Completing the GRH equivalence for average Goldbach representations in arithmetic progressions: excluding $B_q=1$ off the diagonal, removing the Distinct Zero Conjecture and maximal-zero hypotheses

For the summatory function of the weighted Goldbach representation counts $S(x;q,a,b)=\sum_{n\le x}\sum_{\ell+m=n,\,\ell\equiv a,\,m\equiv b\,(\mathrm{mod}\,q)}\Lambda(\ell)\Lambda(m)$, Bhowmik–Halupczok–Matsumoto–Suzuki proved the unconditional asymptotic $S(x;q,a,b)=x^2/(2\varphi(q)^2)+O(x^{1+B_q})$, where $B_q$ is the supremum of the real parts of the zeros of Dirichlet $L$-functions modulo $q$, and converse implications showing that a power-saving error term forces $B_q\le d$ — but each converse carries a caveat: the persistent logical possibility $B_q=1$ (excluded only in the diagonal case $a=b$), the assumed Distinct Zero Conjecture (DZC) on common zeros of same-modulus $L$-functions, and, for sums over $n\equiv c\pmod q$, restrictive hypotheses on a maximal zero (unique associated character, squarefree conductor, $(c,q^*)=1$). The open problem is to remove these caveats and establish (or refute) the full equivalence between the Generalized Riemann Hypothesis (GRH) for Dirichlet $L$-functions modulo $q$ and square-root-cancellation error terms in average Goldbach problems with congruence conditions, in particular by excluding $B_q=1$ when $a\ne b$ and by reconstructing the reverse implications of Granville's estimates for residue-class and multiple-of-$q$ sums without hypotheses on the maximal zero.

Background

That every even integer can be written as a sum of two primes is the classical Goldbach conjecture; a standard weighted form counts representations using the von Mangoldt function $\Lambda(n)$ (equal to $\log p$ when $n$ is a prime power and $0$ otherwise), via $G(n)=\sum_{\ell+m=n}\Lambda(\ell)\Lambda(m)$. Here one studies the average order $\sum_{n\le x}G(n)$, for which the expected main term is $x^2/2$ and the believed pointwise approximation is the singular-series expression $J(n)=nC_2\prod_{p\mid n,\,p>2}\frac{p-1}{p-2}$ for even $n$ and $J(n)=0$ for odd $n$, where $C_2$ is the twin-prime constant. In the congruence-companion problem one fixes a modulus $q$ and residues $a,b$ with $(ab,q)=1$ and studies $S(x;q,a,b)=\sum_{n\le x}G(n;q,a,b)$, where $G(n;q,a,b)=\sum_{\ell+m=n,\,\ell\equiv a,\,m\equiv b\,(\mathrm{mod}\,q)}\Lambda(\ell)\Lambda(m)$. For a Dirichlet character $\chi\pmod q$ (a completely multiplicative function on residue classes, used to decompose arithmetic progressions), $L(s,\chi)$ denotes the associated Dirichlet $L$-function, and $B_q=\sup\{\Re\rho_\chi\}$ over all nontrivial zeros $\rho_\chi$ of all $L(s,\chi)$ with $\chi\pmod q$, so that $1/2\le B_q\le 1$; the Generalized Riemann Hypothesis (GRH) for modulus $q$ is the assertion $B_q=1/2$. For $q=1$ (the Riemann zeta function) it is classical, following work refined by Granville, Bhowmik–Schlage-Puchta, and Bhowmik–Ruzsa, that the square-root-type error bound $\sum_{n\le x}G(n)=x^2/2+O(x^{3/2+\varepsilon})$ for all $\varepsilon>0$ is equivalent to the Riemann Hypothesis. Bhowmik–Halupczok–Matsumoto–Suzuki (Mathematika 65 (2019) 57–97) extended this program to primes in arithmetic progressions. They proved unconditionally that $S(x;q,a,b)=x^2/(2\varphi(q)^2)+O(x^{1+B_q})$, where $\varphi$ is Euler's totient function, and — assuming the Distinct Zero Conjecture (DZC), which asserts that two distinct Dirichlet $L$-functions attached to characters of the same modulus $q$ share no nontrivial zero except possibly a multiple zero at $s=1/2$, together with the separate hypothesis that $\chi(a)+\chi(b)\ne 0$ for all $\chi\pmod q$ (needed to keep the relevant residue from vanishing) — the converse: if $S(x;q,a,b)=x^2/(2\varphi(q)^2)+O_q(x^{1+d+\varepsilon})$ for some $1/2\le d<1$ and all $\varepsilon>0$, then $B_q\le d$ or $B_q=1$. The exceptional alternative $B_q=1$ (all information on nontrivial zeros drowned by a Siegel-type zero at the edge of the critical strip) is excluded, via a square-root argument for the generating Dirichlet series due to I. Ruzsa (exploiting nonnegativity of coefficients), only in the diagonal case $a=b$ and still under DZC. For sums with a condition on the summed integer, $\sum_{n\le x,\,n\equiv c\pmod q}(G(n)-J(n))$, they proved the bound $O(x^{1+B_q})$ unconditionally, and under GRH the bound $O(x^{3/2})$; but the reverse implication — deducing $B_q\le d$ from an error $O_q(x^{1+d+\varepsilon})$ — was obtained only under restrictive hypotheses on a maximal zero $\rho_0$ realizing $B_q$: it must belong to a single character $\chi_1\pmod q$ whose conductor $q^*$ is squarefree with $(c,q^*)=1$. The authors note this cannot handle a Landau–Siegel zero (a real exceptional zero of an $L$-function just left of $s=1$) when complex zeros of the same modulus lie very close to the line $\Re s=1$, and they state explicitly that they could not reconstruct the reverse implications of Granville's estimates for $\sum_{n\le x,\,n\equiv 2\pmod q}(G(n)-J(n))$ and for sums over multiples $q\mid n$. Their explicit formula also requires a truncation $B_q^*=\min(B_q,1-\eta_q(x))$ with $\eta_q(x)$ depending on $q^\varepsilon$ and $\log x$. Later work — the authors' 2020 survey, Cesàro-weighted averages by Cantarini–Gambini–Zaccagnini, the exceptional-zero results of Friedlander–Goldston–Iwaniec–Suriajaya, and the 2024–2026 forward-direction refinements by Ikeda–Suriajaya (multiples of $q$) and Nguyen (distinct moduli) — has extended only the forward direction; every one of the above caveats remains in place.

Problem Statement

Establish, or refute, the full equivalence between the Generalized Riemann Hypothesis for Dirichlet $L$-functions modulo $q$ (the assertion $B_q=1/2$) and square-root-cancellation error terms in average Goldbach problems with congruence conditions, by removing each caveat left by Bhowmik–Halupczok–Matsumoto–Suzuki. Specifically, with $q\ge1$, $(ab,q)=1$, $\chi$ ranging over Dirichlet characters mod $q$, and $\varphi$ Euler's totient function:

(I) (Non-diagonal converse; excluding $B_q=1$.) For $a\ne b$ with $\chi(a)+\chi(b)\ne 0$ for all $\chi\pmod q$, prove or disprove that, under the Distinct Zero Conjecture, the asymptotic

$S(x;q,a,b)=\frac{x^2}{2\varphi(q)^2}+O_q\bigl(x^{3/2+\varepsilon}\bigr)\ \text{for all }\varepsilon>0$

implies $B_q=1/2$, i.e. that the alternative $B_q=1$ can be excluded on the same footing as in the diagonal case $a=b$ (where this is known under the Distinct Zero Conjecture via Ruzsa's square-root argument); removing the Distinct Zero Conjecture itself is the object of (II).

(II) (Removing DZC.) Prove or disprove that the converse implications of (I) and the diagonal case hold without assuming the Distinct Zero Conjecture — establishing or refuting these converses with no common-zero hypothesis on distinct same-modulus $L$-functions, while controlling possible cancellation at common zeros. Two related advances are partial only and do not on their own discharge (II): establishing these converses under a weakened zero-separation hypothesis replaces the DZC caveat rather than removing it, and proving DZC itself discharges that assumption but still leaves the off-diagonal converse of (I) to be proved.

(III) (Residue-class sums; removing the maximal-zero hypotheses.) For integers $q,c$ with $(2,q)\mid c$, prove or disprove that

$\sum_{\substack{n\le x\\ n\equiv c\pmod q}}\bigl(G(n)-J(n)\bigr)\ll_q x^{3/2+\varepsilon}\ \text{for all }\varepsilon>0$

implies $B_q=1/2$ without the auxiliary hypotheses that a maximal zero $\rho_0$ with $\Re\rho_0=B_q$ belongs to a unique character $\chi_1\pmod q$ of squarefree conductor $q^*$ with $(c,q^*)=1$; in particular, settle the cases involving a possible Landau–Siegel zero together with complex zeros of the same modulus very close to $\Re s=1$, and thereby reconstruct the reverse implications of Granville's estimates for the sums over $n\equiv 2\pmod q$ and over multiples $q\mid n$.

These three deliverables jointly serve one objective — the completion (or definitive refutation) of the GRH-equivalence program for average Goldbach representations with congruence conditions, (I)–(III) being the mutually intertwined obstructions identified by the source. Deliverable (I) is the DZC-conditional special case of (II): a proof of (II) yields (I) as a corollary, so the two are not independent proof obligations, although disproving (II) does not by itself settle (I). A resolution of the program either establishes the full equivalence by proving all three implications — with (II) superseding (I) as just noted — or refutes it by disproving any one of them, exhibiting, within the standard framework of analytic number theory, a configuration compatible with the stated error term but violating GRH for modulus $q$.

References

1. Gautami Bhowmik, Karin Halupczok, Kohji Matsumoto, Yuta Suzuki, Goldbach representations in arithmetic progressions and zeros of Dirichlet L-functions, Mathematika 65 (2019), 57–97. DOI: 10.1112/S0025579318000323; arXiv:1704.06103, https://arxiv.org/abs/1704.06103
2. G. Bhowmik, I. Z. Ruzsa, Average Goldbach and the Quasi-Riemann Hypothesis, Analysis Mathematica 44 (2018), 51–56. DOI: 10.1007/s10476-018-0105-4, https://doi.org/10.1007/s10476-018-0105-4
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7. Karin Ikeda, Ade Irma Suriajaya, The average number of Goldbach representations over multiples of q, Functiones et Approximatio Commentarii Mathematici 73 (2025). DOI: 10.7169/facm/240922-30-1; arXiv:2405.04315, https://arxiv.org/abs/2405.04315
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