Monatshefte fur Mathematik最新文献

In this note we prove a simultaneous extension of the author's joint result with M. Harris for critical values of Rankin-Selberg L-functions $L(s,\Pi \times {\Pi }^{\text{'}})$ (Grobner and Harris in J Inst Math Jussieu 15:711-769, 2016, Thm. 3.9) to (i) general CM-fields F and (ii) cohomological automorphic representations ${\Pi }^{\text{'}}={\Pi }_1⊞\cdots ⊞{\Pi }_k$ which are the isobaric sum of unitary cuspidal automorphic representations ${\Pi }_i$ of general linear groups of arbitrary rank over F. In this sense, the main result of these notes, cf. Theorem 1.9, is a generalization, as well as a complement, of the main results in Raghuram (Forum Math 28:457-489, 2016; Int Math Res Not 2:334-372, 2010. https://doi.org/10.1093/imrn/rnp127), and Mahnkopf (J Inst Math Jussieu 4:553-637, 2005).

We develop a theory of $n\times n$ -matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour $\Gamma$ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of $L^p$ -Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

Consider the divisor sum ${\sum }_{n\le N}\tau (n^2+2bn+c)$ for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of $D(-1)$ -quadruples.

Let $K$ be a number field of degree k and let $O$ be an order in $K$ . A generalized number system over $O$ (GNS for short) is a pair $(p,D)$ where $p\in O\left[x\right]$ is monic and $D\subset O$ is a complete residue system modulo p(0) containing 0. If each $a\in O\left[x\right]$ admits a representation of the form $a\equiv {\sum }_{j=0}^{\ell -1}{d}_j{x}^j(modp)$ with $\ell \in N$ and $d_0,\dots ,d_{\ell -1}\in D$ then the GNS $(p,D)$ is said to have the finiteness property. To a given fundamental domain $F$ of the action of $Z^k$ on $R^k$ we associate a class $G_F:=\{(p,D_F):p\in O\left[x\right]\}$ of GNS whose digit sets $D_F$ are defined in terms of $F$ in a natural way. We are able to prove general results on the finiteness property of GNS in $G_F$ by giving an abstract version of the well-known "dominant condition" on the absolute coefficient p(0) of p. In particular, depending on mild conditions on the topology of $F$ we characterize the finiteness property of $(p(x\pm m),D_F)$ for fixed p and large $m\in N$ . Using our new theory, we are able to give general results on the connection between power integral bases of number fields and GNS.

[This corrects the article DOI: 10.1007/s00605-017-1061-y.].

Labyrinth fractals are self-similar dendrites in the unit square that are defined with the help of a labyrinth set or a labyrinth pattern. In the case when the fractal is generated by a horizontally and vertically blocked pattern, the arc between any two points in the fractal has infinite length (Cristea and Steinsky in Geom Dedicata 141(1):1-17, 2009; Proc Edinb Math Soc 54(2):329-344, 2011). In the case of mixed labyrinth fractals a sequence of labyrinth patterns is used in order to construct the dendrite. In the present article we focus on the length of the arcs between points of mixed labyrinth fractals. We show that, depending on the choice of the patterns in the sequence, both situations can occur: the arc between any two points of the fractal has finite length, or the arc between any two points of the fractal has infinite length. This is in stark contrast to the self-similar case.

Let $C_0\left(G\right)$ denote the near-ring of congruence preserving functions of the group G. We investigate the question "When is $C_0\left(G\right)$ a ring?". We obtain information externally via the lattice structure of the normal subgroups of G and internally via structural properties of the group G.