The 2/3 Constant: New Explicit Bounds for Prime Sums under the Riemann Hypothesis

Introduction

The distribution of prime numbers has long been the central mystery of analytic number theory. While the Prime Number Theorem describes the asymptotic density of primes, the Riemann Hypothesis (RH) provides the most refined control over the fluctuations of this density. Recent advancements in the paper arXiv:hal-02079258v1 have shifted focus toward additive prime statistics, specifically the sum of the first n primes, denoted as sigma_n.

The core problem addressed is determining the precise growth of sigma_n and its deviation from smooth approximations. By defining a normalized discrepancy sequence b_n, researchers can translate the vertical distribution of non-trivial zeros of the Riemann zeta function into explicit arithmetic bounds. This article synthesizes the technical framework of arXiv:hal-02079258v1, demonstrating how the assumption of RH leads to a stable asymptotic constant of 2/3 and provides a computational diagnostic for the critical line.

Mathematical Background

To analyze the sum of the first n primes, we define sigma_n = sum_{k=1 to n} p_k, where p_k is the k-th prime. The paper introduces the sequence b_n through the relation: sigma_n = (n^2 / 2) (log n + log log n - b_n). Historically, the behavior of this sequence was known only qualitatively, but arXiv:hal-02079258v1 provides explicit two-sided inequalities.

The analysis relies on the Chebyshev function theta(x) = sum_{p ≤ x} log p and the sum of primes pi_1(x) = sum_{p ≤ x} p. Under the Riemann Hypothesis, the error term in the prime counting function is constrained by the non-trivial zeros rho = 1/2 + i*gamma. The paper establishes a critical connection between pi_1(x) and the logarithmic integral of the squared Chebyshev function, li(theta^2(x)), which serves as a surrogate for the additive growth of primes.

Main Technical Analysis

Spectral Properties and Zero Distribution

The most striking result from the analysis is that the sequence b_n clusters around the constant 2/3. This value emerges from the second-order asymptotics of the prime sum when compared to the squared logarithmic integral. The paper demonstrates that the error in this approximation is dominated by terms of the scale x^{3/2} / log x.

A key technical component is the calculation of the constant c, which depends on the sum over the reciprocal squares of the ordinates of zeta zeros. The paper explicitly bounds the sum sum 1/gamma^2 ≤ 0.0462493. This convergence is essential; it ensures that the oscillatory components induced by the zeros on the critical line do not cause the sequence b_n to diverge. Using an integrated explicit formula, the contribution of each zero rho to the sum pi_1(x) is shown to scale with x^{(rho+2)/2}. Under RH, where Re(rho) = 1/2, these terms are bounded by x^{5/4}, allowing for the sharp constants seen in the final bounds.

Sieve Bounds and Logarithmic Comparisons

The transition from the index n to the additive size sigma_n requires precise control over logarithmic errors. The paper proves that log n and log sigma_k differ by a relative error of only O(sqrt(log sigma / sigma)). This allows for the derivation of a uniform bound for b_n for all n above a threshold n_0. The inequality b_n < 2/3 + c + 0.763 (log log n / log n) captures the primary correction factor to the classical limit. Numerical data provided in the source confirms this, with b_n values trending from 0.797 at 10² toward 0.669 at 10¹⁸.

Novel Research Pathways

1. Quantitative RH Stress Testing via Discrepancy Boundedness

A promising research direction is to formalize the boundedness of the discrepancy B(x) = (pi_1(x) - li(theta^2(x))) * (log x / x^{3/2}) as an equivalent criterion for the Riemann Hypothesis. If a zero were to exist off the critical line with Re(rho) > 1/2, it would theoretically force B(x) to grow without bound. Establishing the exact threshold for this divergence could provide a new numerical method for verifying zero-free regions.

2. r-Lambert Function Inversion for Prime Sums

The relationship between n and sigma_n is a transcendental inversion problem. Future research could utilize the r-Lambert function, the inverse of x*e^x + r*x, to provide a closed-form asymptotic for n(sigma). This would allow for the direct calculation of the number of primes needed to reach a specific sum without the need for iterative summation, incorporating the refined constants derived in arXiv:hal-02079258v1.

3. Generalization to Dirichlet L-functions

The methodology of comparing weighted prime sums to squared integral surrogates can be extended to Dirichlet L-functions. By analyzing pi_{1,chi}(x) = sum chi(p)*p, researchers could investigate if the generalized Riemann Hypothesis (GRH) imposes similar 2/3-type constants on the distribution of primes in arithmetic progressions.

Computational Implementation

(* Section: Prime Sum Analysis and b_n Convergence *)
(* Purpose: Compute b_n for large n and estimate the zeta zero sum constant *)

Module[{nMax = 1000, primes, sigmaN, bn, zeroSum, xVal},
  (* Generate cumulative sum of first n primes *)
  primes = Prime[Range[nMax]];
  sigmaN = Accumulate[primes];
  
  (* Calculate the sequence b_n as defined in arXiv:hal-02079258v1 *)
  (* Relation: sigma_n = (n^2/2)(Log[n] + Log[Log[n]] - b_n) *)
  bn = Table[
    Log[n] + Log[Log[n]] - (2 * sigmaN[[n]]) / (n^2),
    {n, 2, nMax}
  ];
  
  (* Compute the sum of 1/gamma^2 for the first 100 zeros *)
  zeroSum = N[Sum[1/Im[ZetaZero[k]]^2, {k, 1, 100}], 10];
  Print["Zero Sum Constant (Sum 1/gamma^2): ", zeroSum];
  
  (* Plot the convergence of b_n toward the 2/3 + c limit *)
  Print[ListLinePlot[bn, 
    PlotLabel -> "Convergence of Sequence b_n", 
    AxesLabel -> {"n", "b_n"}, 
    PlotRange -> {0.5, 1.0},
    GridLines -> Automatic]];
    
  (* Compare pi_1(x) with the li(theta^2) estimate for x = Prime[1000] *)
  xVal = Prime[nMax];
  Print["Actual pi_1(x): ", sigmaN[[nMax]]];
  Print["li(theta^2) Estimate: ", N[LogIntegral[ChebyshevTheta[xVal]^2]]]
]

Conclusions

The analysis of arXiv:hal-02079258v1 demonstrates that the Riemann Hypothesis dictates the fine-grained behavior of additive prime sums through the sequence b_n. By establishing explicit bounds and the 2/3 asymptotic constant, the authors provide a powerful framework for testing RH via arithmetic means. The most promising avenue for further research lies in the application of the r-Lambert function to invert prime sum relations and the expansion of these techniques to broader L-function families.

References

• arXiv:hal-02079258v1: Explicit estimates for the sum of the first n primes and the Riemann Hypothesis.
• Schoenfeld, L. (1976). Sharper bounds for the Chebyshev functions theta(x) and psi(x). II. Mathematics of Computation.
• Mezo, I. (2022). Some properties of the r-Lambert function. hal-03579652v1.