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Unconditional implications of the Quasi-Riemann Hypothesis

Shubhrajit Bhattacharya

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Abstract

This record identifies published conclusions formerly obtained under RH or GRH whose proofs require only a fixed zero-free half-plane for Dirichlet LL-functions, with further applications of finite-order Hecke LL-functions over Q(−3)\mathbb Q(\sqrt{-3}). The input is the 7/87/8 theorem stated in the recent OpenAI manuscript claiming to prove the quasi-Riemann Hypothesis. We give the uniform analytic deductions explicitly and distinguish them from previously established algebraic and combinatorial results used in the proofs. An accompanying audit records the source verification and rejected candidates. No theorem about all number-field or automorphic LL-functions is inferred from the Dirichlet statement; the pure cubic and cubic-nonresidue applications separately identify the precise fixed-field Hecke input.

Principal conclusions. The following guide distinguishes complete published conclusions from published orders of magnitude whose sharp GRH constants are not recovered. The proofs specify every imported algebraic and computational input.

SourceConclusion recovered here
Montgomery–VaughanCharacter sums ≪qlog⁡log⁡(3q)\ll\sqrt{q}\log\log(3q); Theorem 2.1.
Granville–SoundararajanUniform cancellation when log⁡x/log⁡log⁡q→∞\log x/\log\log q \to\infty; Theorem 3.1.
Klurman–Mangerel–TeräväinenAll four published uniform short-progression conclusions for bounded multiplicative functions, including the hybrid short-interval theorem, with the stated typicality conditions; and L3(q)≪δq2+δL_3(q) \ll_\delta q^{2+\delta} for every modulus; Theorem 4.1 and Corollary 4.2.
WalshThe complete linear Fourier-cancellation conclusion for Möbius and Liouville in intervals with log⁡H/log⁡log⁡X→∞\log H/\log\log X \to\infty; Theorem 5.1.
Green–TaoThe full published power-saving improvement for Möbius correlations with polynomial nilsequences, Om,d(QCm,d(1+∥F∥Lip)N−cm,d)O_{m,d}(Q^{C_{m,d}}(1+\lVert F\rVert_{\mathrm{Lip}})N^{-c_{m,d}}), with cm,d>0c_{m,d}>0 and effective constants; Theorem 6.1.
MangerelQuadratic Gauss-sum rigidity from approximate prime dilations up to NcN^c, for every fixed c>0c>0, with the full published sparse-prime allowance; Theorem 7.1.
Sofos; MontgomeryInfinitely many primes with both the first prime nonresidue and the next gap ≫log⁡plog⁡log⁡p\gg\log p\log\log p; the published Paley clique lower order follows from the same argument; Theorem 8.1 and Corollary 8.2.
Mourtada–MurtyBoth quadratic Euler–Kronecker extreme-value bounds, with leading terms log⁡log⁡x+log⁡log⁡log⁡x\log\log x+\log\log\log x and ≫x\gg\sqrt{x} prime conductors below xx for each sign; Theorem 9.1.
Badzyan; Ford–Luca–MoreeThe full all-conductor bound ∣γQ(ζm)∣≪log⁡mlog⁡log⁡(3m)\lvert\gamma_{\mathbb{Q}(\zeta_m)}\rvert\ll\log m\log\log(3m) and the published GRH error for the second coefficient in counting nn with q∤φ(n)q\nmid\varphi(n); Theorem 10.1 and Corollary 10.3.
HoughExact maximum bias for products of two primes, (log⁡log⁡log⁡x)/log⁡log⁡x(\log\log\log x)/\log\log x; Theorem 11.1.
LittlewoodTwo-sided L(1,χ)L(1,\chi) bounds and the orders of magnitude for ζ\zeta, 1/ζ1/\zeta, and ζ′/ζ\zeta'/\zeta on the line 11.
Darbar–David–Lalín–LumleyBoth extreme-value distribution formulas for cubic L(1,χ)L(1,\chi), throughout the full published GRH ranges with endpoint constant −37-37; Theorem 13.1.
Granville–SoundararajanBoth quadratic L(1,χd)L(1,\chi_d) tail asymptotics with error O(A−1+t−1)O(A^{-1}+t^{-1}), throughout the full published larger range with endpoint constant −20-20; Theorem 14.1.
Hoffstein–LockhartEffective bounds L(1,Ad⁡f)−1L(1,\operatorname{Ad} f)^{-1}, ∣ρf(1)∣2≪log⁡(1+λN)\lvert\rho_f(1)\rvert^2\ll\log(1+\lambda_N) for every Maass newform, including the dihedral case; Theorem 15.1.
Boyd–KisilevskyEffective growth of the imaginary quadratic class-group exponent; Theorem 16.2.
Ellenberg–VenkateshThe exact GRH exponents for torsion in every quadratic and every cubic ideal class group, with effective constants and no exceptional fields, and the specified further families in every fixed degree; Theorem 17.1 and Corollary 17.3.
LamzouriThe full sharper odd-class-number average 154H2/log⁡H+O(H2(log⁡log⁡H)3/(log⁡H)2)\frac{15}{4}H^2/\log H+O(H^2(\log\log H)^3/(\log H)^2); Theorem 18.1.
LouboutinEffective exponent growth in the specified family of simplest cyclic cubic fields; Theorem 24.2.
DailedaThe class-number bound hF≪DFL(DF)/log⁡DFh_F\ll\sqrt{D_F}L(D_F)/\log D_F for all pure cubic fields; Theorem 25.2.
PollackThe exact limiting mean of the least completely split prime in cyclic fields of every fixed odd prime degree; Theorem 26.1.
Elsenhans–Klüners–Nicolae; Euler–KaniComplete 1046-field classification for exponent dividing 8, and the exact 65 idoneal integers; Theorems 19.3 and 19.5.
Chang–Kwon; MiyadaComplete lists of 489 imaginary abelian fields, and 301 multiquadratic fields, with one class per genus.
Ahn–KwonExactly 650 imaginary multiquadratic fields with class-group exponent dividing 2, with independently verified finite computations; Theorem 21.1.
Mollin; LouboutinExactly 107 real quadratic fields of extended Richaud–Degert type with ordinary class number two, and exactly 43 with class number one, using the same fresh complete finite computation; Theorem 22.1 and Corollary 22.4.
LouboutinThe complete 19-value exceptional-case classification for ω(d+x2)≤2\omega(d+x^2)\le2, and hence the full 42-value list for odd dd; Theorem 23.1.
Bourdon–PollackCM torsion in odd degree has upper order (dlog⁡log⁡d)2/3(d\log\log d)^{2/3}; Theorem 27.1.
Clark–Genao–Pollack–SaiaThe exact typical degrees dCM(X0(N))=(log⁡N)(log⁡2)/2+o(1)d_{\mathrm{CM}}(X_0(N))=(\log N)^{(\log2)/2+o(1)} and dCM(X1(N))=N(log⁡N)(log⁡2)/2+o(1)d_{\mathrm{CM}}(X_1(N))=N(\log N)^{(\log2)/2+o(1)}; the least nonsquare which is a square modulo NN has typical size (log⁡N)log⁡2+o(1)(\log N)^{\log2+o(1)}; Theorem 28.1 and Corollary 28.2.
BanwaitAn unconditional algorithm bounding every prime-degree isogeny over each eligible quadratic field; Theorem 29.1.
Rasmussen–TamagawaThe full uniform finiteness theorem for all dimensions and all fixed odd degrees; Theorem 30.1.
Schütt; Elkies–SchüttFinitely many rational CM newforms per weight up to the specified twists; every rational weight-3 newform has a singular K3 realization over Q\mathbb{Q}.
KaniThe exact 21 exceptional types of non-CM elliptic products which are not genus-2 Jacobians; Theorem 32.2.
Pasten; WeiThe all-Heegner-point height error of order log⁡log⁡dK\log\log d_K and the root-discriminant height bound for adjacent CM types; Theorems 33.1 and 34.1.
Binyamini–KowalskiFully effective André–Oort bounds and enumeration for arbitrary subvarieties of Y(1)nY(1)^n; Theorem 35.2; effective polynomial bounds in Theorem 35.3.
Edixhoven–RichardThe André–Oort statement for reductions of full CM orbits modulo any prime; Theorem 36.1.
HabeggerWeakly bounded height on every nonspecial modular-plane curve with unequal projection degrees; Theorem 37.1.
Balestrieri–Johnson–NewtonEffective computation of the Brauer–Manin set for every singular K3 surface and every geometric Kummer surface of a product of CM elliptic curves; Theorem 38.1.
KhayutinJoint equidistribution of CM packets and Galois orbits in the published two-fixed-split-prime setting, including bounded order conductors; Theorem 39.1.
Duke–Schulze-Pillot; DukeEffective primitive-spinor representation thresholds, and effective n>CD337n>CD^{337} squarefree representation; Theorems 41.1 and 41.2.
Kane; Elkies–Ono–YangEffective CM lifting to every supersingular class, with multiplicity bounds and the polynomial congruence; Theorem 42.1.
CampagnaEffective enumeration of singular SℓS_\ell-units for Sℓ={p:p≡1(mod3)}∪{ℓ}S_\ell=\{p:p\equiv1\pmod3\}\cup\{\ell\}, ℓ≥5\ell\ge5; Theorem 40.1.
Achter–Howe; Arul et al.The published GRH orders for split abelian surfaces and for pairs of D4D_4-curves with isogenous Jacobians.

Table 1.

Jeřábek; Adleman–Manders–MillerThe stated total-search complexity memberships for factoring, and deterministic extraction of fixed-degree roots in prime fields.
MillerDeterministic polynomial-time equivalence of factoring, φ\varphi, λ\lambda, and λ′\lambda', with one supplied value; the period-oracle equivalence and graph-recognition conclusions; Theorem 46.1.
Gao; RónyaiPolynomial-time balance decomposition and proper factoring of squarefree completely split polynomials of even degree; Theorem 48.1.
Rónyai; Ivanyos et al.Deterministic proper factoring when three divides the degree of a completely split polynomial; Theorem 49.1.
Pollack–SchornThe exact expected bit complexity for three-square representations without a supplied factorization; Theorem 50.1.
Balaji–Nosan–Shirmohammadi–WorrellSquare-root circuit identity testing in coRP; Theorem 51.1.
Balaji–Perifel–Shirmohammadi–WorrellGeneral cyclotomic identity testing in BPP and the deterministic restricted diagonal theorem; Theorems 52.1 and 52.2.
Castel; KutasA randomized polynomial-time isotropic-vector algorithm in dimension at least five without a factorization oracle; Theorem 53.1.
WesolowskiPolynomial-time prime sampling for every primitive positive definite integral quadratic form of rank at least two, with the published interval and input-size dependence; Theorem 54.1.
Herlédan Le Merdy–WesolowskiThe full worst-case to average-case reduction to supersingular maximal orders for every prime p>3p > 3, including the formerly conditional p≡1(mod8)p \equiv1 \pmod{8} case; Theorem 55.1.

Table 2.

The analytic input and uniform consequences

Throughout, log⁡\log is the natural logarithm, Λ\Lambda is the von Mangoldt function, and sums indexed by n≤xn \le x run over positive integers unless an interval or a different index set is explicitly specified. Put

ϑ=78,δ=1−ϑ=18,ℓ(x)=log⁡log⁡(eex)(x≥1).\vartheta= \frac{7}{8}, \qquad\delta= 1-\vartheta= \frac{1}{8}, \qquad\ell(x) = \log\log(e^{e}x) \quad(x \ge1).

The harmless modification in ℓ\ell makes bounds meaningful also for small arguments. All constants below are absolute and effective unless a dependence is specified.

Theorem 1.1 (Imported zero-free theorem). For every Dirichlet character χ\chi modulo a positive integer qq, its Dirichlet LL-function has no zero in ℜs>7/8\Re s > 7/8. The pole at s=1s = 1 for a principal character is excepted.

This is the Dirichlet part of [58 Theorem 1.1]. We use it as a cited unconditional theorem. We do not claim to have supplied a new proof of it. The companion [59] gives the weaker boundary 11/1211/12; that boundary would suffice for all qualitative conclusions in this record, with larger fixed logarithmic exponents in the intermediate estimates.

For a primitive nonprincipal character, the functional equation relates L(s,χ)L(s,\chi) to L(1−s,χ‾)L(1-s,\overline{\chi}). Since Theorem 1.1 also applies to χ‾\overline{\chi}, every nontrivial zero ρ=β+iγ\rho= \beta+ i\gamma satisfies

δ≤β≤ϑ.(1)\delta\le\beta\le\vartheta. \tag*{(1)}

The same assertion holds for the nontrivial zeros of ζ\zeta. Uniformity in qq below is derived from this zero location and the ordinary explicit formula. It is not inferred from uniformity of the fixed-character estimates in the proof of [58].

Lemma 1.2 (Uniform prime sum). For a primitive nonprincipal character χ\chi modulo q≥3q \ge3 and x≥2x \ge2,

ψ(x,χ):=∑n≤xΛ(n)χ(n)≪x7/8log⁡2(qx).\psi(x,\chi) := \sum_{n \le x} \Lambda(n)\chi(n) \ll\frac{x^{7/8}}{\log^2(qx)}.

Also ψ(x)=x+O(x7/8log⁡2(2x))\psi(x) = x + O\left(\frac{x^{7/8}}{\log^2(2x)}\right).

Proof. We use the classical uniform truncated explicit formula and zero count [60 Chapters 16 and 19]; these analytic identities and bounds are unconditional. At a half-integer xx, choose a permissible height T≍xT \asymp x. For a primitive nonprincipal character the formula is

ψ(x,χ)=−∑∣γ∣≤Txρρ+O(log⁡2(qx)),\psi(x,\chi) = -\sum_{|\gamma| \le T} \frac{x^\rho}{\rho} + O(\log^2(qx)),

where, after (1), the terms from trivial zeros and the logarithmic derivative at zero are included in the error. To justify this last uniformity explicitly, subtract the completed logarithmic derivatives at 1 and 2. The zero contribution has terms ((1−ρ)(2−ρ))−1((1-\rho)(2-\rho))^{-1}, whose absolute sum is Oδ(log⁡(3q))O_\delta(\log(3q)). The absolutely convergent Dirichlet series bounds L′/L(2,χ)L'/L(2,\chi), and the gamma factors and functional equation then bound the finite Laurent coefficient at zero by Oδ(log⁡(3q))O_\delta(\log(3q)).

The usual zero count is

Nχ(U):=#{ρ:∣ℑρ∣≤U}≪(U+1)log⁡(q(U+2)),N_\chi(U) := \#\{\rho: |\Im\rho| \le U\} \ll(U+1)\log(q(U+2)),

with multiplicities. The zeros with ∣γ∣≤1|\gamma| \le1 contribute Oδ(log⁡(3q))O_\delta(\log(3q)) to ∑1/∣ρ∣\sum1/|\rho|; partial summation for the remaining zeros gives

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