Let $t$ be the integral part of the real number $t$.The aim of this short note is to study the distribution of elements of the set $\mathcal{S}(x) := \{\frac{x}{n} : 1\le n\le x\}$ in the ...arithmetical progression $\{a+dq\}_{d\ge 0}$.Our result is as follows: the asymptotic formula\begin{equation}\label{YW:result}S(x; q, a):= \sum_{\substack{m\in \mathcal{S}(x)\\ m\equiv a ({\rm mod}\,q)}} 1 = \frac{2\sqrt{x}}{q} + O((x/q)^{1/3}\log x)\end{equation}holds uniformly for $x\ge 3$, $1\le q\le x^{1/4}/(\log x)^{3/2}$ and $1\le a\le q$,where the implied constant is absolute.The special case of \eqref{YW:result} with fixed $q$ and $a=q$ confirms a recent numeric test of Heyman.
Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-H\"older vectors in certain E-linear representations of Z_p. In the present paper, we define and study ...super-H\"older vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K_\infty / K of a p-adic field K, and compute the super-H\"older vectors in the tilt of K_\infty. We show that these super-H\"older vectors are the perfection of the field of norms of K_\infty / K. By specializing to the case of a Lubin-Tate extension, we are able to recover E((Y)) inside the Y-adic completion of its perfection, seen as a valued E-vector space endowed with the action of O_K^\times given by the endomorphisms of the corresponding Lubin-Tate group.
The evenness and the values modulo 4 of the lengths of the periods of the continued fraction expansions of p and 2p for p ≡ 3 (mod 4) a prime are known. Here we prove similar results for the ...continued fraction expansion of pq, where p, q ≡ 3 (mod 4) are distinct primes.
Let
$N/K$
be a finite Galois extension of p-adic number fields, and let
$\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$
be an r-dimensional unramified ...representation of the absolute Galois group
$G_K$
, which is the restriction of an unramified representation
$\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$
. In this paper, we consider the
$\mathrm {Gal}(N/K)$
-equivariant local
$\varepsilon $
-conjecture for the p-adic representation
$T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$
. For example, if A is an abelian variety of dimension r defined over
${{\mathbb Q}_{p}}$
with good ordinary reduction, then the Tate module
$T = T_p\hat A$
associated to the formal group
$\hat A$
of A is a p-adic representation of this form. We prove the conjecture for all tame extensions
$N/K$
and a certain family of weakly and wildly ramified extensions
$N/K$
. This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.
Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on ...$\mathrm {GL}(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.
This paper introduces the idea of £-Smarandache p seudo fuzzy ideal o f a p seudo BH– algebra. Some suggestions and examples are given to comprehend the features of this idea and to decide the ...connected aspects among them with the £-Smarandache p seudo ideal o f a p seudo BH– algebra.
Let \(G\) be a \(p\)-adic group that splits over an unramified extension. We decompose \(\text{Rep}_{\unicodeSTIX{x1D6EC}}^{0}(G)\), the abelian category of smooth level \(0\) representations of ...\(G\) with coefficients in \(\unicodeSTIX{x1D6EC}=\overline{\mathbb{Q}}_{\ell }\) or \(\overline{\mathbb{Z}}_{\ell }\), into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.
In this note, we prove a general identity between a $q$-multisum $B_N(q)$ and a sum of $N^2$ products of quotients of theta functions. The $q$-multisum $B_N(q)$ recently arose in the computation of a ...probability involving modules over finite chain rings.
![Super-H\"older vectors and ...](http://api.summon.serialssolutions.com/2.0.0/image/issn/XR4PS5YY4K/1937-0652/small "Super-H\\"older vectors and the field of norms")
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