Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs

Victor V. Batyrev

Displaying similar documents to “Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs”

On the sum of the first n values of the Euler function

R. Balasubramanian, Florian Luca, Dimbinaina Ralaivaosaona (2014)

Acta Arithmetica

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Let ϕ(n) be the Euler function of n. We put $E (n) = \sum_{m \leq n} ϕ (m) - (3 / π ²) n ²$ and give an asymptotic formula for the second moment of E(n).

On the range of Carmichael's universal-exponent function

Florian Luca, Carl Pomerance (2014)

Acta Arithmetica

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Let λ denote Carmichael’s function, so λ(n) is the universal exponent for the multiplicative group modulo n. It is closely related to Euler’s φ-function, but we show here that the image of λ is much denser than the image of φ. In particular the number of λ-values to x exceeds $x / {(l o g x)}^{. 36}$ for all large x, while for φ it is equal to $x / {(l o g x)}^{1 + o (1)}$ , an old result of Erdős. We also improve on an earlier result of the first author and Friedlander giving an upper bound for the distribution of λ-values.

On the Euler characteristic of the links of a set determined by smooth definable functions

Krzysztof Jan Nowak (2008)

Annales Polonici Mathematici

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The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ( $C^{\infty}$ ) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism....

Thin vortex tubes in the stationary Euler equation

Alberto Enciso, Daniel Peralta-Salas (2013)

Journées Équations aux dérivées partielles

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In this paper we outline some recent results concerning the existence of steady solutions to the Euler equation in $ℝ^{3}$ with a prescribed set of (possibly knotted and linked) thin vortex tubes.

Inequalities for Taylor series involving the divisor function

Horst Alzer, Man Kam Kwong (2022)

Czechoslovak Mathematical Journal

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Let $T (q) = \sum_{k = 1}^{\infty} d (k) q^{k}, | q | < 1,$ where $d (k)$ denotes the number of positive divisors of the natural number $k$ . We present monotonicity properties of functions defined in terms of $T$ . More specifically, we prove that $H (q) = T (q) - \frac{log (1 - q)}{log (q)}$ is strictly increasing on $(0, 1)$ , while $F (q) = \frac{1 - q}{q} H (q)$ is strictly decreasing on $(0, 1)$ . These results are then applied to obtain various inequalities, one of which states that the double inequality $α \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)} < T (q) < β \frac{q}{1 - q} + \frac{log (1 - q)}{log (q)}, 0 < q < 1,$ holds with the best possible constant factors $α = γ$ and $β = 1$ . Here, $γ$ denotes Euler’s constant. This refines a result of Salem, who...

Average Value of the Euler Function on Binary Palindromes

William D. Banks, Igor E. Shparlinski (2006)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study values of the Euler function φ(n) taken on binary palindromes of even length. In particular, if $ℬ_{2 ℓ}$ denotes the set of binary palindromes with precisely 2ℓ binary digits, we derive an asymptotic formula for the average value of the Euler function on $ℬ_{2 ℓ}$ .

A Menon-type identity using Klee's function

Arya Chandran, Neha Elizabeth Thomas, K. Vishnu Namboothiri (2022)

Czechoslovak Mathematical Journal

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Menon’s identity is a classical identity involving gcd sums and the Euler totient function $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

Libera and Hilbert matrix operator on logarithmically weighted Bergman, Bloch and Hardy-Bloch spaces

Boban Karapetrović (2018)

Czechoslovak Mathematical Journal

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We show that if $α > 1$ , then the logarithmically weighted Bergman space $A_{{log}^{α}}^{2}$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{α - 1}}^{2}$ , while if $α > 2$ and $0 < ε \leq α - 2$ , then the Hilbert matrix operator $H$ maps $A_{{log}^{α}}^{2}$ into $A_{{log}^{α - 2 - ε}}^{2}$ .We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space $ℬ_{{log}^{α}}$ , $α \in ℝ$ , into itself, while $H$ maps $ℬ_{{log}^{α}}$ into $ℬ_{{log}^{α + 1}}$ .In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space $ℬ_{{log}^{α}}^{1}$ , $α > 0$ , into $ℬ_{{log}^{α - 1}}^{1}$ . We show that this result is sharp. We also show that $H$ maps $ℬ_{{log}^{α}}^{1}$ , $α \geq 0$ ,...

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

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Let $[x]$ be an integer part of $x$ and $d (n)$ be the number of positive divisor of $n$ . Inspired by some results of M. Jutila (1987), we prove that for $1 < c < \frac{6}{5}$ , $\sum_{n \leq x} d ([n^{c}]) = c x log x + (2 γ - c) x + O (\frac{x}{log x}),$ where $γ$ is the Euler constant and $[n^{c}]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani (2021)

Communications in Mathematics

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In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that $π {(x)}^{2} < \frac{e x}{log x} π (\frac{x}{e})$ for $x$ sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor $\frac{x}{log x}$ on its right hand side by the factor $\frac{x}{log x - h}$ for a given $h$ , and by replacing the numerical factor $e$ by a given positive $α$ . Finally, we introduce and study inequalities...

On the Riesz means of n/ϕ(n) - III

Ayyadurai Sankaranarayanan, Saurabh Kumar Singh (2015)

Acta Arithmetica

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Let ϕ(n) denote the Euler totient function. We study the error term of the general kth Riesz mean of the arithmetical function n/ϕ(n) for any positive integer k ≥ 1, namely the error term $E_{k} (x)$ where $1 / k! \sum_{n \leq x} n / ϕ (n) {(1 - n / x)}^{k} = M_{k} (x) + E_{k} (x)$ . For instance, the upper bound for |Ek(x)| established here improves the earlier known upper bounds for all integers k satisfying $k ≫ {(l o g x)}^{1 + ϵ}$ .

Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen (2007)

Bulletin de la Société Mathématique de France

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^{n} A$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^{n} A$ carries a divisor with vanishing Beauville-Bogomolov square.

The Complete Monotonicity of a Function Studied by Miller and Moskowitz

Horst Alzer (2009)

Bollettino dell'Unione Matematica Italiana

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Let $S(x)=log(1+x)+\int_{0}^{1}\left[1-\left(\frac{1+t}{2}\right)^{x}\right]\frac{% dt}{\log t}\quad\text{and}\quad F(x)=\log 2-S(x)\,\,(0<x\in\mathbb{R}).$ We prove that $F$ is completely monotonic on $(0,\infty)$ . This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\infty)$ . The sequence $\{S(k)\}$ $(k=1,2,\dots)$ plays a role in information theory.

A direct solver for finite element matrices requiring $O (N log N)$ memory places

Vejchodský, Tomáš

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We present a method that in certain sense stores the inverse of the stiffness matrix in $O (N log N)$ memory places, where $N$ is the number of degrees of freedom and hence the matrix size. The setup of this storage format requires $O (N^{3 / 2})$ arithmetic operations. However, once the setup is done, the multiplication of the inverse matrix and a vector can be performed with $O (N log N)$ operations. This approach applies to the first order finite element discretization of linear elliptic and parabolic problems in triangular...

Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

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For any given positive integer k, and any set A of nonnegative integers, let $r_{1, k} (A, n)$ denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Finiteness of cominuscule quantum $K$ -theory

Anders S. Buch, Pierre-Emmanuel Chaput, Leonardo C. Mihalcea, Nicolas Perrin (2013)

Annales scientifiques de l'École Normale Supérieure

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The product of two Schubert classes in the quantum $K$ -theory ring of a homogeneous space $X = G / P$ is a formal power series with coefficients in the Grothendieck ring of algebraic vector bundles on $X$ . We show that if $X$ is cominuscule, then this power series has only finitely many non-zero terms. The proof is based on a geometric study of boundary Gromov-Witten varieties in the Kontsevich moduli space, consisting of stable maps to $X$ that take the marked points to general Schubert varieties and...