64 citations · 114 across the 16 of their papers we have counts for
16 papers · 1 filter
Improved error estimates for the Davenport-Heilbronn theorems
Manjul Bhargava, Takashi Taniguchi, Frank Thorne
We improve the error terms in the Davenport-Heilbronn theorems on counting cubic fields to . This improves on separate and independent results of the authors and Sh…
The local-global property for bitangents of plane quartics
Yasuhiro Ishitsuka, Tetsushi Ito, Tatsuya Ohshita +2
We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois pr…
Uniform bounds for lattice point counting and partial sums of zeta functions
David Lowry-Duda, Takashi Taniguchi, Frank Thorne
We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice in…
Levels of distribution for sieve problems in prehomogeneous vector spaces
Takashi Taniguchi, Frank Thorne
In a companion paper, we developed an efficient algebraic method for computing the Fourier transforms of certain functions defined on prehomogeneous vector spaces over finite field…
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
Manjul Bhargava, Arul Shankar, Takashi Taniguchi +3
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields (the trivial bound being $O_ε(|{…
Orbital exponential sums for prehomogeneous vector spaces
Takashi Taniguchi, Frank Thorne
Let (G, V) be a prehomogeneous vector space, let O be any G(F_q)-invariant subset of V(F_q), and let f be the characteristic function of O. In this paper we develop a method for ex…
An error estimate for counting -sextic number fields
Takashi Taniguchi, Frank Thorne
In this note, we prove a power-saving remainder term for the function counting -sextic number fields. We also give a prediction on the second main term. We also present numeri…
Orbital L-functions for the space of binary cubic forms
Takashi Taniguchi, Frank Thorne
We introduce the notion of orbital L-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formul…
Secondary terms in counting functions for cubic fields
Takashi Taniguchi, Frank Thorne
We prove the existence of secondary terms of order X^{5/6} in the Davenport-Heilbronn theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic fields t…
Distributions of discriminants of cubic algebras II
Takashi Taniguchi
Let be a number field and the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of and derive some de…
On parameterizations of rational orbits of some forms of prehomogeneous vector spaces
Takashi Taniguchi
In this paper, we consider the structure of rational orbit of some prehomogeneous vector spaces originally motivated by Wright and Yukie. After we give a refinement of Wright-Yukie…
Distributions of discriminants of cubic algebras
Takashi Taniguchi
We study the space of binary cubic and quadratic forms over the ring of integers of an algebraic number field . By applying the theory of prehomogeneous vector spaces founde…
On proportional constants of the mean value of class numbers of quadratic extensions
Takashi Taniguchi
In this article, we slightly refine the mean value theorem for the class number of quadratic extensions obtained by Goldfeld-Hoffstein and Datskovsky. We determine all the proporti…
A mean value theorem for the square of class number times regulator of quadratic extensions
Takashi Taniguchi
Let be a number field. In this paper, we give a formula for the mean value of the square of class numbers times regulators for certain families of quadratic extensions of c…
On the zeta functions of prehomogeneous vector spaces for pair of simple algebras
Takashi Taniguchi
In this paper, we consider the prehomogeneous vector space for pair of simple algebras which are -forms of the type and the type. We mainly study the non-split cases…
A mean value theorem for orders of degree zero divisor class groups of quadratic extensions over a function field
Takashi Taniguchi
Let be a function field of one variable over a finite field with the characteristic not equal to two. In this paper, we consider the prehomogeneous representation of the space…