178 papers
This paper introduces a deterministic fractal constructed from prime numbers modulo 4, analyzes its Hausdorff dimension to distinguish its geometric complexity from the classical Cantor set, and proposes a novel framework linking the density of this prime-driven fractal to the distribution of zeros of the Riemann zeta function.
This paper provides a new Archimedean proof of the Giampietro-Darmon conjecture regarding singular moduli on genus 0 Shimura curves by evaluating Green's functions at CM points, offering an alternative to Daas's previous -adic approach while highlighting the parallels and distinctions between the two methods.
This paper presents an algorithm to compute the image of the adelic Galois representation for elliptic curves over with complex multiplication (excluding -invariants 0 and 1728) and establishes new results regarding the entanglement of their division fields.
This paper investigates the growth rates, asymptotic behaviors, and Gauss congruences of Reidemeister and Nielsen coincidence numbers for endomorphisms of torsion-free nilpotent groups and maps on compact nilmanifolds, while also establishing the rationality of the Nielsen coincidence zeta function.
This paper establishes two new Ramanujan-type congruences for the overpartition function modulo 11 and 13 using the theory of modular forms, while also conjecturing similar results for other prime moduli.
This paper investigates the statistical distributions, including proportions, variance, and maximal proportions, of the number of left prime truncations for integers and irreducible truncations for polynomials over finite fields.
This paper establishes that the -deformed modular group specializes to a finite group at a complex parameter if and only if is a primitive -th root of unity for , in which cases the resulting groups are isomorphic to specific binary polyhedral groups, while the case yields an infinite but "mild" structure with applications to Jones polynomials.
This paper derives the polar-finite decomposition for odd-spin admissible-level characters and establishes new mock theta conjecture-like identities for the corresponding string functions at levels $2/32/5$.
This paper establishes a toric analog of Yuan and Zhang's theory of adelic divisors to show that the arithmetic self-intersection number of a semipositive toric adelic divisor equals the integral of a concave function over a compact convex set, thereby enabling the computation of heights for toric arithmetic varieties with singular metrics.
This paper answers a question posed by Ellenberg regarding primitive elements of small height and uses this result to improve Heath-Brown's upper bounds for the -torsion part of class groups in purely cubic and general pure number fields of arbitrary odd degree.
This paper proposes an equivalent formulation of the Twin Prime Conjecture based on a symmetric property observed within a specific sequence of arithmetic progressions defined for pairs of coprime integers.
This paper generalizes Serre conjecture II to pseudo-reductive groups by proving its equivalence to the original conjecture for semisimple groups, thereby establishing that every torsor under a pseudo-semisimple, simply connected group over global function fields or non-archimedean local fields possesses a rational point.
This paper establishes a de Rham and Hyodo-Kato flip-flopping theorem for dual towers of rigid analytic spaces by utilizing pro-étale comparison theorems, and applies this result to prove the admissibility of the de Rham and Hyodo-Kato cohomologies of finite level coverings of Drinfeld spaces as representations of .
This paper develops an explicit invariant calculus for noncommutative linear differential operators using noncommutative Bell polynomials to derive universal formulas for gauge-covariant Wilczynski invariants and their projective counterparts, subsequently globalizing the framework to construct modular and Siegel modular differential operators on Riemann surfaces.
This paper presents a proof of a Ramanujan identity by expressing it in polar coordinates, which reduces the verification to an elementary trigonometric identity and yields several variations of the original result.
This paper establishes an extreme value theorem for the geodesic flow on the hyperbolic surface associated with the theta group by introducing a spliced continued fraction algorithm, proving its dynamical equivalence to the flow's first return map, and deriving a Galambos-type extreme value law for maximal cusp excursions via spectral analysis of the transfer operator.
This paper constructs simplified oscillatory reconstructions based on the nontrivial zeros of Dirichlet L-functions to visualize how interference patterns act as analytic filters that separate primes into congruence classes, thereby providing a visual bridge between the zero distributions of L-functions and the algebraic structure of cyclotomic fields.
This paper establishes that for sufficiently large , the non-real part of the closure of roots of restricted polynomials outside the unit disk is exactly the closure of a finite-capture locus, achieved by constructing canonical traps and proving uniform inclusion properties within a specific lens-shaped region.
This paper defines and analyzes a class of continuous functions on constructed via an infinite stochastic matrix and specific parameters, establishing criteria for their monotonicity, differentiability, and singularity while investigating the properties of their level sets.
This paper determines the values of the weighted zero-sum constants , , and for the specific case where the weight sets are and in the cyclic group .