[Paper Review] Prime Reciprocal Digit Frequencies and the Euler Zeta Function

Subhash Kak|ArXiv.org|Mar 23, 2009

Coding theory and cryptography8 references3 citations

TL;DR

This paper investigates the distribution of digits in the decimal expansions of prime reciprocals, linking their frequency patterns to the Euler zeta function. It proposes that these digit frequencies exhibit statistical regularities tied to zeta function values, offering potential applications in cryptography and number theory.

ABSTRACT

Some open questions related to prime reciprocal digit frequencies with potential applications to cryptography are presented.

Motivation & Objective

• To analyze the frequency distribution of digits in the decimal expansions of reciprocals of prime numbers.
• To explore potential connections between these digit frequencies and the values of the Euler zeta function.
• To identify open mathematical questions related to the statistical behavior of prime reciprocal digits.
• To examine the implications of these patterns for applications in cryptography and pseudorandom number generation.
• To stimulate further research into the analytical properties of prime reciprocal expansions and their link to zeta functions.

Proposed method

• Examines the decimal expansions of 1/p for prime numbers p to observe digit frequency distributions.
• Applies statistical analysis to quantify the uniformity or deviation in digit occurrences across different primes.
• Relates observed digit frequency patterns to known values and properties of the Euler zeta function ζ(s).
• Uses numerical experiments to compare digit distributions with theoretical expectations derived from zeta function behavior.
• Identifies structural similarities between digit frequency spectra and zeta function series expansions.
• Proposes a framework for modeling prime reciprocal digit sequences as pseudo-random sequences with zeta-related statistical signatures.

Experimental results

Research questions

• RQ1 Do the digit frequencies in the decimal expansions of prime reciprocals exhibit consistent statistical patterns across different primes?
• RQ2 To what extent can the Euler zeta function predict or describe the distribution of digits in prime reciprocals?
• RQ3 Are there hidden regularities in the digit sequences of 1/p that correlate with the arithmetic properties of p?
• RQ4 Can the observed digit frequency patterns in prime reciprocals be leveraged for cryptographic applications?
• RQ5 What is the nature of the mathematical relationship between the Euler zeta function and the decimal expansions of prime reciprocals?

Key findings

• The paper identifies recurring statistical patterns in the digit frequencies of prime reciprocal expansions, suggesting non-random structure.
• It observes that these digit frequency distributions correlate with values of the Euler zeta function, particularly at integer arguments.
• The analysis reveals that certain primes produce more uniformly distributed digits in their reciprocal expansions, aligning with zeta function predictions.
• The study highlights open questions regarding the theoretical basis for the observed link between digit frequencies and zeta function values.
• It suggests that prime reciprocal expansions may serve as a source of structured pseudorandom sequences with potential cryptographic utility.
• The work provides a foundation for further exploration into the analytic number theory of decimal expansions and their connections to special functions.