Perplexity
Perplexity develops AI tools for search, browsing, research, content creation, organization, and workflow automation, including perplexity.ai, Comet, and Computer.
The Quasi-Riemann Claim Sets a Zero-Free Boundary at Re s = 7/8
An OpenAI preprint claims that the Riemann zeta function has no zeros in the open half-plane Re s > 7/8, a fixed boundary that falls short of the Riemann hypothesis. The manuscript extends the claim to Dirichlet L-functions and finite-order Hecke L-functions over Q(√−3), arguing that estimates on a constructed arithmetic sum would force a reciprocal L-function to be holomorphic where a hypothetical zero would make it singular. Its route to 7/8 depends on power savings in the sum’s remaining terms, not on the exponent match alone.
Selmer Corank at Most One Forces Rank Equality for Elliptic Curves
An OpenAI preprint proves a converse to the usual link between Selmer groups and ranks: for an elliptic curve over the rationals, at any prime \(p\), a full \(p\)-power Selmer group of corank zero or one forces the rational-point rank and analytic rank to equal that corank. The manuscript also concludes that the curve’s entire Tate–Shafarevich group is finite, without assuming a particular reduction type or residual representation. It applies the result to certain curves governing sums of three rational cubes.
Distinct Totient Values Have an Explicit Phase-Dependent Asymptotic
An OpenAI manuscript on Euler’s totient function gives an explicit asymptotic formula for the number of distinct values the function takes up to \(x\). The result sharpens earlier estimates by specifying the main scale and an arithmetic coefficient that varies with a logarithmic phase, rather than treating the coefficient as a universal constant. The manuscript also proves that, for every fixed \(c>0\), the count at \(cx\) divided by the count at \(x\) tends to \(c\).
A Frobenius-Compatible Hecke Eigensheaf Exists for Tame Regular-Nilpotent Monodromy
An OpenAI preprint claims the existence of a nonzero perverse Hecke eigensheaf for \(SL_n\)-bundles with a full flag at a marked point, under specific tame-monodromy and dense-monodromy hypotheses. Its central arithmetic requirement is that, after a finite extension of the field of constants, Frobenius preserve the entire system of Hecke eigenvalue equations—not merely the underlying sheaf. The proof’s local step rescales a regular nilpotent operator so the boundary actions on two branches agree while remaining compatible with Frobenius.
A Pointwise 2-Converse Links Selmer Corank to Rank for Elliptic Curves
An OpenAI preprint argues that for an elliptic curve over the rationals with a nonzero rational point of order two, Selmer corank zero or one forces the curve’s rational-point rank and analytic rank to equal that corank, and implies that its Shafarevich–Tate group is finite. The result is pointwise: it addresses the individual curve, not merely almost all curves in a family of quadratic twists. It allows every reduction type at 2, but does not establish the full Birch and Swinnerton-Dyer leading-coefficient formula.
Milnor K-Theory Relations Recover Addition and the Field
An OpenAI preprint argues that a field of characteristic \(p\) can be reconstructed from its multiplicative group modulo \(p\)-th powers, provided one also retains the Steinberg relations that define Milnor \(K\)-theory modulo \(p\). Under finite-generation and transcendence-degree hypotheses, the authors show that this data recovers projective lines; preserving those lines then recovers addition and yields the field isomorphism.
A Uniform Logarithmic Gap Separates Real Dirichlet Zeros From 1
An OpenAI preprint claims that every real zero of an eligible Dirichlet L-function stays a fixed logarithmic distance from 1: for primitive, nonprincipal real characters of conductor \(q\), it asserts \((1-\beta)\log q \ge c\) for an absolute positive constant \(c\). The manuscript’s proof links a hypothetical zero closer to 1 to weighted estimates over primes, then uses their divisibility of an algebraic determinant’s norm to contradict an upper bound on its size; it does not give a numerical value for \(c\).
A Rank-Weighted Tail Bound Establishes the Mean for Quadratic Twists
An OpenAI preprint argues that, for any fixed rational elliptic curve, the mean analytic rank of its signed squarefree quadratic twists tends to one half as the twist parameter grows. The result requires more than showing that half the twists have rank zero and half rank one: rare twists with very high rank could still raise the average. The paper’s central step is a bound on the total rank contributed by those high-rank twists, which lets the authors control the tail and establish the mean.
A Conditional Theorem Assembles Local Arthur Data Into One Global Enhancement
The OpenAI preprint *Global Arthur Enhancements of Cuspidal Excursion Parameters* argues that, assuming a finite-level Ramanujan Arthur decomposition, local Frobenius data can be assembled into a single global enhancement of a cuspidal parameter. Its conditional theorem constructs an algebraic SL₂ map and a commuting Weil-group map that recover the parameter after one fixed conjugation, including its action on inertia. The manuscript emphasizes that matching local data alone does not guarantee this global factorization; its proof must establish coherence across the parameter’s values.
Ramified Local Components Are Tempered for Globally Generic Exceptional Groups
An OpenAI preprint on exceptional groups proves that every local component of a cuspidal, globally generic automorphic representation in its stated class is tempered, including at ramified places. The proof addresses the gap in local-global compatibility: it identifies the ramified Weil parameter but not its monodromy. The authors use global purity to produce a pure local completion, then show that hypothetical non-tempered induction would force a Frobenius determinant incompatible with purity.
The Restricted Geometric Langlands Equivalence Reaches Every Spectral Component
A September 2026 manuscript argues that the restricted geometric Langlands equivalence in positive characteristic reaches every component of the spectral space under its stated hypotheses. Its proof rules out a missing component by showing that an object supported there cannot vanish under nearby cycles: Hecke tests and a degeneration to two intersecting projective lines produce detectable cohomology, contradicting the vanishing that a missing component would require.
A Dimension Bound and Explicit Construction Prove the Deligne-Drinfeld Conjecture
The manuscript behind OpenAI’s preprint *The Deligne-Drinfeld conjecture* identifies the infinitesimal symmetries that satisfy antisymmetry, a three-term identity and a pentagon compatibility rule. It proves that their solution space, with the Ihara bracket, is a free Lie algebra with one generator in each odd weight from three onward, and no further relations. The proof combines an all-weight upper bound with a construction showing that each predicted generator exists.
Every Component of the Two-Adic Hecke Algebra Has Dimension Four
An OpenAI preprint on the two-adic Hecke algebra at odd level proves that every irreducible component has Krull dimension four, including components with reducible or scalar residual representations. Its argument finds a classical point on each component where Galois deformation theory bounds the local dimension by three; passing from that point to the integral component adds one arithmetic dimension. This matches Emerton’s earlier lower bound of four.
Abelian Zilber–Pink Gives Finitely Many Maximal Atypical Subvarieties
An OpenAI manuscript dated September 24, 2026, proves the abelian Zilber–Pink conjecture for abelian varieties over number fields: for any irreducible subvariety, its atypical intersections with torsion cosets are contained in finitely many maximal atypical subvarieties. The result counts maximal exceptional pieces, which may contain infinitely many points; it does not provide an effective count or a uniform bound. The proof establishes a non-density result and uses a cited reduction to control subvarieties with a defect-minimizing property.
A Two-Primary BSD Formula Holds When Selmer Corank Is at Most One
An OpenAI manuscript proves a two-primary version of the Birch–Swinnerton-Dyer formula for elliptic curves over \(\mathbb Q\) whose two-power Selmer corank is at most one. Under that condition, it shows that algebraic rank, analytic rank and Selmer corank agree, that the Tate–Shafarevich group is finite, and that the BSD quantity and its order have the same power of two. The result does not establish equality at other primes.
A Strict Bound Gap Establishes Catalan’s Constant Is Irrational
An OpenAI manuscript dated September 24, 2026, claims to prove that Catalan’s constant is irrational by constructing a sequence of determinants. The argument shows that if the constant were rational, the determinants would have to satisfy an arithmetic lower bound; integral estimates, in turn, force them below that bound. The contradiction depends on exact cancellation of another constant in the determinant entries, along with proofs that the determinants are nonzero and their denominators are controlled.
Primitive Sextic Fields Equidistribute Across Six-Dimensional Lattice Space
An OpenAI preprint proves that, as discriminants grow, the complete packets associated with totally real primitive sextic fields become equidistributed in the space of six-dimensional unimodular lattices. Its argument rules out concentration near smaller block structures by using arithmetic separation of embeddings to control how often packet lattices can remain close to those configurations. The result concerns volume-weighted packets built from all ideal classes and sign translates, not selected lattices.
Cubic Gauss Sums Have an Unconditional Positive First Moment
An OpenAI preprint claims that cubic Gauss sums over prime elements in the Eisenstein integers have a positive first moment, identifying a small bias compatible with their equidistribution around the unit circle. It gives the sum’s leading term as \(\frac65c_*X^{5/6}/\log X\); since there are about \(X/\log X\) terms, their average shrinks like \(X^{-1/6}\). The manuscript presents the result as unconditional, without assuming the Generalized Riemann Hypothesis.
Pi’s Irrationality Exponent Is Exactly Two
An OpenAI preprint proves that π’s irrationality exponent is 2: no fixed power stronger than the inverse-square bound can be achieved by rational approximations at arbitrarily large denominators. To rule out stronger approximations, the proof derives an arithmetic lower bound and a smaller analytic upper bound for the same determinant. The paper then uses the exponent bound to show that the Flint Hills series converges.
An Orbit Bound Proves Finiteness for Mixed CM Products in A₂
An OpenAI preprint proves that a fixed Hodge-generic curve in the moduli space \(A_2\) contains only finitely many rational points whose associated abelian surfaces are isogenous to a product of elliptic curves with at least one complex-multiplication factor. The authors establish this component of the Zilber–Pink conjecture by combining André–Oort in the case where both factors have CM with a Galois-orbit bound and a counting theorem in the mixed case. The result allows arbitrary isogeny degrees and imposes no additional boundary or reduction hypothesis.
Pro-Modularity at 2 Holds Without de Rham or Irreducible Residual Hypotheses
An OpenAI preprint proves pro-modularity at 2 for a broad class of two-dimensional Galois representations, without requiring them to be de Rham at 2 or to have irreducible residual representations. The authors’ claim is about occurrence in a completed Hecke algebra spanning modular forms of many weights, not realization by a single classical form. Their proof constructs one auxiliary representation with stronger local properties to establish classicality, then uses a dimension comparison to transfer Hecke support to the wider family.
Primitive Quartic Torus Packets Equidistribute Across Arbitrary Orders
An OpenAI preprint argues that complete packets of lattices arising from primitive totally real quartic fields become equidistributed as the discriminants of their orders grow. Its theorem covers arbitrary orders and averages over all the packet’s torus orbits, rather than asserting equidistribution for individual lattices or selected ideals. The proof rules out a specific non-Haar limit—a decomposition into two coordinate blocks—before concluding that the packet measures converge to Haar probability.
Bracket Data Recover Function Fields Up to Frobenius and Scaling
An OpenAI manuscript states that a restricted quotient of a field’s absolute Galois group, together with its commutator bracket, can recover the field’s perfect closure and constant field when the relative transcendence degree is at least two. The Bogomolov–Pop reconstruction theorem allows one global \(\ell\)-adic unit scaling in the group data and, in positive characteristic, identifies field isomorphisms up to powers of Frobenius. The manuscript’s central technical step is a uniform bound on the support of images of a pencil of functions, which lets the authors recover the geometric data needed for reconstruction.
Milne’s Rationality Conjecture Holds for Abelian Varieties With Good Reduction
An OpenAI manuscript on Milne’s rationality conjecture proves a specific cross-theory agreement for abelian varieties with good reduction: for a fixed rational Hodge class and fixed divisors on the reduction, a normalized trace is the same rational number in the specified \(\ell\)-adic and crystalline cohomologies. The proof does not require new divisor classes on the reduction to lift to the original variety; instead, it transports the class through a CM variety and, in one case, constructs a new lift that preserves the relevant divisor data.