Related papers: Dirichlet's Lemma in Number Fields

A modified Dirichlet character $f$ is a completely multiplicative function such that for some Dirichlet character $\chi$, $f(p)=\chi(p)$ for all but a finite number of primes $p\in S$, and for those exceptional primes $p\in S$, $|f(p)|\leq…

Let $\Delta\colon G \to \mathrm{GL}(n, K)$ be an absolutely irreducible representation of an arbitrary group $G$ over an arbitrary field $K$; let $\chi\colon G \to K\colon g \mapsto \mathrm{tr}(\Delta(g))$ be its character. In this paper,…

Let $F$ be a real quadratic number field with discriminant $D$ and $\mathcal{O}_F$ the ring of integers in $F$. Let $\chi_F$ be the Dirichlet character associated to $F/\mathbb{Q}$. Write $L(\chi_F,s)$ for the Dirichlet L-function of…

It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain…

For an even Dirichlet character psi, we obtain a formula for L(1,psi) in terms of a sum of Dirichlet L-series evaluated at s=2 and s=3 and a rapidly convergent numerical series involving the central binomial coefficients. We then derive a…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

We obtain the extended genus field of an abelian extension of a rational function field. We follow the definition of Angl\`es and Jaulent, which uses class field theory. First we show that the natural definition of extended genus field of a…

An ordinary character $\chi $ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $\chi $ have the same discriminant. This is the orthogonal discriminant,…

small In this paper, we define $q$-analogues of Dirichlet's beta function at positive integers, which can be written as $\beta_q(s)=\sum_{k\geq1}\sum_{d|k}\chi(k/d)d^{s-1}q^k$ for $s\in\N^*$, where $q$ is a complex number such that $|q|<1$…

Let $\chi$ be a quadratic Dirichlet character. In some literatures, various asymptotic formulae of $L'(1,\chi)$, under the assumption that $L(1,\chi)$ takes a small value, were derived. In this paper, we will give a new treatment unified…

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in…

For any integer $d\geq 3$ such that $-d$ is a fundamental discriminant, we show that the Dirichlet $L$-function associated with the real primitive character $\chi(\cdot)=(\frac{-d}{\cdot})$ does not vanish on the positive part of the…

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is…

For each nonprincipal Dirichlet character $\chi$, let $n_\chi$ be the least $n$ with $\chi(n) \notin \{0,1\}$. We show that as the average of $n_\chi$ over all nonprincipal characters $\chi$ modulo $q$ is $\ell(q) + o(1)$, where $\ell(q)$…

The first and second moments are established for the family of quadratic Dirichlet $L$--functions over the rational function field at the central point $s=\tfrac{1}{2}$ where the character $\chi$ is defined by the Legendre symbol for…

The main object of this paper is to give the generalized von mangoldt function using the L-additive function which can help us to make it possible to calculate The Dirichlet series of the arithmetic derivative $\delta$ and Dirichlet series…

The Dedekind zeta function of a quadratic number field factors as a product of the Riemann zeta function and the $L$-function of a quadratic Dirichlet character. We categorify this formula using objective linear algebra in the abstract…

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are…

We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F(n)^{-s}$, where $F(n)$ denotes the $n$th Fibonacci number. We then give three different…