scholarly journals Riemann zeros from Floquet engineering a trapped-ion qubit

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Ran He ◽  
Ming-Zhong Ai ◽  
Jin-Ming Cui ◽  
Yun-Feng Huang ◽  
Yong-Jian Han ◽  
...  
Keyword(s):  
Zeta Function ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Paul Trap ◽  
Prime Numbers ◽  
Real Component ◽  
Trapped Ion ◽  
Periodically Driven ◽  
Riemann Zeta ◽  
High Degree

AbstractThe non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in random matrix theory and quantum chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped-ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.

The Riemann zeta-function and moment conjectures from Random Matrix Theory

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
Jörn Steuding
Keyword(s):  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Limited Range ◽  
Theory Model ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Discrete Moments

AbstractOn the basis of the Random Matrix Theory-model several interesting conjectures for the Riemann zeta-function were made during the recent past, in particular, asymptotic formulae for the 2kth continuous and discrete moments of the zeta-function on the critical line, $$ \frac{1} {T}\int\limits_0^T {|\zeta (\tfrac{1} {2} + it)|^{2k} dt} and \frac{1} {{N(T)}}\sum\limits_{0 < \gamma \leqslant {\rm T}} {|\zeta (\tfrac{1} {2} + i(\gamma + \tfrac{\alpha } {L}))|^{2k} } $$, by Conrey, Keating et al. and Hughes, respectively. These conjectures are known to be true only for a few values of k and, even under assumption of the Riemann hypothesis, estimates of the expected order of magnitude are only proved for a limited range of k. We put the discrete moment for k = 1, 2 in relation with the corresponding continuous moment for the derivative of Hardy’s Z-function. This leads to upper bounds for the discrete moments which are off the predicted order by a factor of log T.

scholarly journals On the distribution of the zeros of the derivative of the Riemann zeta-function

2014 ◽  
Vol 157 (3) ◽  
pp. 425-442 ◽  
Author(s):  
STEPHEN LESTER
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Horizontal Distribution ◽  
Characteristic Polynomials ◽  
Unitary Matrices ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Random Unitary Matrices

AbstractWe establish an asymptotic formula describing the horizontal distribution of the zeros of the derivative of the Riemann zeta-function. For ℜ(s) = σ satisfying (log T)−1/3+ε ⩽ (2σ − 1) ⩽ (log log T)−2, we show that the number of zeros of ζ′(s) with imaginary part between zero and T and real part larger than σ is asymptotic to T/(2π(σ−1/2)) as T → ∞. This agrees with a prediction from random matrix theory due to Mezzadri. Hence, for σ in this range the zeros of ζ′(s) are horizontally distributed like the zeros of the derivative of characteristic polynomials of random unitary matrices are radially distributed.

A Proof to the Riemann Hypothesis

2020 ◽  
Author(s):  
Sourangshu Ghosh
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Prime Numbers ◽  
Complex Numbers ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

In this paper, we shall try to prove the Riemann Hypothesis which is a conjecture that the Riemann zeta function hasits zeros only at the negative even integers and complex numbers with real part ½. This conjecture is very importantand of considerable interest in number theory because it tells us about the distribution of prime numbers along thereal line. This problem is one of the clay mathematics institute’s millennium problems and also comprises the 8ththe problem of Hilbert’s famous list of 23 unsolved problems. There have been many unsuccessful attempts in provingthe hypothesis. In this paper, we shall give proof to the Riemann Hypothesis.

scholarly journals Two New Equivalent Statements of Riemann Hypothesis

2020 ◽  
Author(s):  
Jamal Salah
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Univalent Functions ◽  
Prime Numbers ◽  
Regular Pattern ◽  
Complete Proof ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Interesting Solution ◽  
Shed Light

The distribution of such prime numbers among all natural numbersdoes not follow any regular pattern; however, the Germanmathematician G. F. B. Riemann (1826-1866) observed that thefrequency of prime numbers is very closely related to the behaviorof an elaborate function called the Riemann zeta function s. Thenontrivial zeroes of zeta function have 1 2 as their real part. This hasbeen checked for the first 1,500,000,000 solutions. A proof that it istrue for every interesting solution would shed light on many ofthe mysteries surrounding the distribution of prime numbers.The celebrated Riemann hypothesis remains unsolved since it wasintroduced in 1859. Miscellaneous approaches have been consideredwithout any exact and complete proof. Furthermore, some equivalentstatements have been established.In this work, we consider the famous Robin inequality and propound aconnection to the theory of univalent functions by the means of Koebefunction.

scholarly journals Sonification of the Riemann Zeta Function

2019 ◽  
Author(s):  
Nick Collins
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Prime Numbers ◽  
Prime Spectrum ◽  
Mathematical Ideas ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Rich ◽  
Infinite Sum

The Riemann zeta function is one of the great wonders of mathematics, with a deep and still not fully solved connection to the prime numbers. It is defined via an infinite sum analogous to Fourier additive synthesis, and can be calculated in various ways. It was Riemann who extended the consideration of the series to complex number arguments, and the famous Riemann hypothesis states that the non-trivial zeroes of the function all occur on the critical line 0:5 + ti, and what is more, hold a deep correspondence with the prime numbers. For the purposes of sonification, the rich set of mathematical ideas to analyse the zeta function provide strong resources for sonic experimentation. The positions of the zeroes on the critical line can be directly sonified, as can values of the zeta function in the complex plane, approximations to the prime spectrum of prime powers and the Riemann spectrum of the zeroes rendered; more abstract ideas concerning the function also provide interesting scope.

scholarly journals RIEMANN ZETA ZEROS AND PRIME NUMBER SPECTRA IN QUANTUM FIELD THEORY

2013 ◽  
Vol 28 (26) ◽  
pp. 1350128 ◽  
Author(s):  
G. MENEZES ◽  
B. F. SVAITER ◽  
N. F. SVAITER
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Zeta Function ◽  
Riemann Hypothesis ◽  
Adjoint Operator ◽  
Zeta Functions ◽  
Prime Numbers ◽  
Quantum Field ◽  
Nontrivial Zeros ◽  
Riemann Zeta

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re (s) = 1/2. Hilbert and Pólya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Using the construction of the so-called super-zeta functions or secondary zeta functions built over the Riemann nontrivial zeros and the regularity property of one of this function at the origin, we show that it is possible to extend the Hilbert–Pólya conjecture to systems with countably infinite number of degrees of freedom. The sequence of the nontrivial zeros of the Riemann zeta function can be interpreted as the spectrum of a self-adjoint operator of some hypothetical system described by the functional approach to quantum field theory. However, if one considers the same situation with numerical sequences whose asymptotic distributions are not "far away" from the asymptotic distribution of prime numbers, the associated functional integral cannot be constructed. Finally, we discuss possible relations between the asymptotic behavior of a sequence and the analytic domain of the associated zeta function.

scholarly journals The zeros of the Riemann zeta-function

1935 ◽  
Vol 151 (873) ◽  
pp. 234-255 ◽  
Author(s):  
Edward Charles Titchmarsh
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Prime Numbers ◽  
Numerical Calculations ◽  
Fundamental Part ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

1―It is well known that the distribution of the zeroes of the Riemann zeta-function ζ( s ) = ∞ Σ n=1 1/ n 8 ( s = σ + it ) plays a fundamental part in the theory of prime numbers. It was conjectured by Riemann that all the complex zeroes of ζ( s ) lie on the line σ = 1/2, but this hypothesis has never been proved or disproved. It is therefore natural to enquiry how far the hypothesis is supported by numerical calculations. The most extensive calculations of this kind have been undertaken by Gram, Backlund, and Hutchinson. The final result obtained by Hutchinson is that ζ( s ) has 138 zeroes on σ = 1/2 between t = 0 and t = 300, and no other zeroes between these values of t .

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