Graph Invariants of Finite Groups via a Theorem of Lagarias

Generalized Bombieri - Lagarias' theorem and generalized Li's criterion

We show that Li's criterion equivalent to the Riemann hypothesis, viz. the statement that the sums k_n=Sum_rho(1-(1-1/rho)^n) over Riemann xi-function zeroes as well as all derivatives lambda_n= (n-1)!^(-1)*d^n/dz^n(z^(n-1)*ln(xi(z))_(z=1)), where n=1, 2, 3..., should be non-negative if and only if the Riemann hypothesis holds true, can be generalized and the non-negativity of certain derivatives of the Riemann xi-function estimated at an arbitrary real point a, except a=1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums k_n=Sum_rho(1-(1-…

An explicit upper bound for the least prime ideal in the Chebotarev density theorem

Lagarias, Montgomery, and Odlyzko proved that there exists an effectively computable absolute constant $A_1$ such that for every finite extension $K$ of ${\mathbb{Q}}$, every finite Galois extension $L$ of $K$ with Galois group $G$ and every conjugacy class $C$ of $G$, there exists a prime ideal $\mathfrak{p}$ of $K$ which is unramified in $L$, for which $\left[\frac{L/K}{\mathfrak{p}}\right]=C$, for which $N_{K/{\mathbb Q}}\,\mathfrak{p}$ is a rational prime, and which satisfies $N_{K/{\mathbb Q}}\,{\mathfrak{p}} \leq 2 {d_L}^{A_1}$. In this paper we show without any restriction that $N_{K/{\…

Théorème de Chebotarev effectif

Let K be a number field, and L be a finite normal extension of K with Galois group G. It is known that the number of Frobenius automorphisms corresponding to prime ideals, whose norms are less than x, is equivalent to the logarithmic integral as x tends to infinity, and these automorphisms are well-distributed among the conjugacy classes of G : this is the Chebotarev theorem. The purpose of this paper is to compute the absolute constants of the error term appearing in a previous work about this theorem, due to Lagarias and Odlyzko.

Chebotarev 密度定理的有效版本

An effective version of Chebotarev's density theorem

本文提出了适用于所有非有理数域的 Chebotarev 密度定理有效版本的显式细化，基于平滑素理想计数函数的显式公式，结合 Dedekind zeta 函数的零点区域和零点估计，给出了更优的界。

Spectra for cubes in products of finite cyclic groups

We consider "cubes" in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the group and is called spectral if it admits an orhogonal basis of characters for the functions supported on the set.) We show an analog of a theorem due to Iosevich and Pedersen, Lagarias, Reeds and Wang, and the third author of this paper, which identified the tiling complements of the unit cube in Euclidean space with the spectra of the same cube.

On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group

Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with i…