Related papers: Dedekind sums in the $p$-adic number field

We define Dedekind sums attached to a totally real number field of class number one. We prove that they satisfy some reciprocity law. Then we relate them to special values of Hecke $L$-functions. We conclude that they are ruled by Stark's…

Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete…

A well-known result, due to Meyer, states that the set P of Pisot numbers, generating a real algebraic number field K, is uniformly discrete and relatively dense in the set of positive real number. In the present paper, we show that P is…

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers.…

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number…

We study the asymptotic behaviour of the classical Dedekind sums $s(s_k/t_k)$ for the sequence of convergents $s_k/t_k$ $k\ge 0$, of the transcendental number \BD \sum_{j=0}^\infty\frac {1}{b^{2^j}},\ b\ge 3. \ED In particular, we show that…

In the present paper, the fundamental aim is to consider a p-adic continuous function for an odd prime to inside a p-adic q-analogue of the higher order modified Dedekind-type sums related to q-Genocchi polynomials with weight alpha by…

The ratio set of a set of positive integers $A$ is defined as $R(A) := \{a / b : a, b \in A\}$. The study of the denseness of $R(A)$ in the set of positive real numbers is a classical topic and, more recently, the denseness in the set of…

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$,…

Let $S(a,b)$ denote the normalized Dedekind sum. We study the range of possible values for $S(a,b)=\frac{k}{q}$ with $\gcd(k,q)=1$. Girstmair proved local restrictions on $k$ depending on $q\pmod{12}$ and whether $q$ is a square and…

For $a\in \Bbb Z$ and $b\in\Bbb N$, $(a,b)=1$, let $s(a,b)$ denote the classical Dedekind sum. We show that Dedekind sums take this value infinitely many times in the following sense. There are pairs $(a_i,b_i)$, $i\in\Bbb N$, with $b_i$…

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \Z{p}$ of sufficiently small density then $$\big| \lambda_{1}A+\lambda_{2}A+...+ \lambda_{k}A \big|…

The quotient set, or ratio set, of a set of integers $A$ is defined as $R(A) := \left\{a/b : a,b \in A,\; b \neq 0\right\}$. We consider the case in which $A$ is the image of $\mathbb{Z}^+$ under a polynomial $f \in \mathbb{Z}[X]$, and we…

In this paper we have discussed convergence of power series both in p-adic norm as well as real norm. We have investigated rational summability of power series with respect to both p-adic norm and real norm under certain conditions. Then we…

In this paper we define the $p$-density of a finite subset $D\subset\ma{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application:…

We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin's continued…

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.

Dedekind sums $s(m,n)$ occur in many fields of mathematics. Since $s(m_1,n)=s(m_2,n)$ if $m_1\equiv m_2$ mod $n$, it is natural to ask which of the Dedekind sums $s(m,n)$, $0\le m<n$, take equal values. So far no simple criterion is known…

We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets…

In a family of $S_{d+1}$-fields ($d=2,3,4$), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For $S_5$-fields, we need to assume the strong Artin conjecture. We also show…