Quadratic Prime Sequences and the Critical Line: Bridging the Hardy-Littlewood Method and the Riemann Hypothesis

Executive Summary

The distribution of prime numbers within quadratic sequences has long remained one of the most challenging frontiers in analytic number theory. The research presented in arXiv:1707.06014v1 offers a sophisticated investigation into the primes represented by the polynomial n^2 + n + p, utilizing a Hardy-Littlewood circle method framework. By coupling a prime-detecting exponential sum with a quadratic theta-type sum, the source paper derives variance estimates that mirror the Barban-Davenport-Halberstam theorem for arithmetic progressions. The central analytic object is an integral over major arcs of a product of exponential sums, yielding an expected main term governed by a singular series.

The primary connection to the Riemann Hypothesis (RH) arises from the error terms in these asymptotic formulas. The fluctuations in the count of primes within these quadratic sequences are governed by the same spectral properties that determine the zero distribution of Dirichlet L-functions. Under the Generalized Riemann Hypothesis (GRH), the paper's dispersion and large-sieve inputs would upgrade from average results to pointwise asymptotics, sharpening the error terms to square-root size. This research provides a concrete bridge between the distribution of zeros on the critical line and observable fluctuations in the density of primes within specific algebraic structures, offering a novel polynomial sieve pathway toward proving RH-related bounds.

Introduction

The Riemann Hypothesis remains the central unsolved problem in number theory, asserting that the non-trivial zeros of the Riemann zeta function lie on the critical line with real part 1/2. While the distribution of primes in linear arithmetic progressions is relatively well-understood through the Prime Number Theorem, quadratic sequences represent a much deeper complexity. Hardy and Littlewood conjectured in 1923 that irreducible quadratic polynomials with no congruence obstructions should represent infinitely many primes, yet even proving this for the simplest case, n^2 + 1, remains an open challenge.

The paper arXiv:1707.06014v1 addresses this by focusing on the family f(n) = n^2 + n + p. This specific polynomial is significant because its discriminant is related to the class number of quadratic fields and the behavior of associated L-functions. The authors employ the circle method to decompose the problem into major and minor arc contributions, providing a way to estimate the density of primes by analyzing the "Major Arc" contributions dominated by a singular series. This analysis is crucial for RH research because the error terms in these sums are essentially a Fourier-Dirichlet transform of L-function data.

This article explicates how the specific exponential-sum decomposition and singular series presented in the source paper connect directly to RH mechanisms. We emphasize how zero-free regions and zero-density inputs enter the error terms and how GRH would complete the argument to a pointwise asymptotic. By turning this circle-method setup into a diagnostic for critical-line phenomena, we propose new avenues for establishing zero-free regions through the study of primes in quadratic sequences.

Mathematical Background

To establish the link between the source paper and the Riemann Hypothesis, we define the key mathematical objects. The central focus is the sum of the von Mangoldt function, Lambda(n), over the values of the polynomial n^2 + n + p. This function assigns a weight of log q to prime powers q^k and zero otherwise. The paper arXiv:1707.06014v1 defines two fundamental exponential sums:

• S1(alpha): The sum of Lambda(m) multiplied by the exponential phase e(alpha * m).
• S2(alpha): The sum of the exponential phase e(-alpha * r^2 / 4) over odd integers r.

The connection to the Riemann zeta function arises when examining the major arc contributions. Major arcs are centered around rational numbers a/q with small denominators. The behavior of these sums is determined by the distribution of primes in arithmetic progressions modulo q. According to the explicit formula in prime number theory, the count of primes in such progressions is equal to a main term minus a sum over the non-trivial zeros of the Dirichlet L-functions. If the Riemann Hypothesis holds, these zeros are restricted to the critical line, leading to an error term of approximately square-root size.

The singular series, S(kappa), arises from local quadratic congruence data. The source paper isolates a key factor, Sigma(q), which is defined using the Jacobi symbol (-kappa/q). This is the value of a quadratic Dirichlet character at q. Consequently, the fluctuations in the singular series are governed by the same analytic phenomena—such as Siegel zeros and critical-line zero distributions—that dominate the study of the Riemann Hypothesis.

Main Technical Analysis

Spectral Properties and Zero Distribution

The analysis in arXiv:1707.06014v1 relies on the singular series being a product over primes involving the quadratic character. This product is directly related to the value of the L-function L(1, chi). Under the Riemann Hypothesis, the fluctuations of these L-function values are strictly bounded, implying that primes in quadratic sequences are distributed with high regularity. The paper derives a variance estimate that mirrors the behavior of the zeta function zeros, suggesting that the "noise" in prime distribution is structured by the underlying spectral properties of the zeta function.

Sieve Bounds and Prime Density

The source paper provides a mean-square estimate for the difference between the actual prime count and the predicted main term. This result, O(y * x^2 / (log x)^B), is essentially a Bombieri-Vinogradov type result for quadratic polynomials. To achieve such a bound unconditionally, one must control the contribution of zeros that might lie close to the line Re(s) = 1. The authors' success in achieving this bound suggests that the aggregate behavior of these sequences is consistent with what would be predicted under the assumption of the Riemann Hypothesis.

Major Arc Integration and Error Analysis

The integral over the major arcs yields a main term plus an error term of size O(x^1/2 * log^c x). The appearance of the x^1/2 factor is highly suggestive, as it corresponds to the square-root cancellation expected under RH. In the context of the circle method, this term arises from the quadratic nature of the sequence, allowing for a precise stationary phase approximation. If the zeros of the zeta function were not on the critical line, resonances would occur that would significantly increase this error term, potentially invalidating the asymptotic formula.

Novel Research Pathways

1. GRH-Conditional Pointwise Asymptotics

A primary research direction is to prove a pointwise asymptotic for a fixed prime p under the assumption of the Generalized Riemann Hypothesis. By replacing the large-sieve mean bounds used in arXiv:1707.06014v1 with GRH-explicit bounds for each character, one could establish that the error term for a specific sequence is always of square-root size. This would turn the averaged theorem into a Bateman-Horn-quality statement that is experimentally testable for individual parameters.

2. Explicit Formulae for Quadratic Sequences

Another pathway involves developing an explicit formula that expresses the quadratic prime counting function directly in terms of the zeros of the associated L-functions. This would involve computing coefficients that depend on both the zeros and the polynomial discriminant. Establishing such a formula would provide a direct translation mechanism between improvements in exponential sum bounds and the location of zeros on the critical line.

3. Spectral Automorphic Enhancements

The treatment of the quadratic exponential sum can be upgraded using a modular or spectral approach. By viewing the sum as an incomplete theta series, researchers can analyze the correlations of the von Mangoldt function with theta coefficients using the spectral theory of automorphic forms. This could potentially outperform classical minor-arc bounds by exploiting cancellation across Kloosterman sums, connecting the problem to the same technology used in subconvexity studies of L-functions.

Computational Implementation

The following Wolfram Language code provides a framework to numerically test the main term suggested by the singular series and visualize the fluctuations of the error term, which are governed by the zeros of the zeta function.

(* Section: Quadratic Prime Distribution and Singular Series *)
(* Purpose: Compare the count of primes in n^2+n+p with the S(kappa) prediction *)

ClearAll[singularSeriesApprox, quadraticPrimeSum, runExperiment];

(* Approximate singular series using the quadratic-character kernel *)
singularSeriesApprox[kappa_Integer, Q_Integer] := Module[
  {term},
  term[q_] := If[OddQ[q], 
    (MoebiusMu[q]/EulerPhi[q]) * JacobiSymbol[-kappa, q], 
    0
  ];
  (* S(kappa) is roughly the product (1 - (chi(p)-1)/(p-1)) *)
  N[Product[1 - (JacobiSymbol[1 - 4*((kappa+1)/4), Prime[i]] - 1)/(Prime[i] - 1), {i, 1, Q}]]
];

(* Sum of Lambda(n^2 + n + p) *)
quadraticPrimeSum[p_Integer, x_Integer] := Sum[
  MangoldtLambda[n^2 + n + p], 
  {n, 0, x}
];

runExperiment[p_Integer, x_Integer] := Module[
  {kappa, sSeries, actualSum, mainTerm, zeros, plot},
  kappa = 4 * p - 1;
  sSeries = singularSeriesApprox[kappa, 50];
  actualSum = quadraticPrimeSum[p, x];
  mainTerm = (x/2) * sSeries;
  
  Print["Polynomial: n^2 + n + ", p];
  Print["Actual Sum: ", N[actualSum]];
  Print["Predicted Main Term: ", N[mainTerm]];
  Print["Error: ", N[actualSum - mainTerm]];

  (* RH Context: Plot Zeta on the critical line *)
  zeros = Table[Im[ZetaZero[k]], {k, 1, 10}];
  plot = Plot[Abs[Zeta[1/2 + I t]], {t, 0, 40}, 
    PlotLabel -> "Zeta Magnitude on Critical Line",
    AxesLabel -> {"t", "|Zeta|"}];
  
  Print["First 5 Zeta Zero Im-parts: ", Take[zeros, 5]];
  plot
];

(* Execute experiment for Euler's prime-generating polynomial p=41 *)
runExperiment[41, 1000]

Conclusions

The study of primes in the form n^2 + n + p as detailed in arXiv:1707.06014v1 provides a robust analytical bridge to the Riemann Hypothesis. The singular series S(kappa) and the associated major arc integrals reveal that the distribution of these primes is not merely a matter of local density, but is deeply regulated by the zeros of Dirichlet L-functions. The most promising avenue for future research lies in refining the error terms in the major arc approximations to satisfy the square-root cancellation property across a wider range of discriminants.

Specific next steps should include restating the paper's major arc bounds explicitly in terms of zero sums and optimizing the circle-method cutoffs under the assumption of GRH. Such advancements would provide a powerful new tool for attacking the Generalized Riemann Hypothesis, as the critical line of the zeta function acts as the ultimate regulator of the randomness observed in prime sequences.

References

• arXiv:1707.06014v1: On the distribution of primes of the form n^2 + n + p.
• Hardy, G. H., and Littlewood, J. E. (1923). Some problems of Partitio Numerorum III: On the expression of a number as a sum of primes.
• Iwaniec, H., and Kowalski, E. Analytic Number Theory.
• Davenport, H. Multiplicative Number Theory.