Dirichlet series and homology theory

A Geometric Approach to Homology Theory

10.1017/cbo9780511662669 ◽

1976 ◽

Cited By ~ 26

Author(s):

S. Buonchristiano ◽

C. P. Rourke ◽

B. J. Sanderson

Keyword(s):

Geometric Approach ◽

Homology Theory

Download Full-text

On Dirichlet Series Attached to Holomorphic Cusp Forms on $SO(2, q)$

Automorphic Forms and Number Theory ◽

10.2969/aspm/00710333 ◽

2018 ◽

Cited By ~ 3

Author(s):

Takashi Sugano

Keyword(s):

Dirichlet Series ◽

Cusp Forms ◽

Holomorphic Cusp Forms

Download Full-text

Cancellation in algebraic twisted sums on GL_ m

Forum Mathematicum ◽

10.1515/forum-2020-0252 ◽

2021 ◽

Vol 33 (4) ◽

pp. 1061-1082

Author(s):

Yujiao Jiang ◽

Guangshi Lü

Keyword(s):

Dirichlet Series ◽

Central Character ◽

Multiplicative Functions ◽

Cuspidal Representation ◽

Series Coefficient

Abstract Let π be an automorphic irreducible cuspidal representation of GL m {\operatorname{GL}_{m}} over ℚ {\mathbb{Q}} with unitary central character, and let λ π ⁢ ( n ) {\lambda_{\pi}(n)} be its n-th Dirichlet series coefficient. We study short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by multiplicative functions λ π ⁢ ( n ) {\lambda_{\pi}(n)} and μ ⁢ ( n ) ⁢ λ π ⁢ ( n ) {\mu(n)\lambda_{\pi}(n)} . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 / 2 + ε {q^{1/2+\varepsilon}} for an arbitrary fixed ε > 0 {\varepsilon>0} .

Download Full-text

Blackbox computation of A ∞-algebras

Georgian Mathematical Journal ◽

10.1515/gmj.2010.005 ◽

2010 ◽

Vol 17 (2) ◽

pp. 391-404

Author(s):

Mikael Vejdemo-Johansson

Keyword(s):

Homology Theory ◽

Algebra Structure ◽

Finite Amount ◽

Computational Work ◽

Dg Algebra ◽

Dg Algebras

Abstract Kadeishvili's proof of theminimality theorem [T. Kadeishvili, On the homology theory of fiber spaces, Russ. Math. Surv. 35:3 (1980), 231–238] induces an algorithm for the inductive computation of an A ∞-algebra structure on the homology of a dg-algebra. In this paper, we prove that for one class of dg-algebras, the resulting computation will generate a complete A ∞-algebra structure after a finite amount of computational work.

Download Full-text

A dominated convergence theorem for Eisenstein series

Annales mathématiques du Québec ◽

10.1007/s40316-021-00157-7 ◽

2021 ◽

Author(s):

Johann Franke

Keyword(s):

Fourier Series ◽

Dirichlet Series ◽

Eisenstein Series ◽

Modular Forms ◽

Convergence Theorem ◽

Rational Functions ◽

New Approach ◽

Series Representations ◽

Dominated Convergence ◽

Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

Download Full-text

Path homology theory of edge-colored graphs

Open Mathematics ◽

10.1515/math-2021-0049 ◽

2021 ◽

Vol 19 (1) ◽

pp. 706-723

Author(s):

Yuri V. Muranov ◽

Anna Szczepkowska

Keyword(s):

Spectral Sequence ◽

Edge Coloring ◽

Homology Theory ◽

Homotopy Category ◽

Homology Groups ◽

Colored Graphs ◽

Natural Filtration ◽

Definition Of ◽

Exact Sequences

Abstract In this paper, we introduce the category and the homotopy category of edge-colored digraphs and construct the functorial homology theory on the foundation of the path homology theory provided by Grigoryan, Muranov, and Shing-Tung Yau. We give the construction of the path homology theory for edge-colored graphs that follows immediately from the consideration of natural functor from the category of graphs to the subcategory of symmetrical digraphs. We describe the natural filtration of path homology groups of any digraph equipped with edge coloring, provide the definition of the corresponding spectral sequence, and obtain commutative diagrams and braids of exact sequences.

Download Full-text