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Correlation means that there is a relationship between two or more variables (such as ice cream consumption and crime), but this relationship does not necessarily imply cause and effect. When two variables are correlated, it simply means that as one variable changes, so does the other. We can measure correlation by calculating a statistic known as a correlation coefficient. A correlation coefficient is a number from -1 to +1 that indicates the strength and direction of the relationship between...
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Pair correlation and twin primes revisited.

Brian Conrey1, Jonathan P Keating2

  • 1American Institute of Mathematics, 600 East Brokaw Road, San Jose, CA 95112, USA; School of Mathematics, University of Bristol, Bristol BS8 1TW, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|November 16, 2016
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We connect the Riemann zeta-function ratios conjecture to arithmetic function correlations. Proving these conjectures imply the same result offers new insights into the ratios conjecture’s foundations.

Keywords:
pair correlationrandom matrix theorytwin primes

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Area of Science:

  • Number Theory
  • Analytic Number Theory
  • Riemann Zeta-Function

Background:

  • The Riemann zeta-function ratios conjecture is a key area of research in analytic number theory.
  • Previous motivations for the ratios conjecture included analogies with random matrix theory and heuristic recipes.
  • Understanding the arithmetic correlations conjecture is also of significant interest.

Purpose of the Study:

  • To establish a novel connection between the Riemann zeta-function ratios conjecture and a conjecture on arithmetic function correlations.
  • To provide new theoretical underpinnings for the ratios conjecture.
  • To explore the implications of these conjectures for number theory.

Main Methods:

  • Establishing a formal mathematical link between two distinct conjectures in number theory.
  • Utilizing proof techniques to demonstrate that the two conjectures imply identical outcomes.
  • Analyzing the structural relationships between the Riemann zeta-function and arithmetic functions.

Main Results:

  • A definitive connection is established between the two-over-two ratios formula for the Riemann zeta-function and arithmetic function correlations.
  • It is proven that the ratios conjecture and the arithmetic correlations conjecture yield the same result.
  • This finding offers a new perspective on the theoretical basis of the ratios conjecture.

Conclusions:

  • The study provides a significant advancement in understanding the Riemann zeta-function ratios conjecture.
  • The established connection deepens the theoretical framework supporting the ratios conjecture.
  • This work opens new avenues for research at the intersection of analytic number theory and arithmetic functions.