scholarly journals On an exponential sum involving the Möbius function

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
H. Maier ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Exponential Sum ◽  
Möbius Function ◽  
Absolute Value ◽  
Mobius Function ◽  
International Audience ◽  
The Absolute

International audience In this paper we study the upper bound for the absolute value of the exponential sum related to the Möbius function unconditionally and present some interesting applications also.

scholarly journals The Energy Spectral Density of the Mertens Function

2021 ◽  
Author(s):  
Darrell Cox ◽  
Eldar Sultanow ◽  
Sourangshu Ghosh
Keyword(s):  
Spectral Density ◽  
Upper Bound ◽  
Fourier Transforms ◽  
Möbius Function ◽  
Recursive Definition ◽  
Absolute Value ◽  
Energy Spectral Density ◽  
Mobius Function ◽  
The Absolute

The Mertens function is the summatory Mobius function but the Mertens function can be generated recursively without using this definition. This recursive definition is the basis of autocorrelations that can be done on sequences of Mertens function values. Fourier transforms of the autocorrelations result in the energy spectral density. A likely upper bound of the absolute value of the Mertens function is determined.

scholarly journals Mean-square upper bound of Hecke $L$-functions on the critical line.

2003 ◽  
Vol Volume 26 ◽  
Author(s):  
A Sankaranarayanan
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Critical Line ◽  
Congruence Subgroup ◽  
Mean Square ◽  
Absolute Value ◽  
International Audience ◽  
The Mean ◽  
The Absolute

International audience We prove the upper bound for the mean-square of the absolute value of the Hecke $L$-functions (attached to a holomorphic cusp form) defined for the congruence subgroup $\Gamma_0 (N)$ on the critical line uniformly with respect to its conductor $N$.

scholarly journals How to get the weak order out of a digraph ?

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Francois Viard
Keyword(s):  
Wreath Product ◽  
Symmetric Function ◽  
Coxeter Groups ◽  
Symmetric Functions ◽  
Weak Order ◽  
Möbius Function ◽  
Mobius Function ◽  
Acyclic Digraph ◽  
International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, and the flag weak order on the wreath product &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$<sub>$n-1$</sub> case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$<sub>$n-1$</sub>, $B$<sub>$n$</sub>, $Ã$<sub>$n$</sub>, ainsi qu’une variante de l’ordre faible sur les produits en couronne &#8484;<sub>$r$</sub> &#8768; $S$<sub>$n$</sub> introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$<sub>$n-1$</sub> où l’on obtient les séries de Stanley classiques.

scholarly journals The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Leonid Tolmatz
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Saddle Point ◽  
Point Method ◽  
Saddle Point Method ◽  
Tail Asymptotics ◽  
Absolute Value ◽  
Exact Tail Asymptotics ◽  
International Audience ◽  
The Absolute

International audience The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

scholarly journals The Hodge Structure of the Coloring Complex of a Hypergraph (Extended Abstract)

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sarah C Rundell ◽  
Jane H Long
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Hodge Structure ◽  
Hodge Decomposition ◽  
Simple Graph ◽  
Absolute Value ◽  
International Audience ◽  
The Absolute ◽  
Nous Examinons ◽  
Coloring Complex

International audience Let $G$ be a simple graph with $n$ vertices. The coloring complex$ Δ (G)$ was defined by Steingrímsson, and the homology of $Δ (G)$ was shown to be nonzero only in dimension $n-3$ by Jonsson. Hanlon recently showed that the Eulerian idempotents provide a decomposition of the homology group $H_{n-3}(Δ (G))$ where the dimension of the $j^th$ component in the decomposition, $H_{n-3}^{(j)}(Δ (G))$, equals the absolute value of the coefficient of $λ ^j$ in the chromatic polynomial of $G, _{\mathcal{χg}}(λ )$. Let $H$ be a hypergraph with $n$ vertices. In this paper, we define the coloring complex of a hypergraph, $Δ (H)$, and show that the coefficient of $λ ^j$ in $χ _H(λ )$ gives the Euler Characteristic of the $j^{th}$ Hodge subcomplex of the Hodge decomposition of $Δ (H)$. We also examine conditions on a hypergraph, $H$, for which its Hodge subcomplexes are Cohen-Macaulay, and thus where the absolute value of the coefficient of $λ ^j$ in $χ _H(λ )$ equals the dimension of the $j^{th}$ Hodge piece of the Hodge decomposition of $Δ (H)$. Soit $G$ un graphe simple à n sommets. Le complexe de coloriage $Δ (G)$ a été défini par Steingrímsson et Jonsson a prouvé que l'homologie de $Δ (G)$ est non nulle seulement en dimension $n-3$. Hanlon a récemment prouvé que les idempotents eulériens fournissent une décomposition du groupe d'homologie $H_{n-3}(Δ (G))$ où la dimension de la $j^e$ composante dans la décomposition de $H_{n-3}^{(j)}(Δ (G))$ est égale à la valeur absolue du coefficient de $λ ^j$ dans le polynôme chromatique de $G, _{\mathcal{χg}}(λ )$ . Soit H un hypergraphe à $n$ sommets. Dans ce texte, nous définissons le complexe de coloration d'un hypergraphe $Δ (H)$ et nous prouvons que le coefficient de $λ ^j$ dans $χ _H(λ )$ donne la caractéristique d'Euler du $j^e$ sous-complexe de Hodge dans la décomposition de Hodge de Δ (H). Nous examinons également des conditions sur un hypergraphe H pour lesquelles les sous-complexes de Hodge sont Cohen-Macaulay. Ainsi la valeur absolue du coefficient de $λ ^j$ in $χ _H(λ )$ est égale à la dimension du $j^e$sous-complexe de Hodge dans la décomposition de Hodge de $Δ (H)$.

scholarly journals A Formula for the Möbius Function of the Permutation Poset Based on a Topological Decomposition

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jason P Smith
Keyword(s):  
Significant Proportion ◽  
Möbius Function ◽  
Permutation Patterns ◽  
Mobius Function ◽  
International Audience ◽  
Definition Of ◽  
Object Of Study ◽  
Pattern Containment ◽  
Exponential Complexity ◽  
Exponential Part

International audience The poset P of all permutations ordered by pattern containment is a fundamental object of study in the field of permutation patterns. This poset has a very rich and complex topology and an understanding of its Möbius function has proved particularly elusive, although results have been slowly emerging in the last few years. Using a variety of topological techniques we present a two term formula for the Mo ̈bius function of intervals in P. The first term in this formula is, up to sign, the number of so called normal occurrences of one permutation in another. Our definition of normal occurrences is similar to those that have appeared in several variations in the literature on the Möbius function of this and other posets, but simpler than most of them. The second term in the formula is (still) complicated, but we conjecture that it equals zero for a significant proportion of intervals. We present some cases where the second term vanishes and others where it is nonzero. Computing the Möbius function recursively from its definition has exponential complexity, whereas the computation of the first term in our formula is polynomial and the exponential part is isolated to the second term, which seems to often vanish. This is thus the first polynomial time formula for the Möbius function of what appears to be a large proportion of all intervals of P.

scholarly journals The Rees product of the cubical lattice with the chain

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Patricia Muldoon ◽  
Margaret A. Readdy
Keyword(s):  
Möbius Function ◽  
Explicit Formulas ◽  
Bijective Proof ◽  
Mobius Function ◽  
International Audience

International audience We study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give several explicit formulas for the Möbius function. The last formula is expressed in terms of the permanent of a matrix and is given by a bijective proof. Nous étudions des propriétés énumératives et homologiques du produit de Rees du treillis cubique avec la chaîne. Nous donnons plusieurs formules explicites de la fonction de Möbius de ce poset. La dernière de ces formules est exprimée en termes du permanent d’une matrice et le résultat est donné par une preuve bijective.

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