scholarly journals Freezing transitions and extreme values: random matrix theory, and disordered landscapes

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120503 ◽  
Author(s):  
Yan V. Fyodorov ◽  
Jonathan P. Keating
Keyword(s):  
Matrix Theory ◽  
Extreme Values ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Constant Length ◽  
Energy Models ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Transition Scenario ◽  
Unitary Ensemble

We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

Number theory

2018 ◽  
Author(s):  
Thomas Spencer
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix ◽  
Riemann Zeta Function ◽  
Matrix Theory ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Eigenvalues Of Random Matrices ◽  
Areas Of Interest

This article examines some of the connections between random matrix theory (RMT) and number theory, including the modelling of the value distributions of the Riemann zeta function and other L-functions as well as the statistical distribution of their zeros. Number theory has been used in RMT to address seemingly disparate questions, such as modelling mean and extreme values of the Riemann zeta function and counting points on curves. One thing in common among the applications of RMT to number theory is the L-function. The statistics of the critical zeros of these functions are believed to be related to those of the eigenvalues of random matrices. The article first considers the truth of the generalized Riemann hypothesis before discussing the values of the Riemann zeta function, the values of L-functions, and further areas of interest with respect to the connections between RMT and number theory

scholarly journals Maxima of log-correlated fields: some recent developments*

2022 ◽  
Vol 55 (5) ◽  
pp. 053001
Author(s):  
E C Bailey ◽  
J P Keating
Keyword(s):  
Recent Progress ◽  
Symmetric Functions ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Compact Groups ◽  
Gaussian Fields ◽  
The Riemann Zeta Function ◽  
Recent Developments ◽  
Explicit Formulae ◽  
Riemann Zeta

Abstract We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.

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 Extreme values of the Riemann zeta function and its argument

2018 ◽  
Vol 372 (3-4) ◽  
pp. 999-1015 ◽  
Author(s):  
Andriy Bondarenko ◽  
Kristian Seip
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Extreme Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Jensen polynomials for the Riemann zeta function and other sequences

2019 ◽  
Vol 116 (23) ◽  
pp. 11103-11110 ◽  
Author(s):  
Michael Griffin ◽  
Ken Ono ◽  
Larry Rolen ◽  
Don Zagier
Keyword(s):  
Partition Function ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
General Theorem ◽  
Hermite Polynomials ◽  
Gaussian Unitary Ensemble ◽  
Random Matrix Model ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Unitary Ensemble

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This hyperbolicity has been proved for degrees d≤3. We obtain an asymptotic formula for the central derivatives ζ(2n)(1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d≤8. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.

scholarly journals Statistics of the largest geomagnetic storms per solar cycle (1844-1993)

1997 ◽  
Vol 15 (6) ◽  
pp. 719-728 ◽  
Author(s):  
D. M. Willis ◽  
P. R. Stevens ◽  
S. R. Crothers
Keyword(s):  
Statistical Analysis ◽  
Solar Cycle ◽  
Geomagnetic Storm ◽  
Extreme Values ◽  
Geomagnetic Storms ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Relative Dispersion ◽  
Solar Cycles ◽  
Aa Index

Abstract. A previous application of extreme-value statistics to the first, second and third largest geomagnetic storms per solar cycle for nine solar cycles is extended to fourteen solar cycles (1844–1993). The intensity of a geomagnetic storm is measured by the magnitude of the daily aa index, rather than the half-daily aa index used previously. Values of the conventional aa index (1868–1993), supplemented by the Helsinki Ak index (1844–1880), provide an almost continuous, and largely homogeneous, daily measure of geomagnetic activity over an interval of 150 years. As in the earlier investigation, analytic expressions giving the probabilities of the three greatest storms (extreme values) per solar cycle, as continuous functions of storm magnitude (aa), are obtained by least-squares fitting of the observations to the appropriate theoretical extreme-value probability functions. These expressions are used to obtain the statistical characteristics of the extreme values; namely, the mode, median, mean, standard deviation and relative dispersion. Since the Ak index may not provide an entirely homogeneous extension of the aa index, the statistical analysis is performed separately for twelve solar cycles (1868–1993), as well as nine solar cycles (1868–1967). The results are utilized to determine the expected ranges of the extreme values as a function of the number of solar cycles. For fourteen solar cycles, the expected ranges of the daily aa index for the first, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude, contrary to the situation for the half-daily aa index over nine solar cycles. The observed range of the first extreme daily aa index for fourteen solar cycles is 159–352 nT and for twelve solar cycles is 215–352 nT. In a group of 100 solar cycles the expected ranges are expanded to 137–539 and 177–511 nT, which represent increases of 108% and 144% in the respective ranges. Thus there is at least a 99% probability that the daily aa index will satisfy the condition aa < 550 for the largest geomagnetic storm in the next 100 solar cycles. The statistical analysis is used to infer that remarkable conjugate auroral observations on the night of 16 September 1770, which were recorded during the first voyage of Captain Cook to Australia, occurred during an intense geomagnetic storm.

scholarly journals Extreme Values of the Riemann Zeta Function on the 1-Line

2017 ◽  
Vol 2019 (22) ◽  
pp. 6924-6932 ◽  
Author(s):  
Christoph Aistleitner ◽  
Kamalakshya Mahatab ◽  
Marc Munsch
Keyword(s):  
Lower Bound ◽  
Extreme Values ◽  
Mean Square ◽  
Self Similarity ◽  
New Variant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean ◽  
Resonator Method ◽  
Optimal Lower Bound

Abstract We prove that there are arbitrarily large values of t such that $|\zeta (1+it)| \geq e^{\gamma } (\log _{2} t +\log _{3} t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the “long resonator” method. While earlier implementations of this method crucially relied on a “sparsification” technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.

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.

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