Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

The classical Dedekind sums s(d, c) can be represented as sums over the partial quotients of the continued fraction expansion of the rational (frac{d}{c}). Hardy sums, the analog integer-valued sums arising in the transformation of the logarithms of (theta )-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial quotients of these continued fractions. As an application, we prove that the graph of the Hardy sums is dense in (textbf{R}times textbf{Z}).

At the international congress of mathematicians in 1900, Hilbert claimed that the Riemann zeta function (zeta (s)) is not the solution of any algebraic ordinary differential equations on its region of analyticity. Let T be an infinite order linear differential operator introduced by Van Gorder in 2015. Recently, Prado and Klinger-Logan [9] showed that the Hurwitz zeta function (zeta (s,a)) formally satisfies the following linear differential equation

Then in [6], by defining (T_{p}^{a}), a p-adic analogue of Van Gorder’s operator T,  we constructed the following convergent infinite order linear differential equation satisfied by the p-adic Hurwitz-type Euler zeta function (zeta _{p,E}(s,a))

In this paper, we consider this problem in the positive characteristic case. That is, by introducing (zeta _{infty }(s_{0},s,a,n)), a Hurwitz type refinement of Goss zeta function, and an infinite order linear difference operator L, we establish the following difference equation

We extend previous work by constructing a universal abelian tensor category (textbf{T}_t) generated by two objects X, Y equipped with finite filtrations (0subsetneq X_0subsetneq ...subsetneq X_{t+1}= X) and (0subsetneq Y_0subsetneq ... subsetneq Y_{t+1}= Y), and with a pairing (Xotimes Yrightarrow mathbbm {1}), where (mathbbm {1}) is the monoidal unit. This category is modeled as a category of representations of a Mackey Lie algebra (mathfrak {gl}^M(V,V_*)) of cardinality (2^{aleph _t}), associated to a diagonalizable pairing between two vector spaces (V,V_*) of dimension (aleph _t) over an algebraically closed field ({{mathbb {K}}}) of characteristic 0. As a preliminary step, we study a tensor category ({{mathbb {T}}}_t) generated by the algebraic duals (V^*) and ((V_*)^*). The injective hull of the trivial module ({{mathbb {K}}}) in ({{mathbb {T}}}_t) is a commutative algebra I, and the category (textbf{T}_t) consists of all free I-modules in ({{mathbb {T}}}_t). An essential novelty in our work is the explicit computation of Ext-spaces between simples in both categories (textbf{T}_t) and ({{mathbb {T}}}_t), which had been an open problem already for (t=0). This provides a direct link from the theory of universal tensor categories to Littlewood-Richardson-type combinatorics.

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.

Let (zeta (s)), (s={sigma }+it), be the Riemann zeta function. We use Fourier analysis to obtain, after a preliminary study of quadratic Riemann sums, a precise formula of the local integrals (int _n^{n+1} |zeta ({sigma }+it ) |^2 textrm{d}t), for (frac{1}{2}<{sigma }<1). We also study related (mathcal {S}^{2})-Stepanov norms of (zeta (s)) in connection with the strong Voronin Universality Theorem.

We state and prove a formula for the adjoint of the nullwert map from spaces of Jacobi cusp forms of lattice index to spaces of modular forms. Furthermore, we prove a nonvanishing result for the image of the adjoint of the nullwert map.

In this paper we study the theta lifting of a weight 2 Bianchi modular form ({mathcal {F}}) of level (Gamma _0({mathfrak {n}})) with ({mathfrak {n}}) square-free to a weight 2 holomorphic Siegel modular form. Motivated by Prasanna’s work for the Shintani lifting, we define the local Schwartz function at finite places using a quadratic Hecke character (chi ) of square-free conductor ({mathfrak {f}}) coprime to level ({mathfrak {n}}). Then, at certain 2 by 2 g matrices (beta ) related to ({mathfrak {f}}), we can express the Fourier coefficient of this theta lifting as a multiple of (L({mathcal {F}},chi ,1)) by a non-zero constant. If the twisted L-value is known to be non-vanishing, we can deduce the non-vanishing of our theta lifting.

To any V in the Grassmannian (textrm{Gr}_k({mathbb R}^n)) of k-dimensional vector subspaces in ({mathbb {R}}^n) one can associate the diagonal entries of the ((ntimes n)) matrix corresponding to the orthogonal projection of ({mathbb {R}}^n) to V. One obtains a map (textrm{Gr}_k({mathbb {R}}^n)rightarrow {mathbb {R}}^n) (the Schur–Horn map). The main result of this paper is a criterion for pre-images of vectors in ({mathbb {R}}^n) to be connected. This will allow us to deduce connectivity criteria for a certain class of subspaces of the real Stiefel manifold which arise naturally in frame theory. We extend in this way results of Cahill et al. (SIAM J Appl Algebra Geom 1:38–72, 2017).

Let X be a K3 surface, let C be a smooth curve of genus g on X, and let A be a line bundle of degree d on C. Then a line bundle M on X with (Motimes {mathcal {O}}_C=A) is called a lift of A. In this paper, we prove that if the dimension of the linear system |A| is (rge 2), (g>2d-3+(r-1)^2), (dge 2r+4), and A computes the Clifford index of C, then there exists a base point free lift M of A such that the general member of |M| is a smooth curve of genus r. In particular, if |A| is a base point free net which defines a double covering (pi :Clongrightarrow C_0) of a smooth curve (C_0subset {mathbb {P}}^2) of degree (kge 4) branched at distinct 6k points on (C_0), then, by using the aforementioned result, we can also show that there exists a 2:1 morphism ({tilde{pi }}:Xlongrightarrow {mathbb {P}}^2) such that ({tilde{pi }}|_C=pi ).

Let K be a number field and G a finitely generated torsion-free subgroup of (K^times ). Given a prime (mathfrak {p}) of K we denote by ({{,textrm{ind},}}_mathfrak {p}(G)) the index of the subgroup ((Gbmod mathfrak {p})) of the multiplicative group of the residue field at (mathfrak {p}). Under the Generalized Riemann Hypothesis we determine the natural density of primes of K for which this index is in a prescribed set S and has prescribed Frobenius in a finite Galois extension F of K. We study in detail the natural density in case S is an arithmetic progression, in particular its positivity.