A Comprehensive Theoretical Investigation of a Modular Aliasing Signature (MAS)-like Term in the Goldbach Counting Function \(R_2(n)\) and its Implications for the Strong Goldbach Conjecture

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Abstract (Click to Expand/Collapse)

This comprehensive report consolidates the findings of a three-phase theoretical investigation into a hypothesized structured error term, termed the Modular Aliasing Signature (MAS)-like term, within the error term \(E(n)\) of the Goldbach counting function \(R_2(n) = \sum_{k_1+k_2=n} \Lambda(k_1)\Lambda(k_2)\). The study aimed to derive this term's structure, refine its coefficient, and assess its implications for the Strong Goldbach Conjecture (SGC). Phase 1 employed the Hardy-Littlewood circle method to demonstrate that major arc contributions can indeed produce an MAS-like term of the form \(E_{\text{MAS-like}}(n) \approx K_0 \cdot \chi_X(n) \cdot X(n)\), arising from specific interactions involving an off-critical-line zero \(\rho_0 = \sigma_0+it_0\) (\(\sigma_0 > 1/2\)) of a Dirichlet L-function \(L(s,\chi_X)\), where \(\chi_X\) is a real, non-principal, primitive character modulo \(k\), and \(X(n) = \operatorname{Re} \left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right)\). Phase 2 focused on the theoretical refinement of the coefficient \(K_0\). Initial derivations based on specific major arc terms led to forms like \(K_0 \approx -2 \mu(k)k/\varphi(k)^2 \cdot \prod_{p \nmid k} (1 + \frac{\chi_X(p)}{(p-1)^2})\), which presented issues for non-square-free \(k\) and predicted vanishing if \(\chi_X(2)=-1\) (for odd \(k\)). An alternative, \(K_0 = \operatorname{Re} \left( -2 \frac{\chi_X(-1)\tau(\chi_X)}{\varphi(k)} \right)\), proposed based on analogies with established explicit formulas, robustly handles all \(k\) and predicts \(K_0=0\) if \(\chi_X(-1)=-1\). The definitive determination of \(K_0\) for \(R_2(n)\) was identified as a critical area requiring further validation from authoritative literature or more exhaustive derivations. Qualitatively, if \(K_0 \neq 0\), the MAS-like term's \(n^{\sigma_0}\) growth makes it dominant over contributions from on-critical-line zeros but still \(o(n)\) for \(\sigma_0 < 1\). Phase 3 analyzed the implications for SGC. Given that established zero-free regions ensure \(\sigma_0 < 1\), the MAS-like term is \(o(n)\) and therefore cannot, by itself, cause an asymptotic failure of SGC for large \(n\) (as \(R_2(n) = \mathfrak{S}(n)n + o(n) > 0\)). The primary significance of identifying the MAS-like term (if \(K_0 \neq 0\)) lies in providing a more detailed structural understanding of the components within the \(o(n)\) error term \(E(n)\). The report concludes by emphasizing that the validation of the precise form and vanishing conditions of \(K_0\) is paramount for any quantitative application of this MAS framework to the SGC problem.

Table of Contents

1. Introduction

The Strong Goldbach Conjecture (SGC), one of the oldest and most profound unsolved problems in number theory, asserts that every even integer \(n \geq 4\) can be expressed as the sum of two prime numbers. This conjecture is often studied analytically via the weighted sum over prime powers, \(R_2(n) = \sum_{k_1+k_2=n} \Lambda(k_1)\Lambda(k_2)\), where \(\Lambda\) is the von Mangoldt function. The Hardy-Littlewood circle method provides a powerful analytical tool, predicting that \(R_2(n) = M_2(n) + E(n)\). The main term, \(M_2(n) = \mathfrak{S}(n)n\) (where \(\mathfrak{S}(n)\) is the singular series), is positive and of order \(n\) for even \(n\). The truth of the SGC hinges on demonstrating that the error term \(E(n)\) satisfies \(|E(n)| < M_2(n)\) for all relevant \(n\).

This project undertakes a detailed theoretical exploration of a specific, hypothesized structured component within the error term \(E(n)\). This component, termed the Modular Aliasing Signature (MAS)-like term, is conjectured to arise from the influence of hypothetical off-critical-line zeros of Dirichlet L-functions.

SGC Hypothesis 1.1 (MAS for \(R_2(n)\))

The error term \(E(n)\) for \(R_2(n)\) contains a significant component of the form: \[ E_{\text{MAS}}(n) = K(n) \cdot \chi_X(n \pmod k) \cdot X(n), \] where:

The investigation is structured into three phases, detailed in the subsequent sections, covering derivation, coefficient refinement, significance assessment, bounding, and SGC implications.

2. Phase 1: Derivation of the MAS-like Term's Structure

This phase aimed to establish whether a term consistent with SGC Hypothesis 1.1 could be derived from the standard machinery of the Hardy-Littlewood circle method.

2.1 The Hardy-Littlewood Circle Method Framework

The starting point is the integral representation of \(R_2(n)\): \[ R_2(n) = \int_0^1 S(\alpha)^2 e^{-2\pi i n \alpha} d\alpha, \quad \text{where } S(\alpha) = \sum_{m \leq n} \Lambda(m) e^{2\pi i \alpha m}. \] The unit interval \([0,1]\) is dissected into major arcs \(\mathfrak{M}\) and minor arcs \(\mathfrak{m}\). The major arcs \(\mathfrak{M}_{a,q}\) are typically defined for \(1 \le q \le Q\) and \(1 \le a \le q\) with \((a,q)=1\) as the intervals: \[ \mathfrak{M}_{a,q} = \left\{ \alpha : \left|\alpha - \frac{a}{q}\right| \le \beta_0 \right\}. \] Common choices for the parameters are \(Q = (\ln n)^A\) for some \(A>0\), and \(\beta_0 = Q/(qn)\).

2.2 Approximation of \(S(\alpha)\) on Major Arcs

For \(\alpha = a/q + \beta \in \mathfrak{M}_{a,q}\), the exponential sum \(S(\alpha)\) can be effectively approximated. By sorting the sum modulo \(q\), \(S(\alpha)\) is expressed in terms of characters \(\chi \pmod q\) and their associated sums \(S_\chi(\beta) = \sum_{m \leq n} \Lambda(m)\chi(m)e^{2\pi i m \beta}\). The standard approximation is (cf. \cite{Davenport_MNT_grand_ref, Vaughan_HLM_grand_ref}): \[ S(a/q + \beta) = \frac{1}{\varphi(q)} \sum_{\chi \pmod q} \tau(\bar{\chi},a) S_\chi(\beta), \] where \(\tau(\bar{\chi},a) = \sum_{h=1}^q \bar{\chi}(h)e^{2\pi i ha/q}\) is the Gauss sum. If \(\chi\) is primitive, \(\tau(\bar{\chi},a) = \bar{\chi}(a)\tau(\bar{\chi})\). The sums \(S_\chi(\beta)\) are related to \(\psi(u,\chi) = \sum_{m \le u} \Lambda(m)\chi(m)\) and its explicit formula:

Thus, \(S(a/q + \beta)\) can be written as: \[ S(a/q + \beta) \approx \frac{\mu(q)}{\varphi(q)} S_P(\beta) + \frac{1}{\varphi(q)} \sum_{\substack{\chi \pmod q \\ \chi \neq \chi_0}}^* \bar{\chi}(a)\tau(\bar{\chi}) S_\chi(\beta), \] where \(\sum^*\) denotes a sum over primitive characters.

2.3 Derivation from Cross-Term Interaction

The main term \(M_2(n) \approx \mathfrak{S}(n)n\) arises from the \((\frac{\mu(q)}{\varphi(q)}S_P(\beta))^2\) part. We seek an MAS-like error term from the interaction (cross-term) of the principal character component of one \(S(\alpha)\) factor and a non-principal character component of the other. Let \(\chi_X\) be a real, non-principal, primitive character modulo \(k\), and \(\rho_0 = \sigma_0 + it_0\) an off-critical-line zero of \(L(s, \chi_X)\). Let \(S_{\chi_X,\text{eff}}(\beta, \rho_0, \bar{\rho}_0) \approx -\left( I(\rho_0, \beta) + I(\bar{\rho}_0, \beta) \right)\). The relevant part of \(S(a/k+\beta)^2\) for the cross-term (for \(q=k\)) is: \[ 2 \left( \frac{\mu(k)}{\varphi(k)} S_P(\beta) \right) \left( \frac{1}{\varphi(k)} \bar{\chi_X}(a)\tau(\bar{\chi_X}) S_{\chi_X,\text{eff}}(\beta, \rho_0, \bar{\rho}_0) \right). \] Integrating this over \(\mathfrak{M}_{a,k}\) and summing over \(a\): The character-dependent pre-integral factor becomes \(2 \frac{\mu(k)k}{\varphi(k)^2} \chi_X(n)\). The integral \(J_{\text{conv}}(n, \rho_0, \bar{\rho}_0) = \int_{-\beta_0}^{\beta_0} S_P(\beta) S_{\chi_X,\text{eff}}(\beta, \rho_0, \bar{\rho}_0) e^{-2\pi i n \beta} d\beta\) is a convolution which yields approximately \(-\left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right)\). This leads to a contribution from \(q=k\): \[ E_{k, \chi_X, \rho_0}(n) \approx -2 \frac{\mu(k) k}{\varphi(k)^2} \chi_X(n) \operatorname{Re}\left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right). \] This is \(K_k \cdot \chi_X(n) \cdot X(n)\) with \(K_k = -2 \frac{\mu(k) k}{\varphi(k)^2}\) and \(X(n) = \operatorname{Re}\left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right)\).

Phase 1 established that a term structurally consistent with the MAS hypothesis can be derived from the Hardy-Littlewood circle method, arising from a specific cross-term interaction on major arcs ($q=k$). The coefficient $K_0$ was initially estimated based on this $q=k$ term. This initial $K_k = -2 \mu(k)k/\varphi(k)^2$ depends on $\mu(k)$, which presents issues for non-square-free $k$. The overall coefficient $K_0$ requires summing contributions from all relevant $q=mk$, leading to the form $K_0^{\text{prod}}$ (see Equation \eqref{eq:K0_prod_grand_doc_main_consolidated_ref_fixed_again_html_final} below), which also faced challenges regarding vanishing conditions and applicability to all $k$.

3. Phase 2: Theoretical Refinement of \(K_0\) and Qualitative Significance

Phase 2 focused on addressing the limitations of the initial \(K_0\) estimates and assessing the MAS-like term's qualitative role.

3.1 Refining \(K_0\) by Summing Over \(q=mk\)

If $\chi_X$ is primitive modulo $k$, it induces characters $\chi'_X = \chi_X \psi_{0,m}$ modulo $q=mk$ (where $(m,k)=1$ and $\psi_{0,m}$ is principal mod $m$). A careful derivation of how such terms sum over $q=mk \le Q$ leads to a coefficient for $\chi_X(n)X(n)$ of the form: \[ K_0^{\text{prod}} = -2 \frac{\mu(k) k}{\varphi(k)^2} \sum_{\substack{m \geq 1 \\ \gcd(m, k) = 1 \\ m \text{ sq-free} \\ m \le Q/k}} \frac{\chi_X(m)}{\varphi(m)^2} \approx -2 \frac{\mu(k) k}{\varphi(k)^2} \prod_{p \nmid k} \left(1 + \frac{\chi_X(p)}{(p-1)^2}\right). \label{eq:K0_prod_grand_doc_main_consolidated_ref_fixed_again_html_final} \] This \(K_0^{\text{prod}}\) vanishes if \(\mu(k)=0\) (e.g., \(k=4\)) or if the product term is zero. A factor \((1 + \frac{\chi_X(p)}{(p-1)^2})\) in the product is zero if \(p=2\) (and \(2 \nmid k\)) and \(\chi_X(2)=-1\). This occurs if \(k\) is odd and \(k \equiv 3, 5 \pmod 8\). These vanishing conditions, particularly for \(\mu(k)=0\), indicated that this derivation path might be oversimplified for certain moduli.

3.2 Alternative \(K_0\) from Structural Analogies and Implications

Coefficients in explicit formulas for sums like \(\psi(N,\chi)\) or related binary problems often take a general structural form. By analogy, a plausible coefficient for the complex amplitude of the term involving \(\chi_X(n) (n^{\rho_0}/\rho_0 + n^{\bar{\rho}_0}/\bar{\rho}_0)\) in \(R_2(n)\) is \(C(\chi_X) \approx -2 \frac{\chi_X(-1)\tau(\chi_X)}{\varphi(k)}\). This leads to a real coefficient \(K_0 = \operatorname{Re}(C(\chi_X))\).

Proposition 3.1 (Hypothesized Form of \(K_0\) from Analogy)

A structurally common coefficient form for the MAS-like term is: \[ K_0^{\text{lit-analogy}} = \operatorname{Re} \left( -2 \frac{\chi_X(-1)\tau(\chi_X)}{\varphi(k)} \right). \label{prop:K0_hyp_form_grand_consolidated_ref_fixed_again_html_final} \] This yields:

Key Challenge 3.2 (Definitive \(K_0\) for \(R_2(n)\))

The conflicting predictions for \(K_0\) from \(K_0^{\text{prod}}\) (Equation \eqref{eq:K0_prod_grand_doc_main_consolidated_ref_fixed_again_html_final}) and \(K_0^{\text{lit-analogy}}\) (Proposition \ref{prop:K0_hyp_form_grand_consolidated_ref_fixed_again_html_final}) highlight a critical uncertainty. A definitive \(K_0\) for \(R_2(n)\) must account for all contributing major arc terms and character interactions, or be extracted from an established, detailed explicit formula for \(R_2(n)\) in the literature.

Remark 3.3

For composite \(k\) like \(k=8\), multiple primitive real characters exist. For example, if \(k=8\), one type (e.g., related to Jacobi symbol \((n/8)\)) has \(\chi_X(-1)=1\), leading to \(K_0^{\text{lit-analogy}} \neq 0\). Another type (e.g., related to \((-1)^{(n^2-1)/8}\)) has \(\chi_X(-1)=-1\), leading to \(K_0^{\text{lit-analogy}} = 0\).

3.3 Qualitative Significance Assessment of \(E_{\text{MAS-like}}(n)\)

Assuming a scenario where \(K_0 \neq 0\), the MAS-like term \(E_{\text{MAS-like}}(n)\) has magnitude \(\sim |K_0| n^{\sigma_0}/|\rho_0|\).

A quantitative example using \(K_0^{\text{lit-analogy}} \approx -1.118\) (for \(\chi_5\)), \(\rho_0 = 0.75+10i\), for \(n=10^6\), yields \(|E_{\text{MAS-like}}(10^6)| \approx 7071\). This is numerically comparable to an \(O(n(\ln n)^{-2})\) minor arc bound (which is \(\approx 5241\) at \(n=10^6\)), suggesting potential numerical significance for moderate \(n\) in shaping the error term \(E(n)\).

Phase 2 critically examined the coefficient \(K_0\). The proposed \(K_0^{\text{lit-analogy}}\) (Proposition \ref{prop:K0_hyp_form_grand_consolidated_ref_fixed_again_html_final}) offers a plausible structure with strong vanishing predictions based on \(\chi_X(-1)\). If \(K_0 \neq 0\), the MAS-like term represents a significant \(o(n)\) component of \(E(n)\), potentially dominant among structured error terms from off-line zeros. The definitive determination of \(K_0\) for \(R_2(n)\), resolving conflicting predictions from different derivation paths, is essential for quantitative analysis and was identified as the paramount next step for future research.

4. Interactive Visualization of MAS-like Terms (Hypothetical Parameters)

To better understand the behavior of the derived terms, interactive visualizations can be invaluable. Below, we outline conceptual plots that could be implemented using Plotly.js. The JavaScript code for these plots is included at the end of this document.

4.1 Visualizing the Oscillatory Term \(X(n)\)

This plot shows \(X(n) = \operatorname{Re} \left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right)\) versus \(n\). Users can adjust \(\sigma_0\) and \(t_0\) to see how the amplitude envelope \( \sim n^{\sigma_0}/|\rho_0|\) and the oscillation frequency (related to \(t_0 \ln n\)) change.

4.2 Interactive \(K_0\) Calculator (Conceptual)

Calculated \(K_0^{\text{lit-analogy}}\) for selected k:

Properties: \(\chi_X(-1)\)=, \(\tau(\chi_X)\)=, \(\varphi(k)\)=

This tool calculates \(K_0^{\text{lit-analogy}}\) based on the selected \(k\) and its associated real primitive character \(\chi_X\), displaying intermediate values. Note: \(K_0 = 0\) if \(\chi_X(-1)=-1\).

4.3 Visualizing \(E_{\text{MAS-like}}(n)\)

Uses parameters from 4.1 (for \(\rho_0\)) and 4.2 (for \(k\) and \(K_0\)).

This plot shows \(E_{\text{MAS-like}}(n) = K_0 \cdot \chi_X(n) \cdot X(n)\) versus \(n\). It demonstrates how the character \(\chi_X(n)\) further modulates \(X(n)\) and how \(K_0\) scales (and potentially nullifies) the term. If \(K_0=0\) for the selected k, this plot will be zero.

Phase 3 focused on the consequences of the MAS-like term for SGC, particularly given \(\sigma_0 < 1\).

5.1 Bounding \(E_{\text{MAS-like}}(n)\) using Zero-Free Regions

Known zero-free regions for Dirichlet L-functions \(L(s,\chi_X)\) (e.g., de la Vallée Poussin's \(\sigma > 1 - c/\log(k(|t|+2))\) for non-exceptional zeros, and Siegel's bound for potential exceptional real zeros) ensure that any off-critical-line zero \(\rho_0 = \sigma_0+it_0\) must have \(\sigma_0 < 1\). This fundamental constraint implies \(n^{\sigma_0} = o(n)\). Therefore, \(|E_{\text{MAS-like}}(n)| = O(n^{\sigma_0})\) is also an \(o(n)\) term.

5.2 Implications for the Strong Goldbach Conjecture

The SGC requires \(R_2(n) = M_2(n) + E(n) > 0\). The main term is \(M_2(n) = \mathfrak{S}(n)n = \Theta(n)\).

  1. Asymptotic SGC: Since \(|E_{\text{MAS-like}}(n)| = o(n)\) and \(M_2(n) = \Theta(n)\), this specific term, considered in isolation, cannot cause \(R_2(n)\) to become zero or negative for asymptotically large \(n\).
  2. Total Error \(E(n)\): The SGC's truth for large \(n\) depends on the total error \(E(n)\) being \(o(n)\). Standard unconditional bounds for \(R_2(n)\) (e.g., \(E(n) = O(n (\ln n)^{-A})\) or \(E(n) = O(n e^{-c\sqrt{\ln n}})\)) indeed satisfy \(E(n)=o(n)\). This implies \(R_2(n) = \mathfrak{S}(n)n (1 + o(1))\), which is positive for \(n \ge n_0\). This establishes SGC for sufficiently large \(n\), a result already known from the work of Hardy-Littlewood and subsequent improvements on error term bounds.
  3. Role of the MAS Framework: If SGC is true for large \(n\) due to \(E(n)=o(n)\), the significance of \(E_{\text{MAS-like}}(n)\) (if \(K_0 \neq 0\) and \(\rho_0\) exists with \(\sigma_0\) being the largest real part among exponents of structured \(o(n)\) error terms) is that it would be the dominant structured component of \(E(n)\). It would describe the principal oscillatory behavior and character-dependent patterns within the error term. Understanding such structured components is crucial for refining estimates of \(E(n)\) and for potentially lowering the threshold \(n_0\) above which SGC is analytically proven. The (still to be definitively confirmed) vanishing conditions for \(K_0\) (e.g., if \(K_0^{\text{lit-analogy}}=0\) when \(\chi_X(-1)=-1\)) would specify which characters \(\chi_X\) do not contribute this specific type of MAS signature via the analyzed cross-term mechanism.

The MAS-like term, constrained by \(\sigma_0 < 1\), is \(o(n)\) and does not alter the known asymptotic truth of SGC ($R_2(n)>0$ for sufficiently large $n$). Its primary value lies in potentially elucidating the fine structure of the error term $E(n)$. The quantitative impact and the very presence of this term for specific characters $\chi_X$ depend critically on the definitive value and properties of $K_0$.

6. Overall Project Conclusions and Future Work

6.1 Summary of Findings

This three-phase theoretical investigation has systematically explored the possibility and implications of a Modular Aliasing Signature (MAS)-like term in the error function \(R_2(n)\) for the Goldbach problem.

  1. Structural Derivation (Phase 1): A term of the form \(E_{\text{MAS-like}}(n) \approx K_0 \cdot \chi_X(n) \cdot X(n)\) was plausibly derived from the Hardy-Littlewood circle method, arising from specific major arc interactions involving an off-critical-line zero \(\rho_0\) of an \(L(s,\chi_X)\).
  2. Coefficient Challenge (Phase 2): The precise coefficient \(K_0\) proved theoretically challenging. Different derivation paths led to conflicting forms (e.g., \(K_0^{\text{prod}}\) vs. \(K_0^{\text{lit-analogy}}\)) with different vanishing conditions. The form \(K_0^{\text{lit-analogy}} = \operatorname{Re} \left( -2 \frac{\chi_X(-1)\tau(\chi_X)}{\varphi(k)} \right)\) was proposed as more robust and consistent with general structures, implying \(K_0=0\) if \(\chi_X(-1)=-1\). The definitive determination of \(K_0\) for \(R_2(n)\) was identified as a critical unresolved issue.
  3. SGC Implications (Phase 3): Given that any off-critical-line zero \(\rho_0\) must have \(\operatorname{Re}(\rho_0) = \sigma_0 < 1\), the MAS-like term is \(o(n)\). It therefore cannot, in isolation, cause an asymptotic failure of SGC. Its significance is in characterizing a potentially dominant structured component of the \(o(n)\) error term \(E(n)\).

The project successfully establishes a theoretical basis for investigating such structured error terms. The primary unresolved issue is the definitive determination of \(K_0\).

6.2 Future Work

The crucial next step, prerequisite for any reliable quantitative application or further development of this MAS framework for \(R_2(n)\), is:

  1. Definitive Determination of \(K_0\) (Top Priority):
    • (a) Conduct an exhaustive literature search for explicit point-wise formulas for \(R_2(n)\) in the works of specialists (e.g., Fujii; Perelli, Kaczorowski, Languasco, Zaccagnini; Goldston, Brüdern) to extract the precise coefficient of terms of the form \(\chi_X(n) \operatorname{Re}\left( \frac{n^{\rho_0}}{\rho_0} + \frac{n^{\bar{\rho}_0}}{\bar{\rho}_0} \right)\).
    • (b) If literature is insufficient or doesn't cover all cases (e.g., specific forms for \(k=4\)), undertake a more exhaustive first-principles derivation of \(K_0\), correctly accounting for all relevant character interactions and sums over \(q\) for general \(k\). This may involve analyzing other cross-terms in \(S(\alpha)^2\) or interactions between different major arcs beyond the simplified model used in Section \ref{sec:phase1_derivation_grand}.
  2. Validate Vanishing Conditions for \(K_0\): Based on the definitive \(K_0\), confirm under what precise arithmetic conditions (e.g., \(\chi_X(-1)=-1\), or other properties of \(\chi_X, k\)) the MAS-like term is genuinely absent or requires a different formulation.
  3. Numerical Validation: With a robustly determined \(K_0\), perform numerical simulations of \(E_{\text{MAS-like}}(n)\) to visualize its magnitude and behavior for various \(n, k, \chi_X, \rho_0\), comparing it against \(M_2(n)\) and bounds for other error terms.
  4. Exploration of Other Error Structures: If the specific MAS-like term (from the analyzed principal \(\times\) non-principal interaction) vanishes for certain key characters (like \(\chi_4\)), investigate other potential structured error terms in \(R_2(n)\) involving those characters (e.g., from \((\text{non-principal})^2\) interactions).
  5. Refined Error Analysis for SGC: Incorporate the validated and quantified MAS-like term (and other identified structured errors) into a comprehensive analysis of \(E(n)\) to see if it allows for sharper overall bounds on \(|E(n)|\), which could contribute to lowering the threshold \(n_0\) above which SGC is analytically proven.

7. References

  1. Davenport, H. (2000). \emph{Multiplicative Number Theory} (3rd ed., revised by H. L. Montgomery). Springer. (ISBN: 978-0387950976)
  2. Vaughan, R. C. (1997). \emph{The Hardy-Littlewood Method} (2nd ed.). Cambridge University Press. (ISBN: 978-0521573474)
  3. Montgomery, H. L., \& Vaughan, R. C. (2007). \emph{Multiplicative Number Theory I: Classical Theory}. Cambridge University Press. (ISBN: 978-0521849036)
  4. Iwaniec, H., \& Kowalski, E. (2004). \emph{Analytic Number Theory}. AMS Colloquium Publications. (ISBN: 978-0821836330)
  5. Fujii, A. (1991). \emph{An additive problem of prime numbers, II}. Acta Arithmetica, 58, 173--199. (Key reference to check)
  6. Brüdern, J., Kaczorowski, J., \& Perelli, A. (2019). \emph{Explicit formulae for averages of Goldbach representations}. Transactions of the American Mathematical Society, 372(9), 6529--6555. (Key reference to check, though for averages)