Displaying similar documents to “Inequalities for Taylor series involving the divisor function”

On the divisor function over Piatetski-Shapiro sequences

Hui Wang, Yu Zhang (2023)

Czechoslovak Mathematical Journal

Similarity:

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 < 6 5 , n x d ( [ n c ] ) = c x log x + ( 2 γ - c ) x + O 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.

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

Boban Karapetrović (2018)

Czechoslovak Mathematical Journal

Similarity:

We show that if α > 1 , then the logarithmically weighted Bergman space A log α 2 is mapped by the Libera operator into the space A log α - 1 2 , while if α > 2 and 0 < ε 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 maps the logarithmically weighted Bloch space log α , α , into itself, while H maps log α into log α + 1 .In Pavlović’s paper (2016) it is shown that maps the logarithmically weighted Hardy-Bloch space log α 1 , α > 0 , into log α - 1 1 . We show that this result is sharp. We also show that H maps log α 1 , α 0 ,...

On sum-product representations in q

Mei-Chu Chang (2006)

Journal of the European Mathematical Society

Similarity:

The purpose of this paper is to investigate efficient representations of the residue classes modulo q , by performing sum and product set operations starting from a given subset A of q . We consider the case of very small sets A and composite q for which not much seemed known (nontrivial results were recently obtained when q is prime or when log | A | log q ). Roughly speaking we show that all residue classes are obtained from a k -fold sum of an r -fold product set of A , where r log q and log k log q , provided the...

Dimension of weakly expanding points for quadratic maps

Samuel Senti (2003)

Bulletin de la Société Mathématique de France

Similarity:

For the real quadratic map P a ( x ) = x 2 + a and a given ϵ &gt; 0 a point x has good expansion properties if any interval containing x also contains a neighborhood  J of x with P a n | J univalent, with bounded distortion and B ( 0 , ϵ ) P a n ( J ) for some n . The ϵ -weakly expanding set is the set of points which do not have good expansion properties. Let α denote the negative fixed point and M the first return time of the critical orbit to [ α , - α ] . We show there is a set of parameters with positive Lebesgue measure for which the Hausdorff...

Remarks on Ramanujan's inequality concerning the prime counting function

Mehdi Hassani (2021)

Communications in Mathematics

Similarity:

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x ) 2 < e x log x π 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 x log x on its right hand side by the factor 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...

A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

Similarity:

Let K be an algebraic number field with non-trivial class group G and K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) denote the number of non-zero principal ideals a K with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F k ( x ) behaves for x → ∞ asymptotically like x ( l o g x ) 1 - 1 / | G | ( l o g l o g x ) k ( G ) . We prove, among other results, that ( C n C n ) = n + n for all integers n₁,n₂ with 1 < n₁|n₂.

Limits of log canonical thresholds

Tommaso de Fernex, Mircea Mustață (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let 𝒯 n denote the set of log canonical thresholds of pairs ( X , Y ) , with X a nonsingular variety of dimension n , and Y a nonempty closed subscheme of X . Using non-standard methods, we show that every limit of a decreasing sequence in 𝒯 n lies in 𝒯 n - 1 , proving in this setting a conjecture of Kollár. We also show that 𝒯 n is closed in 𝐑 ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in...

Uniform algebras and analytic multi­functions

Zbigniew Slodkowski (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Similarity:

Dati due elementi f e g in un'algebra uniforme A , sia G = f ( M A / f ( A ) . Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione λ log max g ( f - 1 ( λ ) ) è subarmonica su G e che l’applicazione λ g ( f - 1 ( λ ) ) è analitica nel senso di Oka.

Equilibrium states for interval maps: the potential - t log | D f |

Henk Bruin, Mike Todd (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let f : I I be a C 2 multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential φ t : x - t log | D f ( x ) | for t close to 1 , and also that the pressure function t P ( φ t ) is analytic on an appropriate interval near t = 1 .

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

Similarity:

For any positive integer k and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying r 1 , k ( A , n ) = r 1 , k ( A , n ) for all n ≥ n₀, we have r 1 , k ( A , n ) as n → ∞.

Asymmetric covariance estimates of Brascamp–Lieb type and related inequalities for log-concave measures

Eric A. Carlen, Dario Cordero-Erausquin, Elliott H. Lieb (2013)

Annales de l'I.H.P. Probabilités et statistiques

Similarity:

An inequality of Brascamp and Lieb provides a bound on the covariance of two functions with respect to log-concave measures. The bound estimates the covariance by the product of the L 2 norms of the gradients of the functions, where the magnitude of the gradient is computed using an inner product given by the inverse Hessian matrix of the potential of the log-concave measure. Menz and Otto [Uniform logarithmic Sobolev inequalities for conservative spin systems with super-quadratic single-site...

Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

Similarity:

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 ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

A generalized Kahane-Khinchin inequality

S. Favorov (1998)

Studia Mathematica

Similarity:

The inequality ʃ l o g | a n e 2 π i φ n | d φ 1 d φ n C l o g ( | a n | 2 ) 1 / 2 with an absolute constant C, and similar ones, are extended to the case of a n belonging to an arbitrary normed space X and an arbitrary compact group of unitary operators on X instead of the operators of multiplication by e 2 π i φ .

The mean square of the divisor function

Chaohua Jia, Ayyadurai Sankaranarayanan (2014)

Acta Arithmetica

Similarity:

Let d(n) be the divisor function. In 1916, S. Ramanujan stated without proof that n x d ² ( n ) = x P ( l o g x ) + E ( x ) , where P(y) is a cubic polynomial in y and E ( x ) = O ( x 3 / 5 + ε ) , with ε being a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis (RH), E ( x ) = O ( x 1 / 2 + ε ) . In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce E ( x ) = O ( x 1 / 2 ( l o g x ) l o g l o g x ) . In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we prove E ( x ) = O ( x 1 / 2 ( l o g x ) ) . ...