1,862 research outputs found
Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros
Odlyzko has computed a data set listing more than successive Riemann
zeros, starting at a zero number beyond . The data set relates to
random matrix theory since, according to the Montgomery-Odlyzko law, the
statistical properties of the large Riemann zeros agree with the statistical
properties of the eigenvalues of large random Hermitian matrices. Moreover,
Keating and Snaith, and then Bogomolny and collaborators, have used random unitary matrices to analyse deviations from this law. We contribute
to this line of study in two ways. First, we point out that a natural process
to apply to the data set is to thin it by deleting each member independently
with some specified probability, and we proceed to compute empirical two-point
correlation functions and nearest neighbour spacings in this setting. Second,
we show how to characterise the order correction term to the spacing
distribution for random unitary matrices in terms of a second order
differential equation with coefficients that are Painlev\'e transcendents, and
where the thinning parameter appears only in the boundary condition. This
equation can be solved numerically using a power series method. Comparison with
the Riemann zero data shows accurate agreement.Comment: 22 pages, 10 figures, Version 2 added some new references in
bibliography, Version 3 corrected the scaling on the spacing distribution and
some typo
Linear law for the logarithms of the Riemann periods at simple critical zeta zeros
Each simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn).
Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis
Gaps between zeros of the derivative of the Riemann \xi-function
Assuming the Riemann hypothesis, we investigate the distribution of gaps
between the zeros of \xi'(s). We prove that a positive proportion of gaps are
less than 0.796 times the average spacing and, in the other direction, a
positive proportion of gaps are greater than 1.18 times the average spacing. We
also exhibit the existence of infinitely many normalized gaps smaller (larger)
than 0.7203 (1.5, respectively).Comment: 15 page
Linear law for the logarithms of the Riemann periods at simple critical zeta zeros
Each simple zero 1/2 + iγn of the Riemann zeta function on the critical line with γn > 0 is a center for the flow s˙ = ξ(s) of the Riemann xi function with an associated period Tn. It is shown that, as γn →∞, log Tn ≥ π/4 γn + O(log γn).
Numerical evaluation leads to the conjecture that this inequality can be replaced by an equality. Assuming the Riemann Hypothesis and a zeta zero separation conjecture γn+1 − γn≥ γn-θ for some exponent θ > 0, we obtain the upper bound log Tn ≤ γn2 + θ Assuming a weakened form of a conjecture of Gonek, giving a bound for the reciprocal of the derivative of zeta at each zero, we obtain the expected upper bound for the periods so, conditionally, log Tn = π/ 4 γn +O(log γn). Indeed, this linear relationship is equivalent to the given weakened conjecture, which implies the zero separation conjecture, provided the exponent is sufficiently large. The frequencies corresponding to the periods relate to natural eigenvalues for the Hilbert–Polya conjecture. They may provide a goal for those seeking a self-adjoint operator related to the Riemann hypothesis
Roots of the derivative of the Riemann zeta function and of characteristic polynomials
We investigate the horizontal distribution of zeros of the derivative of the
Riemann zeta function and compare this to the radial distribution of zeros of
the derivative of the characteristic polynomial of a random unitary matrix.
Both cases show a surprising bimodal distribution which has yet to be
explained. We show by example that the bimodality is a general phenomenon. For
the unitary matrix case we prove a conjecture of Mezzadri concerning the
leading order behavior, and we show that the same follows from the random
matrix conjectures for the zeros of the zeta function.Comment: 24 pages, 6 figure
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Eigenvalue Density, Li’s Positivity, and the Critical Strip
We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and also offer a physical interpretation of the result and discuss the Hilbert-Polya approach
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