The Arithmetic Rhythm of Prime Sums: Connecting Goldbach Convolutions to the Critical Line

Introduction

The study of additive prime number theory has long sought to reconcile the additive structure of the integers with the multiplicative nature of the primes. Central to this field is the Goldbach conjecture, which posits that every even integer greater than two can be expressed as the sum of two primes. While the binary Goldbach conjecture remains one of the most famous unsolved problems in mathematics, analytic number theorists have made significant progress by studying the average behavior of the Goldbach generating function.

The research paper arXiv:hal-00289430 provides a profound investigation into the summatory function of Goldbach weights. Specifically, it examines the function G(n), defined via the von Mangoldt function Λ(n), which captures the number of ways to write n as a sum of two primes or prime powers. The paper focuses on the asymptotic expansion of the sum of G(n) for n up to x, revealing a main term of x^2 / 2 and a secondary term H(x) that fluctuates based on the distribution of the non-trivial zeros of the Riemann zeta function ζ(s).

The significance of this work lies in its explicit connection between the error terms of additive problems and the Riemann Hypothesis (RH). By employing the circle method and Gallagher-type large sieve density estimates, the analysis demonstrates that fluctuations in the Goldbach sum are not merely random noise but are fundamentally constrained by the location of the zeros of ζ(s) in the critical strip. This provides a bridge between the local behavior of prime distributions and the global analytic properties of the zeta function.

Mathematical Background

To analyze the summatory behavior of Goldbach's function, we define several key mathematical objects. Let Λ(n) be the von Mangoldt function, which is log p if n is a power of a prime p, and zero otherwise. The Goldbach generating function G(n) is defined as the additive self-convolution:

G(n) = ∑ Λ(k)Λ(l) where the sum is over k + l = n.

The primary object of study in arXiv:hal-00289430 is the summatory function Φ(x) = ∑ G(n) for n ≤ x. Using the Circle Method, this sum can be expressed as an integral over the unit interval: ∫ T(-α) S(α)^2 dα, where T(α) is a sum over integers and S(α) is an exponential sum weighted by the von Mangoldt function.

The paper decomposes S(α) into a main term T(α) and a remainder R(α). This expansion reveals that Φ(x) is approximately x^2 / 2 plus a term H(x). This H(x) is directly related to the error term in the Prime Number Theorem: ψ(x) - x, which is governed by the non-trivial zeros ρ = σ + iγ of the Riemann zeta function.

Main Technical Analysis

Spectral Properties and the Identity of Integrated Prime Errors

The technical core of the analysis revolves around the precise estimation of the remainder terms. The paper establishes the following asymptotic relation:

∑ G(n) = x^2 / 2 + H(x) + O(x log^5 x)

The term H(x) arises from the interaction of the main signal with the fluctuations of the primes. The authors derive a crucial identity that links H(x) directly to the summatory error of the von Mangoldt function:

H(x) = 2 ∑ (ψ(n) - n) for n ≤ x - 1.

Under the Riemann Hypothesis, the discrepancy ψ(n) - n is bounded by O(n^1/2 log^2 n). Consequently, the oscillations in H(x) would be of the order x^3/2. If RH is false, any zero with a real part σ > 1/2 would force H(x) to grow at a significantly faster rate, making the Goldbach sum highly sensitive to the horizontal distribution of zeta zeros.

Sieve Bounds and the Ω-Result

The paper utilizes Gallagher's large sieve density estimates near the line σ = 1 to control character sums. A significant finding is the Ω-result, which shows that the error term cannot be made arbitrarily small:

Φ(x) - x^2 / 2 - H(x) = Ω(x log log x)

This result is derived by considering values of n that are highly composite (divisible by many small primes). For such n, G(n) exhibits spikes that prevent a smoother asymptotic fit. This suggests that even if the Riemann Hypothesis were proven, the discrete nature of prime distribution would still enforce a minimum level of irregularity in additive problems.

Novel Research Pathways

1. Explicit Formula Expansion for H(x)

One promising direction is the application of the explicit formula for ψ(x) to H(x) to identify specific harmonics associated with zeta zeros. By representing H(x) as a sum over ρ, researchers could potentially prove that RH is equivalent to a specific growth bound on the Goldbach error term, such as O(x^1.5 + ε).

2. Hybrid Sieve-Analytic Methods

Combining sieve weights with the circle method analysis could yield new bounds for primes in arithmetic progressions. By studying a sieve-weighted version of G(n), it may be possible to reduce the impact of arithmetic spikes and reveal the underlying analytic oscillations driven by the zeta function more clearly.

Computational Implementation

The following Wolfram Language code demonstrates the relationship between the summatory Goldbach function and the fluctuations predicted by the zeros of the Riemann zeta function.

(* Section: Goldbach Summatory Function and Zeta Fluctuations *)
(* Purpose: Compare the error in G(n) sums with zeta-zero-based H(x) *)

ClearAll[vonMangoldt, goldbachG, sumGoldbach, zetaH];

(* Define the von Mangoldt function *)
vonMangoldt[n_] := If[n <= 1, 0, Log[FactorInteger[n][[1, 1]]] * 
    If[Length[FactorInteger[n]] == 1, 1, 0]];

(* Compute G(n) as the convolution of von Mangoldt weights *)
goldbachG[n_] := Sum[vonMangoldt[k] * vonMangoldt[n - k], {k, 1, n - 1}];

(* Summatory function of G(n) *)
sumGoldbach[x_] := Sum[goldbachG[n], {n, 1, x}];

(* Approximation of H(x) using the first N non-trivial zeta zeros *)
zetaH[x_, numZeros_] := Module[{zeros, rho, t},
  zeros = Table[ZetaZero[k], {k, 1, numZeros}];
  -2 * Re[Sum[
    rho = zeros[[i]];
    (x^(rho + 1) / (rho * (rho + 1))),
    {i, 1, numZeros}
  ]]
];

(* Generate comparison data for x up to 150 *)
With[{limit = 150, nZeros = 25},
  data = Table[{x, sumGoldbach[x] - x^2/2}, {x, 10, limit, 5}];
  approx = Table[{x, zetaH[x, nZeros]}, {x, 10, limit, 5}];
  
  ListLinePlot[{data, approx}, 
    PlotLegends -> {"Actual Error", "Zeta Zero Model (H(x))"},
    AxesLabel -> {"x", "Fluctuation Value"},
    PlotLabel -> "Goldbach Sum Error vs Zeta Zero Explicit Formula",
    PlotTheme -> "Scientific"]
]

Conclusions

The analysis of arXiv:hal-00289430 confirms that the Goldbach conjecture is not an isolated problem of arithmetic but is a single facet of the grander mystery of the Riemann Hypothesis. The identification of H(x) as the integrated error of the prime number theorem provides a rigorous technical bridge between additive number theory and the spectral properties of the zeta function. While the Ω-term highlights inherent arithmetic obstructions, the most promising avenue for further research lies in refining the error terms through improved zero-free regions for Dirichlet L-functions.

References

• arXiv:hal-00289430: Bhowmik, G., & Schlage-Puchta, J.-C. "Goldbach numbers in arithmetic progressions."
• Gallagher, P. X. (1970). "A Large Sieve Density Estimate near σ = 1." Invent. Math. 11, 329-339.
• Fujii, A. (1991). "An additive problem of prime numbers." Acta Arithmetica, 58, 173-179.