Digit Cycles and Prime Reciprocals: A Geometric Path to the Riemann Hypothesis

Executive Summary

The research paper arXiv:2105.04400v1 introduces a fascinating intersection between the decimal representation of reciprocal primes and the structural distribution of the prime number sequence. By decomposing the fraction 1/P into specific geometric progressions, the analysis reveals an underlying arithmetic symmetry that governs the period length and remainder of prime expansions. The core insight lies in the formulation of 1/P as a convergent series where the coefficients are determined by the prime's multiplicative order.

The connection to the Riemann Hypothesis (RH) emerges through the density and distribution of full reptend primes—those for which the period of the decimal expansion is maximized at P minus 1. In analytic number theory, the frequency of such primes is tied to Artin's Constant and is conditional upon the Generalized Riemann Hypothesis (GRH). This article bridges the gap between the local digit-periodicity of 1/P and the global distribution of the zeta function's zeros, proposing that fluctuations in prime density can be modeled as interference patterns within these periodic expansions.

Introduction

The study of prime numbers has long been divided between the additive properties of their distribution and the multiplicative nature of their identities. The Riemann Hypothesis, formulated in 1859, remains the most significant bridge between these perspectives, asserting that the non-trivial zeros of the Riemann zeta function lie on the critical line where the real part of s equals 1/2. While modern research often focuses on complex analysis, arXiv:2105.04400v1 redirects attention to the arithmetic structures inherent in the reciprocals of primes.

The specific problem addressed is the representation of 1/P as a periodic fraction and its decomposition into geometric series. For any prime P, the decimal expansion of 1/P displays a periodicity determined by the order of 10 modulo P. The source paper demonstrates that 1/7 can be expressed as a sum of (14 * 2^n) / 10^(2(n+1)). This is not merely an observation of base-10 arithmetic; it represents a fundamental decomposition of a prime's reciprocal into a starting value, a ratio, and a block length. By formalizing these parameters as indicators of prime density, we explore how the constraints on these sequences correlate with the bounds of the prime-counting function.

Mathematical Background

To understand the connection to the Riemann Hypothesis, we must define the core mathematical objects involved in the reciprocal expansion of primes as presented in arXiv:2105.04400v1.

The Geometric Series Representation

Any reciprocal prime 1/P can be expressed through the general formula: 1/P = sum from n=0 to infinity of (s * r^n) / (base^(L(n+1))). In this context, s is the initial significant digit block, r is the common ratio or remainder, and L is the length of the digit block. For the prime 7 in base 10, using a block length of 2, we find s = 14 and r = 2, leading to the identity 1/7 = 14 / (100 - 2).

Full Reptend Primes and Artin's Conjecture

A prime P is a full reptend prime in base 10 if the period of its decimal expansion is exactly P - 1. This occurs if and only if 10 is a primitive root modulo P. Examples include 7, 17, 19, and 23. Artin's conjecture states that the density of such primes among all primes is approximately 0.3739. Crucially, proving this conjecture depends on the Generalized Riemann Hypothesis. The frequency with which primes achieve full reptend status is thus directly linked to the distribution of zeros of Dedekind zeta functions.

Main Technical Analysis

Spectral Properties and Zero Distribution

In the expansion of 1/7, the sequence is generated by the ratio r divided by the base raised to the power L. We can view this ratio as a frequency component of the prime P. If we generalize this to all primes, each prime possesses a characteristic ratio. The distribution of these ratios as P approaches infinity provides a spectral map of the primes. The Riemann Hypothesis implies that primes are distributed with a specific form of pseudo-randomness; in the geometric series model, this randomness must manifest in the distribution of the remainders r. Any significant bias in these remainders would imply a clustering of primes that would shift the zeros of the zeta function off the critical line.

Sieve Bounds and Period Length

The length L is the smallest integer such that 10^L is congruent to r modulo P. The value of L is always a divisor of P - 1. The distribution of these divisors is a central theme in sieve theory. According to the Riemann Hypothesis, the number of primes up to x with a specific period length L should follow a distribution dictated by the density of the zeros of the zeta function. The source paper's categorization of primes into harmonic bins based on their period length provides a discrete path toward understanding the fluctuations in the prime-counting function pi(x).

Algebraic Structures and the Residue Formula

The identity P = (base^L - r) / s suggests that every prime is a factor of a near-power of the base. In the context of the Riemann Hypothesis, the zeros of the zeta function are often interpreted as the eigenvalues of a quantum operator. Here, P acts as an eigenvalue of the base-system under a transformation defined by the parameters s, r, and L. The distribution of P is constrained by the requirement that these parameters satisfy the cyclic properties of the group of integers modulo P.

Novel Research Pathways

1. Multi-Base Spectral Analysis

By computing geometric series representations across multiple bases, we can create spectral signatures for each prime. The spectral function derived from these convergence radii provides a discrete approximation to continuous measures associated with L-functions. The zeros of the zeta function should manifest as resonances in this spectral signature at specific heights on the critical line.

2. Remainder Stability and Zero Spacing

The remainder r in the source paper's formula varies for different primes. A promising research direction is to analyze the distribution of the ratio sigma = log(r) / log(base^L). We hypothesize that the distribution of sigma is statistically identical to the distribution of normalized spacings between the zeros of the Riemann zeta function, as described by the Montgomery Pair Correlation Conjecture.

Computational Implementation

The following Wolfram Language implementation calculates the parameters s, r, and L described in arXiv:2105.04400v1 and explores their relationship to the distribution of primes.

(* Section: Reciprocal Prime Geometry Analysis *)
(* Purpose: Extract s, r, L parameters and identify full reptend primes *)

Module[{primes, base = 10, maxP = 500, results},
  primes = Prime[Range[2, PrimePi[maxP]]];
  
  results = Table[
    Module[{L, s, r},
      (* Multiplicative order determines the period length L *)
      L = MultiplicativeOrder[base, p];
      (* Calculate s and r based on the paper's geometric identity *)
      s = Floor[base^L / p];
      r = Mod[base^L, p];
      {p, L, s, r}
    ],
    {p, primes}
  ];

  (* Filter for Full Reptend Primes where L == p - 1 *)
  fullReptend = Select[results, #[[2]] == #[[1]] - 1 &];

  Print["First 5 Full Reptend Primes (P, L, s, r): ", Take[fullReptend, 5]];

  (* Visualize the distribution of period lengths relative to P *)
  ListPlot[results[[All, {1, 2}]], 
    PlotLabel -> "Period Length L vs Prime P", 
    AxesLabel -> {"Prime P", "Length L"}, 
    PlotStyle -> Directive[PointSize[0.015], Blue]]
]

Conclusions

The investigation into the geometric series expansions of reciprocal primes reveals a profound structural regularity within the prime numbers. By defining primes through the parameters of starting value, remainder, and period length, we move away from seeing primes as isolated entities and toward viewing them as components of a harmonic system. The mathematical link between the period length of 1/P and the Generalized Riemann Hypothesis provides a clear path for further inquiry. The most promising avenue for further research is the spectral analysis of the remainder r across multiple bases, which may quantify the exact relationship between remainder fluctuations and the secondary terms of the prime counting function. This approach suggests that the Riemann Hypothesis is a fundamental requirement for the consistency of arithmetic expansions in periodic systems.

References

• Source Paper: arXiv:2105.04400v1
• Artin, E. (1927). Uber die Zetafunktionen gewisser algebraischer Zahlkorper.
• Hooley, C. (1967). On Artin's conjecture. Journal fur die reine und angewandte Mathematik.
• Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function.