183 papers
This paper resolves Skolem's problem for -generalized Lucas sequences by characterizing their zero-distribution at negative indices and proving that the zero-multiplicity is exactly for all .
This short note refutes two natural conjectures regarding polynomial count varieties by demonstrating that a smooth variety with a point count of is not necessarily isomorphic to affine space, and that such varieties do not necessarily have vanishing Hodge numbers for .
This paper establishes that quadratic congruences modulo hold for a wide range of half-integral weight cusp forms with the eta multiplier and an arbitrary Dirichlet character, extending previous results limited to real characters through the study of modular Galois representations and a new theorem on the existence of specific conjugacy classes in their images.
This paper investigates sign changes in summatory functions of arithmetic functions and their Dirichlet inverses, demonstrating that convolving with specific "magic partition function" encodings can smooth oscillatory behavior to produce sequences with predictable sign properties under reasonable asymptotic bounds.
This paper establishes the decidability of the Skolem Problem for linear recurrence sequences over finitely generated commutative rings of positive characteristic by proving that their zero sets form effectively computable finite unions of -normal sets, leveraging recent advances in solving S-unit equations and linear equations over multiplicatively independent numbers.
This paper establishes that the surjectivity of the joint distribution of leftmost digits across multiple integer bases implies the rational independence of their logarithms, proving the converse for two bases and for three or more bases under the assumption of Schanuel's conjecture.
This paper computationally investigates the principle that congruences between holomorphic cuspforms induce congruences between the special values of their associated Rankin–Selberg -functions, leading to the formulation of a precise general conjecture.
This paper empirically investigates the oscillatory behavior of determinants associated with Ramanujan functions on small primes, exploring potential regularities in their complex point sets as a means to address Lehmer's conjecture regarding the non-vanishing of Ramanujan's tau function.
This paper establishes non-trivial bounds for bilinear sums of trace functions below the Pólya-Vinogradov range by leveraging a general stratification theorem and a robust Goursat-Kolchin-Ribet criterion, thereby extending previous results from special cases to a broader class of -adic sheaves with specific geometric monodromy properties.
This paper establishes new properties of the -representation of integers, including a proof of Kimberling's 2012 conjecture, utilizing the Walnut theorem-prover and ChatGPT 5 as computational aids.
This paper characterizes isogeny classes of abelian surfaces over finite fields that contain no curves of genus 3 or less by completing the classification for genus up to 2, establishing a link between genus 3 curves and degree 4 polarizations, and describing the specific genus 3 curves that exist on such surfaces.
This paper proves that, under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the unique smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group, thereby significantly strengthening previous results on the Shafarevich conjecture.
This paper proves the Shafarevich conjecture for very irregular varieties of dimension less than half that of their Albanese variety by combining the Lawrence-Venkatesh method with a big monodromy criterion.
This paper generalizes previous work on toroidal averages by studying the cubic moments of special values of -functions for Dirichlet characters of prime modulus, while highlighting connections to bilinear forms of trace functions and the number of solutions to monoidal equations in finite fields.
This paper constructs a -adic -function for ordinary elliptic curves over global function fields of characteristic associated with -extensions, establishes its interpolation properties and functional equations, and proves the Iwasawa main conjecture in several cases, including a reduction criterion for the scenario.
This paper advances the theory of -adic continued fractions by proving that palindromic and quasi-periodic expansions converge to either transcendental numbers or quadratic irrationals without prior restrictions on partial quotients, while also establishing a quantitative -adic version of Ridout's theorem and analyzing the growth of denominators for algebraic numbers.
This paper establishes linear independence results for multiple Eisenstein series in positive characteristic and proves that their associated -shuffle algebra is isomorphic to the tensor square of the multiple zeta value algebra, thereby confirming a conjecture that it forms an associative algebra.
This paper proves that for every integer , the interval between consecutive squares contains an integer with at most three prime factors, improving the previous bound of four by combining finite verification up to $10^{31}$ with explicit sieve-theoretic arguments.
This paper investigates the discrete convolution of the Liouville function, deriving an explicit formula for its weighted averages to reveal new properties regarding its Dirichlet and power series as well as its generalizations to multiple factors.
This paper extends the emerging theory of hook length biases to -core partitions by employing combinatorial methods to establish specific inequalities comparing the counts of hooks of different lengths within these partitions.