Some updates about Riemann hypothesis

By on 30/04/2017
20170424-riemann_critical_strip

The following article is an updated version of a post previously published on DocMadhattan.


The Rieman hypothesis was stated following the 1859 Riemann’s paper On the Number of Primes Less Than a Given Magnitude. This is the begin of the paper [1]Translated by David R. Wilkins (pdf):

I believe that I can best convey my thanks for the honour which the Academy has to some degree conferred on me, through my admission as one of its correspondents, if I speedily make use of the permission thereby received to communicate an investigation into the accumulation of the prime numbers; a topic which perhaps seems not wholly unworthy of such a communication, given the interest which Gauss and Dirichlet have themselves shown in it over a lengthy period.
For this investigation my point of departure is provided by the observation of Euler that the product
\[\prod \frac{1}{1-\frac{1}{p^s}} = \sum \frac{1}{n^s}\] if one substitutes for $p$ all prime numbers, and for $n$ all whole numbers. The function of the complex variable $s$ which is represented by these two expressions, wherever they converge, I denote by $\zeta (s)$. Both expressions converge only when the real part of $s$ is greater than 1; at the same time an expression for the function can easily be found which always remains valid.

20170424-riemann_critical_strip

The Riemann zeta function is connected to the prime numbers distribution, in particular Riemann argued that all of its non trivial zeros [2]In this case, for trivial zero I intend a negative even integer number. have the form $z = \frac{1}{2} + bi$, where $z$ is complex, $b$real,$i = \sqrt{-1}$. There’s also a general form of the zeros: $z = \sigma + bi$, where $\sigma$ belong to the critical strip (see below and the image at the right).
In the story of the search of the zeta-zeros, Hugh Montgomery has an important part [3]Montgomery, Hugh L. “The pair correlation of zeros of the zeta function.” In Proc. Symp. Pure Math, vol. 24, pp. 181-193. 1973. (pdf): in 1972 he investigated the distance between two zeta-zeros, finding a function of this difference. After this paper, in 1979, with Norman Levinson [4]Levinson, Norman, and Hugh L. Montgomery. “Zeros of the derivatives of the Riemann zeta-function.” Acta mathematica 133, no. 1 (1974): 49-65. doi:10.1007/BF02392141 he established some others zeta properties, investigating in particular the zeros of zeta derivatives. Obviosly he first of all proofed an equivalence relation between the zeros of Riemann zeta function and the zeros of the derivatives: in particular also these zeros belong to the critical strip, $0 < \sigma < \frac{1}{2}$.
The analitical research around zeta-zeros is not the only way: the first was Lehmer (1956 and 1957) who performed the first computational attempt in order to proof the hypothesis. An example of this kind of researches is given by Richard Brent [5]Brent, Richard P. “On the zeros of the Riemann zeta function in the critical strip.” Mathematics of Computation 33, no. 148 (1979): 1361-1372. doi:10.1090/S0025-5718-1979-0537983-2: in his work he try to evaluate Riemann zeta using the Gram points, that are the points in which the zeta change its sign [6]We can also define a Gram point as the value of $b$ such that
\[\zeta \left ( \frac{1}{2} + bi \right ) = Z(b) e^{-i \theta (b)}\] is a non-zero real.
In the equation
\[\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} – \frac{t}{2} – \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots\] is the Riemann-Siegel theta function and $Z (t)$is the Hardy Z-function.
. Brent focused his research on the first 70000000 Gram blocks, veryfing the hypothesis.
But there’s another approach to the problem: physics. In the end of 1990s Alain Connes [7]Connes, Alain. “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.” Selecta Mathematica, New Series 5, no. 1 (1999): 29-106. doi:10.1007/s000290050042 (arXiv) proofed the link of Rieman hypotesis with quantum chaos.
Quantum chaos studies chaotic classical dynamical systems using quantum laws. In particular Connes found a particular chaotic system in which quantum numbers are prime numbers and the energy levels of the system correspond to the zeta-zeros on the critical line $\sigma = \frac{1}{2}$. In physics it could be the better (but not only) suspect to resolve the hypothesis.
So it is interesting to observe the recently paper [8]Thanks to Roberto Natalini for the suggestions about this second part of the article published on Physical Review Letters [9]Bender, Carl M., Dorje C. Brody, and Markus P. Müller. “Hamiltonian for the zeros of the Riemann zeta function.” Physical Review Letters 118, no. 13 (2017): 130201. doi:10.1103/PhysRevLett.118.130201: starting from the Hilbert-Pólya conjecture, Carl Bender, Dorje Brody and Markus Müller proposed a new non-hermitian hamiltonian $\hat H$ which eigenvalues exactly correspond to the non-trivial zeros of the Riemann’s function.
This result isn’t conclusive. Indeed the three authors write in the conclusion:

We hope that further analysis of the properties of $\hat H$, such as identifying its domain and establishing its self-adjointness, will prove the reality of the eigenvalues, and thus the veracity of the Riemann hypothesis

At the other hand, Jean Bellissard (Georgia Tech) writes a breaf comment about Bender, Brody and Müller’s paper, finding some problems:

  1. the space of some mathematical objects is not well defined;
  2. the operator $\hat p$ proposed in the paper doesn’t admits a selfadjoint extension, so it hasn’t the characteristics required by the authors;
  3. the eigenvalues don’t occur on the line of the solution of the Riemann hypotesis.

As attractive this idea looks, it does not hold when checking the analysis part of the problem.

Another interesting analysis of the same paper is written by Alessandro Zaccagni for the italian site MaddMaths! (Google translation).
So we can only wait for further results. In the meanwhile I suggest the following review: Physics of the Riemann Hypothesis by Daniel Schumayer and David A. W. Hutchinson.


Read also: Wikipedia, The Clay Mathematics Institute, MathWorld

References   [ + ]

1. Translated by David R. Wilkins (pdf)
2. In this case, for trivial zero I intend a negative even integer number.
3. Montgomery, Hugh L. “The pair correlation of zeros of the zeta function.” In Proc. Symp. Pure Math, vol. 24, pp. 181-193. 1973. (pdf)
4. Levinson, Norman, and Hugh L. Montgomery. “Zeros of the derivatives of the Riemann zeta-function.” Acta mathematica 133, no. 1 (1974): 49-65. doi:10.1007/BF02392141
5. Brent, Richard P. “On the zeros of the Riemann zeta function in the critical strip.” Mathematics of Computation 33, no. 148 (1979): 1361-1372. doi:10.1090/S0025-5718-1979-0537983-2
6. We can also define a Gram point as the value of $b$ such that
\[\zeta \left ( \frac{1}{2} + bi \right ) = Z(b) e^{-i \theta (b)}\] is a non-zero real.
In the equation
\[\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} – \frac{t}{2} – \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots\] is the Riemann-Siegel theta function and $Z (t)$is the Hardy Z-function.
7. Connes, Alain. “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function.” Selecta Mathematica, New Series 5, no. 1 (1999): 29-106. doi:10.1007/s000290050042 (arXiv)
8. Thanks to Roberto Natalini for the suggestions about this second part of the article
9. Bender, Carl M., Dorje C. Brody, and Markus P. Müller. “Hamiltonian for the zeros of the Riemann zeta function.” Physical Review Letters 118, no. 13 (2017): 130201. doi:10.1103/PhysRevLett.118.130201

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