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A stochastic interpretation of the Riemann zeta function.

K S Alexander1, K Baclawski, G C Rota

  • 1Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA.

Proceedings of the National Academy of Sciences of the United States of America
|January 15, 1993
PubMed
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We introduce a novel stochastic process where the Riemann zeta function

Area of Science:

  • Number Theory
  • Probability Theory
  • Stochastic Processes

Background:

  • The Riemann zeta function is a complex function of profound importance in number theory.
  • Understanding its properties often involves advanced analytical techniques.
  • Connecting number theory concepts to probability theory remains an active area of research.

Purpose of the Study:

  • To establish a novel stochastic process.
  • To demonstrate that the terms of the Riemann zeta function emerge as probability distributions.
  • To bridge number theoretic functions with probabilistic frameworks.

Main Methods:

  • Construction of a specific stochastic process.
  • Identification of elementary random variables within this process.

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  • Analysis of the probability distributions associated with these random variables.

    • Main Results:

    • The probability distributions of the elementary random variables are shown to be the terms of the Riemann zeta function.
    • This provides a probabilistic interpretation for the values of the Riemann zeta function.

    Conclusions:

    • A new stochastic framework is presented for studying the Riemann zeta function.
    • This approach offers a potentially intuitive way to explore the properties of the Riemann zeta function.
    • The findings open avenues for interdisciplinary research between number theory and probability.