Density Hypothesis for Riemann-Zeta Zeros

Density Hypothesis for Riemann-Zeta Zeros

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 003; 2026-10-05), the asserted result is that zeta, Dirichlet functions, and finite-order Eisenstein-field Hecke families admit a zero-free region with boundary $11/12$; In manuscript 2 (family 003; 2026-09-30), the asserted result is that zeta, Dirichlet $L$-functions, and finite-order Hecke $L$-functions for $\mathbb Q(\sqrt{-3})$ have no zeros with real part greater than $7/8$, with poles excluded.

Mathematical statement of the claimed result

Manuscript 1.

For $K=\mathbb Q(\sqrt{-3})$ and every finite-order Hecke character $\chi$ over $K$, the claimed zero-free assertion, with meromorphic poles excluded from evaluation, is \[ \operatorname{Re}s>11/12\quad\Longrightarrow\quad L_K(s,\chi)\ne0. \] It includes every Dirichlet $L$-function and the Riemann zeta function. The accompanying prime-progression claim uses $\pi(x;q,a)$ and asserts, with an absolute effective constant, \[ \sup_{\substack{1\le q\le x\$a,q)=1}} \left|\pi(x;q,a)-\frac1{\varphi(q)}\int_2^x\frac{dt}{\log t}\right| \ll x^{11/12}\log x\quad(x\ge2). \] The critical-line requirement $\operatorname{Re}\rho=1/2$ is not a consequence of this half-plane claim.

Manuscript 2.

For every finite-order Hecke character of $F=\mathbb Q(\sqrt{-3})$, every Dirichlet character and the zeta function, the principal claim is \[ \operatorname{Re}s>7/8\quad\Longrightarrow\quad L(s)\ne0, \] where a principal character's pole at $s=1$ is allowed. The weaker displayed $11/12$ theorem follows within this scope. The cited arithmetic consequence gives absolute $A,C>0$ such that every odd prime $p$ has least positive quadratic nonresidue \[ n(p)\le C(\log p)^A. \] Thus $n(p)\ll_{\delta} p^{\delta}$ for each fixed $\delta>0$. Given binary $p$ and $a\in\mathbb F_p$, a deterministic algorithm of time polynomial in $\log p$ is further claimed to return a square root of $a$ or correctly declare it nonexistent. The half-plane assertion leaves zeros between the critical line and $7/8$ untreated.

Outline of the argument

For manuscript 1 (family 003; 2026-10-05), twisted Möbius sums are enlarged to a family before Poisson summation. A recursive estimate for the dual mean square, followed by inversion of cube factors, supplies the saving used to exclude a reciprocal-function singularity.

For manuscript 2 (family 003; 2026-09-30), smoothed character sums are estimated in two ways: directly by large-sieve and recursive moment bounds, and through a Mellin transform involving the reciprocal of an $L$-function. Poisson summation and sextic residue characters align the two expressions so that their savings force continuation of the reciprocal.

Formal verification

For manuscript 2 (family 003; 2026-09-30), OpenAI lists Lean statements for this assertion: zeta, Dirichlet $L$-functions, and finite-order Hecke $L$-functions for $\mathbb Q(\sqrt{-3})$ have no zeros with real part greater than $7/8$, with poles excluded.

This coverage does not establish a formal proof of the complete problem as stated here.

No Lean formalization is listed at this snapshot for manuscript 1 (family 003; 2026-10-05).

The linked family Comparator checks are DirichletSevenEighths.lean, HeckeSevenEighths.lean, QuasiRiemannHypothesis.lean, SiegelZeros.lean.

Their named declarations include OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SevenEighths.HeckeFamily.LFunction_ne_zero_of_seven_eighths_lt_re, OAI.SiegelZeros.WeightedTorusJets.dirichletRealZeroBound_proof, OAI.SiegelZeros.WeightedTorusJets.exists_absolute_real_zero_gap, OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, The Quasi-Riemann Hypothesis, 2026-10-05; citation key OAI:The-Quasi-Riemann-Hypothesis-October-5-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI, The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re (s)>7/8, 2026-09-30; citation key OAI:The-Quasi-Riemann-Hypothesis-September-30-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• Family 003: Lean scope.
• Comparator DirichletSevenEighths.json.
• Comparator DirichletSevenEighths.lean.
• Comparator HeckeSevenEighths.json.
• Comparator HeckeSevenEighths.lean.
• Comparator QuasiRiemannHypothesis.json.
• Comparator QuasiRiemannHypothesis.lean.
• Comparator SiegelZeros.json.
• Comparator SiegelZeros.lean.
• OpenAI catalogue entry.

Density Hypothesis for Riemann-Zeta Zeros

Definitions and mathematical statement

Count zeta zeros with multiplicity by \[ N(\sigma,T)=\#\{\rho=\beta+i\gamma:\zeta(\rho)=0,\ 0<\beta<1, \beta\ge\sigma,\ |\gamma|\le T\}. \] For each fixed $1/2\le\sigma\le1$ and $\varepsilon>0$, the density hypothesis predicts \[ N(\sigma,T)\ll_{\sigma,\varepsilon}T^{2(1-\sigma)+\varepsilon}. \] The bound is uniform in large $T$, with dependence on $\sigma,\varepsilon$ permitted. It limits the number of off-line zeros rather than asserting that they do not exist.

Formulation and concept references: [S1].

Short English statement

Are zeta zeros to the right of any fixed vertical line as sparse as the density hypothesis predicts?

Research attempt

Current frontier and scope.

The zero count includes multiplicities and both signs of the ordinate. Guth--Maynard's Theorem 1.2 gives $N(\sigma,T)\leq T^{15(1-\sigma)/(3+5\sigma)+o(1)}$; combining it with Ingham's estimate gives the uniform benchmark $T^{30(1-\sigma)/13+o(1)}$. The version consulted was revised on 7 April 2026. Its Section 13 carefully separates zero detection from large-value estimation. Neither displayed exponent is the conjectural $2(1-\sigma)$ throughout the strip. [E1] These are the relevant established benchmarks for this attempt, not a claim that one formula is the strongest available at every $\sigma$.

Attack map.

A large-values approach needs stronger control of the particular Dirichlet polynomials produced by zero detection, not of an arbitrary coefficient sequence without hypotheses. A mollifier approach tries to make $\zeta M$ close to $1$ in mean square: every zero forces a unit-sized defect, which should consume some of that mean square. A direct zero-repulsion approach would need a spacing theorem stronger than the elementary local zero count; simple distinctness or multiplicity bounds would not control the total. The mollifier route is developed below using area integrals, which makes the zero-detection implication particularly transparent and exposes its still-unproved analytic input.

Attempt I: an area-budget zero detector.

For $T\geq4$, set \[ M_T(s)=\sum_{d\leq\lfloor T\rfloor}\frac{\mu(d)}{d^s},\qquad g_T(s)=\zeta(s)M_T(s)-1. \] For each fixed $1/2<u_0<1$, define the genuine analytic quantity \begin{equation}\label{wsref-0-33-30-30-30-35-32-37-36-3a-61-72-65-61} \mathcal E(u_0,T)=\int_{u_0}^{2}\int_{T-1}^{2T+1} |g_T(u+it)|^2\,dt\,du. \end{equation} No Dirichlet-series expansion in the critical strip is assumed in this definition. The following is a sufficient, but unproved, estimate: \begin{equation}\label{wsref-0-33-30-30-30-35-32-37-36-3a-6d-69-73-73-69-6e-67} \forall u_0\in(1/2,1)\ \forall\eta>0:\qquad \mathcal E(u_0,T)\ll_{u_0,\eta}T^{2(1-u_0)+\eta}. \end{equation} The mollifier length here grows with $T$. A theorem for a fixed mollifier with constants depending arbitrarily on its length would not establish this assertion.

Area-detection lemma. Estimate (30005276:missing) implies the density hypothesis in the exact form stated in this entry.

Proof. Fix $1/2<\sigma<1$ and $\varepsilon>0$. Choose \[ 0<r<\min\{(\sigma-1/2)/2,1/4,\varepsilon/8\},\qquad u_0=\sigma-r. \] The standard local zero bound is $O(\log T)$ zeros, counted with multiplicity, in any horizontal strip of height one at ordinates of size $T$; this is explicitly used in [E1], Section 13.1. From the zeros with $\beta\geq\sigma$ and $T\leq\gamma\leq2T$, greedily select centers whose ordinates are at least $2r$ apart. Each choice removes only $O(\log T)$ zeros, including repeated zeros. Thus, if the original count is $Z$, there are at least $Z/(C\log T)$ selected centers.

For each selected $\rho$, the disk $D(\rho,r)$ lies inside the rectangle in (30005276:area). The disks are pairwise disjoint up to boundaries. The function $g_T$ is analytic there, since the pole of $\zeta$ is far away, and $g_T(\rho)=-1$. The submean inequality for the subharmonic function $|g_T|^2$ gives \[ \int_{D(\rho,r)}|g_T(s)|^2\,dA(s)\geq\pi r^2. \] Summing these nonnegative integrals yields \begin{equation}\label{wsref-0-33-30-30-30-35-32-37-36-3a-64-65-74-65-63-74-69-6f-6e} Z\ll r^{-2}(\log T)\mathcal E(\sigma-r,T) \ll_{\sigma,r,\eta}T^{2(1-\sigma)+2r+\eta}\log T. \end{equation} Take $\eta=\varepsilon/4$ and absorb the logarithm into $T^{\varepsilon/4}$ for large $T$. Dyadic summation bounds the positive ordinates up to any height, with the finitely many small ordinates absorbed in the constant; complex conjugation handles negative ones. At $\sigma=1/2$, the standard total zero bound $O(T\log T)$ suffices. At $\sigma=1$, the stipulated nontrivial zeros have $\beta<1$, so the count is empty. All constants may depend on $\sigma,\varepsilon$, as allowed in the original statement. $\square$

This proof accounts for multiplicity through the local count, rather than pretending that one disk detects a multiplicity proportional amount of area. It also avoids evaluating a function only on a measure-zero set without a justified neighborhood estimate.

Attempt II: why the required exponent looks plausible.

For $\Re s>1$, absolute convergence permits the exact expansion \begin{equation}\label{wsref-0-33-30-30-30-35-32-37-36-3a-63-6f-65-66-66-69-63-69-65-6e-74-73} \zeta(s)M_T(s)-1=\sum_{n>X}a_X(n)n^{-s},\qquad X=\lfloor T\rfloor,\quad a_X(n)=\sum_{\substack{d\mid n\\d\leq X}}\mu(d). \end{equation} For $2\leq n\leq X$, the omitted coefficient is zero by $\sum_{d\mid n}\mu(d)=0$; the constant coefficient has been subtracted. Also $|a_X(n)|\leq\tau(n)$. For fixed $u>1/2$ and sufficiently small $\eta>0$, the elementary divisor bound $\tau(n)\ll_\eta n^{\eta/2}$ gives \begin{equation}\label{wsref-0-33-30-30-30-35-32-37-36-3a-64-69-61-67-6f-6e-61-6c} \sum_{n>X}|a_X(n)|^2n^{-2u} \ll_{u,\eta}X^{1-2u+\eta}. \end{equation} If a mean-square theorem with time length $T$ retained only this coefficient-square term, it would give $T X^{1-2u+\eta}=T^{2-2u+\eta}$, exactly the exponent needed by (30005276:missing). Integrating over $u$ would not cost a new power when $u\geq u_0>1/2$.

This is a motivation, not an application of a mean-square theorem. Equation (30005276:coefficients) was proved only for $\Re s>1$. Square summability of its coefficients with a weight does not establish ordinary convergence of that Dirichlet series at the desired points, nor does it justify a mean-square approximation uniform in $X=T$. For fixed $X$, its summatory coefficients have main term \[ \sum_{n\leq Y}\sum_{\substack{d\mid n\\d\leq X}}\mu(d) =Y\sum_{d\leq X}\frac{\mu(d)}d+O(X), \] so absolute convergence across the line $1$ certainly has not followed from the cancellation of the first $X$ coefficients. Subtracting the single constant coefficient does not change this issue.

Attempt III: an explicit off-diagonal obstruction.

Even for a finite polynomial $D(t)=\sum_{L<n\leq2L}b_n n^{-it}$, expanding the integral gives off-diagonal factors \[ \int_T^{2T}e^{it\log(m/n)}\,dt, \qquad \left|\int_T^{2T}e^{it\log(m/n)}\,dt\right| \leq\min\{T,2/|\log(m/n)|\}. \] Distinct integers do not make this integral vanish. In particular, adjacent integers near $L\gg T$ can be coherent throughout a substantial part of the averaging interval.

A direct counterexample to a uniform "diagonal only" bound is $b_n=n^{iT}$. For $t=T+v$ with $0\leq v\leq1/4$, \[ e^{iv\log L}D(T+v)=\sum_{L<n\leq2L}e^{-iv\log(n/L)}. \] Every summand has real part at least $\cos((\log2)/4)>0$. Hence \[ \int_T^{2T}|D(t)|^2\,dt\gg L^2, \qquad T\sum_{L<n\leq2L}|b_n|^2=TL. \] Taking $L/T\to\infty$ disproves a bound by an absolute constant times the second expression. This does not disprove (30005276:missing): the coefficients here are not the arithmetic coefficients of the mollifier. It proves that the desired estimate would require their arithmetic structure or a different cancellation mechanism, rather than a general orthogonality principle.

Stress test.

The attempt separates three levels: the area lemma is proved; (30005276:diagonal) is a valid estimate of a formal diagonal expression; and the passage from that expression to (30005276:area) is unproved. An interchange of an infinite sum with an integral, or a passage from fixed mollifier length to length $T$, is not asserted. The loss $2r$ in (30005276:detection) is made arbitrarily small only after $\sigma,\varepsilon$ are fixed. No uniformity as $\sigma\downarrow1/2$ is needed or claimed.

The companion computation checks the coefficient cancellation in (30005276:coefficients) for finite $X$ and records exact finite coefficient-square sums at rational exponents where possible. It also checks the coherent-polynomial construction numerically, explicitly as a numerical check of an already proved elementary bound. No computed list of zeta zeros is promoted to a density theorem. Even the full density hypothesis would permit isolated zeros off the critical line; this argument is not a proposed proof of the Riemann hypothesis.

Outcome.

Outcome: Conditional reduction.

The density hypothesis has been reduced to the explicit area estimate (30005276:missing) for a growing mollifier. The exponent is explained by a valid coefficient calculation, and the tempting invalid mean-square step is falsified for general coefficients. A proof of the arithmetic off-diagonal estimate, with the required uniformity in the mollifier length, is still missing. No improved unconditional zero density exponent or historical novelty claim is made.

Sources

• S1 Terence Tao, "A computation-outsourced discussion of zero density theorems for the Riemann zeta function", What's new, 7 July 2024. https://terrytao.wordpress.com/tag/riemann-zeta-function/. Catalogue formulation source.
• E1 Larry Guth and James Maynard. New large value estimates for Dirichlet polynomials, arXiv:2405.20552v2, 7 April 2026. Theorem 1.2 and Section 13.1, including the local zero count and zero-detection reduction. https://arxiv.org/html/2405.20552v2. Consulted 27 September 2026 for the indicated result or scope.