression of the Laplace transform that links the two operators and binds the roots of the functions in such a manner that the period of the oscillation is defined (and derived) by the eigenvalues of one and the elements of another. A proof then of the Riemann hypothesis is obtained with a set of algebraic paradoxes that unmanageably occur for the single incident of any non-trivial real part greater or less than a rational one half."
Z. Elhadj, "A positive answer to the Riemann hypothesis: A new result predicting the location of zeros" (09/2012)
[abstract:] "In this paper, a positive answer to the Riemann hypothesis is given by using a new result that predict the exact location of zeros of the alternating zeta function on the critical strip."
J. Bredakis, "Proof by
contradiction of the negation of Riemann Hypothesis, with graphical demonstration" (blog post, 2012)
D. Pozdnyakov, "Physical interpretation of the Riemann hypothesis" (03/2012)
[abstract:] "An equivalent formulation of the Riemann hypothesis is given. The formulation is generalized. The physical interpretation of the Riemann hypothesis generalized formulation is given in the framework of quantum theory terminology. An axiom is laid down on the ground of the interpretation taking into account the observed properties of the surrounding reality. The Riemann hypothesis is true according to the axiom. **It is shown that it is unprovable.**"
H. Shinya, "A Proof of the Riemann Hypothesis (07/2011)
M.V. Atovigba, "Proof of the Riemann Hypothesis" (2011)
Despite no mention of the fact in the abstract(!), this preprint contains a purported proof of the RH:
L. Fekih-Ahmed, "On the zeros of the Riemann zeta function" (04/2010)
[Here are G. Müller's comments on why this 'proof' is flawed.]
Xian-Shun Luo, "Riemann Hypothesis and Levison
Theorem" (12/2008)
[abstract:] "In this paper we will give a simple proof of Riemann Hypothesis, considered to be one of the greatest unsolved problem in mathematics,
related to inverse scattering problem and radom matrices. The important relationship between Riemann Hypothesis and random matrices was found by Freeman J.
Dyson (1972). Dyson [wrote] a paper [in] 1975 [which] related random matrices and inverse scattering problem. Under this explanation, the famous Riemann
Hypothesis is equivalent to Levison theorem of scattering phase-shifts. We will prove this relation."
Xian-Shun Luo, "Riemann Hypothesis and Selberg eigenvalues conjecture" (12/2008)
[abstract:] "In this paper we will give a direct proof of Riemann Hypothesis and Selberg eigenvalue conjecture, [using] some results of the classical complex
analysis before 1900 [...] and explain with a little Algebra and Physics. "
email exchange between Xian-Shun Luo and others regarding these purported proofs
M.M. Anthony, "A Simple Proof of the Riemann Hypothesis" (11/2008)
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2018)
Yuan-You Fu-Rui Cheng, "Proof of the Riemann Hypothesis" (10/2008)
[abstract:] "The Riemann zeta function is a meromorphic function on the whole complex plane. It has infinitely many zeros and a unique pole at $s = 1$. Those zeros at $s = –2, –4, –6, ...$ are known as trivial zeros. The Riemann hypothesis is a conjecture of Riemann, from 1859, claiming that all nontrivial zeros of $\zeta(s)$ lie on the line $\Re(s) =\tfrac{1}{2}$. Let $x \ge2$. Define $\Lambda(n) =\log p$ whenever $n =p\sp{m}$ for a prime number $p$ and a positive integer $m$, and zero otherwise. Then, the Riemann hypothesis is equivalent to the $\varpi$-form of the prime number theorem as $\varpi(x) =O(x\sp{1/2} \log\sp{2} x)$, where $\varpi(x) =\sum\sb{n\le x}\ \bigl(\Lambda(n) –1\big)$ with the sum running through the set of all natual integers. Similarly that in the literature, we use a classical integral formula for the step function of $x$. But, we apply Cauchy's Residue Theorem to the funtion ${\mathcal Z}(s) \cdot \tfrac{x\sp{s}} {s}$ instead, where ${\mathcal Z}(s) = –\tfrac{\zeta\sp{\prime}(s)} {\zeta(s)} –\zeta(s)$. Starting from the estimate on $\varpi(x)$ from the known zero-free region of ${\mathcal Z}(s)$, we utilize the induction on the exponent in $O(x\sp{\theta} \log\sp{2} x)$. For each fixed $\theta$, we also use the induction on $x$. With the help of {\em the strong density hypothesis} proved recently, we prove the Riemann hypothesis."
A. Bergstrom, "Proof of Riemann's zeta-hypothesis" (09/2008)
[abstract:] "Make an exponential transformation in the integral formulation of Riemann's zeta-function $\zeta(s)$ for $Re(s) > 0$. Separately, in addition make the substitution $s \rightarrow 1 – s$ and then transform back to s again using the functional equation. Using residue calculus, we can in this way get two alternative, equivalent series expansions for $\zeta(s)$ of order $N$, both valid inside the "critical strip", i.e. for $0 < Re(s) < 1$. Together, these two expansions embody important characteristics of the zeta-function in this range, and their detailed behavior as $N$ tends to infinity can be used to prove Riemann's zeta-hypothesis that the nontrivial zeros of the zeta-function must all have real part $1/2$."
In addition to the preprint, the arXiv file also contains a discussion of some thirty-five Frequently Asked Questions from readers. Further questions not adequately dealt with in the existing FAQ are welcome.
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2018)
N.A. Carella, "Divisor and Totient Functions Estimates" (06/2008)
[G. Caveney comments:] "*Despite the modest title, this paper's Theorem 7 if true would amount to a proof of RH, because it would
establish the Nicolas inequality for all sufficiently large numbers, and Nicolas proved his inequality is false for infinitely many numbers if RH is false.
* *The author's claimed proof of Theorem 7 takes up only about one page! At the very start of the proof, he refers to a Proposition 8-i, but said proposition
is nowhere to be found in his paper. So that's the first flaw in his proof.*"
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2018)
Boris Kupershmidt, "Remarks on a Nicolas Inequality",
and "Remarks on Robin's and Nicolas Inequalities" (and
a revision of the latter).
[G. Caveney comments:] "*Maybe this is a joke, but it's strange that a math professor posted a joke that makes him look bad as a serious paper
on his own faculty web page.
* *The first paper and the first version of the second paper claim to prove the Nicolas inequality, thus proving RH.
*
*The second version of the second paper claims to disprove the Nicolas inequality. But at the end of the paper, rather than claim to have proved RH false, the author claims he has proved that Nicolas' results were mistaken.
*
*Very strange.*"
L. Aizenberg, "Lindelöf's hypothesis is true and Riemann's one is not" (12/2007)
[abstract:] "We present an elementary, short and simple proof of the validity of the Lindelöf hypothesis about the Riemann zeta-function.
The obtained estimate and classical results by Bohr-Landau and Littlewood disprove Riemann's hypothesis."
Y. Choie, N. Lichiardopol, P. Moree and P. Solé, "On Robin's criterion for the Riemann Hypothesis", *J. de Theories des Nombres de Bordeaux* **19** (2007), 357–372.
G. Herrington, "Problem with an implicit claim of a proof of the Riemann Hypothesis" (04/2019)
C. Castro, "The Riemann Hypothesis is a
consequence of CT-invariant quantum mechanics" (submitted to *J. Phys. A*, 02/2007)
[abstract:] "The Riemann's hypothesis (RH) states that the nontrivial zeros of the
Riemann zeta-function are of the form $s_n =1/2 + i lambda_n$.
By constructing a continuous family of scaling-like operators involving
the Gauss-Jacobi theta series, and by invoking a novel CT-invariant
Quantum Mechanics, involving a judicious charge conjugation $C$
and time reversal $T$ operation, we show why the Riemann Hypothesis is true."
This follows earlier attempts:
C. Castro, A. Granik, and J. Mahecha,
"On SUSY-QM, fractal strings
and steps towards a proof of the Riemann hypothesis" (2001)
Despite the humility of the title, this preprint does contain a (purported)
proof of the RH. The following preprint examines the strategy proposed.
E. Elizalde,
V. Moretti, and
S. Zerbini,
"On recent
strategies proposed for proving the Riemann hypothesis"
(abstract) "We comment on some apparently weak points in the novel
strategies recently developed by various authors aiming at a proof of
the Riemann hypothesis. After noting the existence of relevant
previous papers where similar tools have been used, we refine some of
these strategies. It is not clear at the moment if the problems we
point out here can be resolved rigorously, and thus a proof of the RH
be obtained, along the lines proposed. However, a specific suggestion
of a procedure to overcome the encountered difficulties is made, what
constitutes a step towards this goal."
This is the latest:
C. Castro, "On two strategies towards
the Riemann Hypothesis: Fractal Supersymmetric QM and a trace formula" (06/2006)
[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the
Riemann zeta-function are of the form $s_n =1/2+i lambda_n$. An improvement of our previous
construction to prove the RH is presented by implementing the Hilbert-Pólya proposal and
furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum
reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung
potential (that capture the average level density of zeros) by recurring to a weighted superposition of
Weierstrass functions $W(x,p,D)$ and where the summation has to be performed over
all primes $p$ in order to recapture the connection between the distribution of zeta zeros and prime
numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an
ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential)
has the same functional form as the fluctuating part of the level density of zeros.
The second approach to prove the RH relies on the existence of a continuous family of scaling-like
operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of
eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another
test of the Riemann Hypothesis."
A. Madrecki, "One page proof of Riemann Hypothesis"
(09/2007)
[abstract:] "We give a short Wiener measure proof of the Riemann hypothesis based on a surprising, unexpected and
deep relation between the Riemann zeta $\zeta(s)$ and the trivial zeta $\zeta_{t}(s):= Im(s)(2Re(s)-1)$."
C. Picard refutes Madrecki's proof
Jinzhu Han, Zaizhu Han, "A proof of the Riemann Hypothesis"
(06/2007)
[abstract:] "In this paper, a proof of the Riemann Hypothesis is given. We first obtain the ratio limited value of
the Riemann zeta functions at non-trivial zero points by L'Hospital Rule. And then we prove that all non-trivial zero
points of have real part 1/2. Thus, we provide evidence for the correctness of the Riemann Hypothesis."
Loreno Heer comments "*There may be an error but I am not sure. In equation 14 they use L'Hospital Rule to calculate the **m*th derivative of
the zeta function at a non-trivial zero. I think this is not allowed because the first derivative does already exist and then the rule can not be applied any
further. Is this correct?"
[errors pointed out by Felix Pahl]
G. Spencer-Brown, best known as the author of the cult
book *Laws of Form*, has made public (06/2006) a purported proof of
the Riemann Hypothesis, apparently destined to appear as an appendix in a new edition of *LoF*.
G. Spencer Brown, "A proof of Riemann's
hypothesis via Denjoy's equivalent theorem" - another (2008) supposed RH proof from the reclusive author of *Laws
of Form*.
A. Palmer, Proposed Proof of Riemann Hypothesis
(03/2006)
The author of this preprint believes he has found a proof:
J.J. Garcia Moreta, "Chebyshev Partition function: A connection between statistical
physics and Riemann Hypothesis" (09/2006)
[abstract:] "In this paper we present a method to obtain a possible self-adjoint Hamiltonian
operator so its energies satisfy Z(1/2+iE_n)=0, which is an statement equivalent to Riemann
Hypothesis, first we use the explicit formula for the Chebyshev function Psi(x) and apply the change
x=exp(u), after that we consider an Statistical partition function involving the Chebyshev function
and its derivative so Z=Tr(exp(-BH), from the integral definition of the partition function Z we try
to obtain the Hamiltonian operator assuming that H=P^{2}+V(x) by proposing a Non-linear integral
equation involving Z(B) and V(x)."
Jiang Chun-Xuan, "Disproofs of Riemann
Hypothesis", *Algebras, Groups and Geometries*
**22** (2005) 123–136
[abstract:] "As it is well known, the Riemann hypothesis on the zeros of the $\zeta(s)$
function has been assumed to be true in various basic developments of the 20th century mathematics, although it has never been proved to
be correct. The need for a resolution of this open historical problem has been voiced by several distinguished mathematicians. By using
preceding works, in this paper we present comprehensive disproofs of the Riemann hypothesis. Moreover, in 1994 the author discovered the
arithmetic function $J_n(\omega)$ that can replace Riemann's $\zeta(s)$ function in view of its proved features: if, then the function has infinitely many
prime solutions; and if $J_n(\omega) \neq 0$, then the function has finitely many prime solutions. By using the Jiang function $J_n(\omega)$ we prove the twin
prime theorem, Goldbach's theorem and the prime theorem of the form $x^2 + 1$. Due to the importance of resolving the historical open nature
of the Riemann hypothesis, comments by interested colleagues are here solicited."
[errors pointed out by Felix Pahl]
H. Delille, who in 2004 began making
and retracting claims that he had a proof of the Riemann hypothesis,
provided this summary of his work, which
involves the analytic continuation of Beurling zeta functions
(August 2004).
K. Shi, "A geometric
proof of Riemann Hypothesis" (07/2003)
[abstract:] "Beginning from the formal resolution of Riemann Zeta function, by using the formula of inner
product between two infinite-dimensional vectors in the complex space, the author proved the world's baffling problem
- Riemann hypothesis raised by German mathematician B. Riemann in 1859."
[errors pointed out by Felix Pahl]
J. Perry, purported proof submitted to this site,
10/2002
[errors pointed out by Felix Pahl]
C. Musès, "Some new considerations on the Bernoulli numbers, the
factorial function, and Riemann's zeta function", *Applied Mathematics
and Computation* **113** (2000) 1-21. Curiously, although published,
this includes a supposed proof of the RH on page 21. Musès recently died.
You should be able to view a PDF version of this article at
this site
(just login as a guest).
Chengyan Liu, "Riemann Hypothesis" (09/1999)
[abstract:] "Through an equivalent condition on the Farey series set forth by Franel and Landau, we prove Riemann Hypothesis for the Riemann zeta-function and the Dirichlet L-function. "
**Two**
proofs of the Riemann Hypothesis by the illustrious Archimedes (formerly Ludwig) Plutonium (08/1993)
B. Conrad's purported
refutations of these alleged proofs
R. Herrera, "On the real part of non-trivial zeros of Riemann's zeta function" (unpublished manuscript, 1987)
Prof. Herrera, of the School of Mathematics at the University of Costa Rica, passed away recently. One of his former students brought this manuscript to my attention, asking "*How can I help this former professor of mine fulfill his dream of having his paper published and if proven correctly receive recognition for his life's work?*"
Prof. Hwang's daughter supplied the following three unpublished manuscripts recently found amongst her deceased father's
papers:
J.S. Hwang, "On the first exceptional zero
of Riemann's zeta-function"
J.S. Hwang, "The quickest disproof of
Riemann's Hypothesis"
J.S. Hwang, "On the falsity of Riemann's
Hypothesis"
G. Herrington, "Problem with a claimed proof of the Riemann Hypothesis
" (2018)
S. Allen, "Symmetry argument for the Riemann Hypothesis, universality and
broken symmetry "
C. Pradas,
"Proof of the Riemann Hypothesis -
preliminary notes"
suggested refutation of Pradas' alleged proof
J. Constant, "Proof of Riemann's Hypothesis"
J. Constant, "Some Extended Zeta Functions Provide Easy Proofs of Riemann's Hypothesis"
S. Huang, "Prime numbers as lawful creatures of the mind"
(purports to contain a sort of philosophical/psychological 'proof' of the RH)
What?
**related links**
The respected French economist Henri Berliocchi
(who also seems to have extensive interests in homeopathy and mathematics) has brought out a book called *Infirmation
de l'hypothese Riemann* (Economica, Paris, 2001: ISBN 2-7178342-6) in which he claims
to have disproved the Riemann Hypothesis. It appears that Berliocchi has not produced a counterexample (*i.e.* a nontrivial zero of the zeta function off the critical line), but instead used an argument by contradiction. An English translation was made available in 2016.
Littlewood's opinion that these people are
probably wasting their time
C.A. Feinstein, "The Collatz 3*n*+1
conjecture is unprovable" (02/2004)
[abstract:] "In this paper, we show that any proof of the Collatz 3*n*+1 Conjecture must have an infinite number of lines.
Therefore, no formal proof is possible. We also discuss whether the proof strategy in this paper has any promise for proving that the
Riemann Hypothesis is also unprovable."
"Gödel's result was a major body blow to mathematicians
everywhere. There were so many statements about numbers, and
especially prime numbers, which appeared to be true but we had no
idea how to prove. Goldbach: that every number is the sum of two
prime numbers; Twin Primes: that there are infinitely many primes
differing by 2, such as 17 and 19. Were these going to be statements
that we couldn't prove from the existing axiomatic foundations?
There is no denying that it was an unnerving state of affairs.
Maybe the Riemann Hypothesis was simply unprovable within our
current axiomatic description of what we think we mean by arithmetic.
Many mathematicians consoled themselves with the belief that anything
that is really important should be provable, that it is only tortuous
statements with no valuable mathematical content that will end up
being one of Gödel's unprovable statements.
But Gödel was not so sure. In 1951, he questioned whether
our current axioms were sufficient for many of the problems of number
theory.
*"One is faced with an infinite series of axioms which can be
extended further and further, without any end being visible...It is
true that in the mathematics of today the higher levels of this
hierarchy are practically never used...it is not altogether unlikely
that this character of present-day mathematics may have something to
do with its inability to prove certain fundamental theorems, such as,
for example, Riemann's Hypothesis."* "
from M. du Sautoy, *The Music of the
Primes* (Fourth Estate, 2003)
notes on C. Deninger's
approach to the RH and a talk of his which "gave...the impression that a proof of
the Riemann hypothesis is just around the corner..."
Louis de Branges attempts to clarify
his "proof" of the Riemann Hypothesis "...explains the mathematical motivation for
his Riemann Hypothesis proof and reveals that he proved the Bieberbach conjecture so
that he could get funding to work on the Riemann Hypothesis."
In June 2004, Louis de Branges announced
another proof of the Riemann Hypothesis. "*However*", cautions
Eric Weisstein's *Mathworld*,
"*both the 23-page preprint cited in the release (which is actually from 2003) and a longer preprint
from 2004 on de Branges's home page seem to lack an actual proof. Furthermore, a counterexample to de Branges's
approach due to Conrey and Li has been
known since 1998. The media coverage therefore appears to be much ado about nothing.*"
"Comment l'hypothèse de Riemann ne fut pas prouvee" ["How the Riemann Hypothesis was not proved" (Excerpts from two letters from P. Cartier to A. Weil, dated 12th and 15th September 1979)], *Theorie des nombres*, Semin. Delange-Pisot-Poitou, Paris 1980–81, *Prog. Math.* 22 (1982) 35–48
(April fool?)
"proof" of RH by Shalosh B. Ekhad
Bombieri's RH proof April fool
what appears to be
a sort of prophecy that the RH will be proven by 2003
J. Caveney, "TI-30XA calculator confirms proof of Riemann Hypothesis
based on ZetaGrid data"
(JC: "*No, it's not serious. But it's amusing.*")
T. McAlee's theological
"proof" of the RH
A **very** silly proof of the
RH
J. Partington, "How I Proved the Riemann Hypothesis"
(very British humour!)
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