Will a physicist prove the Riemann hypothesis?

Wolf, Marek

doi:10.1088/1361-6633/ab3de7

Cited by 23 publications

(17 citation statements)

References 146 publications

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“…where v = (pq) 2/3 t −1 . 4 More precisely, minimally we need q = t. Although p can be arbitrary for this case, the index is known to be independent of p.…”

Section: N = 4 Sym and Riemann Hypothesismentioning

confidence: 99%

“…While this is originally a purely mathematical problem, its physical realization -since the Hilbert-Pólya conjecture [1,2] -has been discussed in various problems of physics (see e.g. review [3][4][5]), including quantum mechanics, statistical mechanics, random matrix theory and string theory [6][7][8][9][10]. In this paper, we point out new relations among the Riemann hypothesis, string theory and gauge theory.…”

Section: Introductionmentioning

confidence: 99%

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String theory, $\mathcal{N}=4$ SYM and Riemann hypothesis

Honda¹,

Yoda²

2022

Preprint

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We discuss new relations among string theory, four-dimensional N = 4 supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis. It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function σ(n). Based on previous results in literature, we focus on the fact that σ(n) appears in a problem of counting supersymmetric states in the N = 4 SYM with SU (3) gauge group: the Schur limit of the superconformal index plays a role of a generating function of σ(n). Then assuming the Riemann hypothesis gives bounds on information on the 1/8-BPS states in the N = 4 SYM. The AdS/CFT correspondence further connects the Riemann hypothesis to the type IIB superstring theory on AdS5 ×S 5 . In particular, the Riemann hypothesis implies a miraculous cancellation among Kaluza-Klein modes of the supergravity multiplet and D3-branes wrapping supersymmetric cycles in the string theory. We also discuss possibilities to gain new insights on the Riemann hypothesis from the physics side.YITP-22-30, KUNS2920 * masazumi.honda(at)yukawa.kyoto-u.ac.jp † t.yoda(at)gauge.scphys.kyoto-u.ac.jp

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“…where v = (pq) 2/3 t −1 . 4 More precisely, minimally we need q = t. Although p can be arbitrary for this case, the index is known to be independent of p.…”

Section: N = 4 Sym and Riemann Hypothesismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

String theory, $\mathcal{N}=4$ SYM and Riemann hypothesis

Honda¹,

Yoda²

2022

Preprint

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“…This approach has attracted for many years the attention of theoretical physicists, for the simple reason that it is deeply related to the spectral theory of quantum mechanics. Originally stated by Pólya and Hilbert around 1910, this approach has given rise to an important series of works on the Riemann ζ-function by Berry, Keating, Connes, Sierra, Srednicki, Bender and many others [16][17][18][19][20][21][22][23][24][25][26] (for a more complete list of references, see the reviews [27,28]). In a nutshell, this approach can be iconically rephrased as…”

Section: A Quantum Approachmentioning

confidence: 99%

Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis

Mussardo,

LeClair

2021

Preprint

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The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function M (x) = x k=1 µ(k), where µ(k) is the Möbius coefficient of the integer k: the RH is indeed true if the Mertens function goes asymptotically as M (x) x 1/2+ . We show that this behavior can be established on the basis of a new probabilistic approach based on the global properties of Mertens function. To this aim, we focus the attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős-Kac theorem for square-free numbers, etc. These results lead us to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk, therefore with an asymptotic behaviour given by x 1/2+ . This represents a theoretical advance in the field. We also argue how the Riemann Hypothesis implies the Generalised Riemann Hypothesis for the Dirichlet L-functions. Next we study the local properties of the Mertens function dictated by the Möbius coefficients restricted to the square-free numbers. Motivated by the natural curiosity to see how close to a purely random walk is any sub-sequence extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity: together with several frequency tests within a block, the list of our tests include those for the longest run of ones in a block, the binary matrix rank test, the Discrete Fourier Transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc. The successful outputs of all these tests (with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive "experimental" confirmations of the brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however ridiculously improbable.

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“…When J. Derbyshire asked A. Odlyzko about his opinion on the validity of RH he replied "Either it's true, or else it isn't" [11, p. 357-358]. There were some attempts to prove RH using the physical methods, see [31] or [39].…”

Section: Introductionmentioning

confidence: 99%

The Disproof of the Riemann Hypothesis

Dumitrescu,

Wolf

2021

Preprint

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Will a physicist prove the Riemann hypothesis?

Cited by 23 publications

References 146 publications

String theory, $\mathcal{N}=4$ SYM and Riemann hypothesis

String theory, $\mathcal{N}=4$ SYM and Riemann hypothesis

Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis

The Disproof of the Riemann Hypothesis

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