# ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

### ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

FYI: 'NOTES ON PAIR CORRELATION OF ZEROS AND PRIME NUMBERS' by Prof. D. A. GOLDSTON,

"These notes are based on my four lectures given at the Newton Institute in April 2004 during
the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose
is to introduce the reader to the analytic number theory necessary to understand Montgomery’s
work on the pair correlation of the zeros of the Riemann zeta-function and subsequent work on
how this relates to prime numbers. A very brief introduction to Selberg’s work on the moments
of S(T) is also given.
"

https://arxiv.org/pdf/math/0412313.pdf.

Enjoy! 'Montgomery's Pair Correlation Conjecture',

https://en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture.
Attachments "So if you could be the Devil and offer a mathematician to sell his soul for the proof of one theorem - what theorem would most mathematicians ask for? I think it would be the Riemann Hypothesis." -- Professor H. L. Montgomery.
Plotting the Pair Correlation Function for the Zeta Zeros-GUE....png (5.64 KiB) Viewed 144 times
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### Re: ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

No need to sell your soul to the devil. First acknowledge that numbers are composed of binary peices. Dimension, and value. That zero is absent value and is dimension only.

Next rework basic arithmetic accordingly, coincidentally solveing for division by zero. Then you will have your framework for zero and its relationship to primes.

I'm still working on the first part, so it's a waste of time to attempt the second part.

Until zero is defined by something other than hogwash, its relationship to primes will not be understood. Regardless of the validity of the prerequisite I set forth for zero.
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### Re: ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

FYI: 'On Montgomery's pair correlation conjecture to the zeros of Riedmann zeta function', by Prof. Pei Li,

http://summit.sfu.ca/system/files/iritems1/9751/etd1821.pdf;

'From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond' by Profs. Owen Barrett, Frank W. K. Firk, Steven J. Miller, and Caroline Turnage-Butterbaugh,

"Abstract: The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery’s work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities."

https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/NTandQM_BMTB60arxiv.pdf.
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### Re: ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

FYI: 'An introduction to random matrix theory' by Prof. Gaëtan Borot,

"These are lectures notes for a 4h30 mini-course held in Ulaanbaatar, National University of Mongolia, August 5-7th 2015, at the summer school "Stochastic Processes and Applications". It aims at presenting an introduction to basic results of random matrix theory and some of its motivations, targeted to a large panel of students coming from statistics, finance, etc. Only a small background in probability is required."

https://arxiv.org/pdf/1710.10792.pdf;

https://en.wikipedia.org/wiki/Random_matrix.
Attachments Eigenvectors of Matrices.jpg (16.65 KiB) Viewed 107 times "In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems." -- Wikipedia.
Random Matrix.gif (15.34 KiB) Viewed 107 times
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### Re: ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

Professor H. L. Montgomery's great work on the pair correlation of zeta zeros of the Riemann Zeta Function and the prime numbers is a vital gateway to the understanding of quantum physics.

All matter (mass) - energy interactions are more (useful work) or less (entropy) about nature's attempt to form more robust structures over time and over space. We have gained valuable insight into that idea via their (Montgomery and Dyson) works...

Moreover, what are the fundamental components of nature? How do they work? Why?
Attachments Montgomery's Pair Correlation Conjecture is true! Wow!
Please keep an open mind..jpg (11.67 KiB) Viewed 77 times
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### Re: ON THE PAIR CORRELATION OF ZEROS AND PRIME NUMBERS...

FYI: 'Pair correlation and twin primes revisited' by Prof. Brian Conrey and Prof. Jonathan P. Keating:

"1. Introduction:
Montgomery in his famous pair correlation paper
 used heuristics based on the Hardy–Littlewood
conjecture concerning the distribution of prime pairs 
to conclude that pairs of zeros of the Riemann zeta function have the same scaled statistics, in the limit
in which their height up the critical tends to infinity,
as pairs of eigenvalues of large random Hermitian
matrices (or of unitary matrices with Haar measure).
Montgomery did not give the details of the calculation
involving twin primes in his paper, but that calculation
has been repeated with variations several times in
the literature (e.g. [3–7]). Goldston & Montgomery 
proved rigorously that the pair correlation conjecture is
equivalent to an asymptotic formula for the variance
of the number of primes in short intervals, and
Montgomery & Soundararajan  proved that this
variance formula follows from the Hardy–
Littlewood prime-pair conjecture, under certain
assumptions.
In a slightly different vein, Bogomolny & Keating
[10,11] and later Conrey & Snaith  developed
methods to give more precise estimates for the
pair correlation (and higher correlations) of Riemann zeta zeros...
"

Remark: We strongly believe H. L. Montgomery's Pair Correlation Conjecture is true!
Attachments Plotting the Pair Correlation Function for the Zeta Zeros-GUE....png (5.64 KiB) Viewed 28 times
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