scholarly journals On Zudilin's $q$-Question about Schmidt's Problem

10.37236/2231 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Victor J. W. Guo ◽  
Jiang Zeng
Keyword(s):  
Number Theory ◽  
Recent Result ◽  
Acta Arith ◽  
New Approach

We propose an elemantary approach to Zudilin's $q$-question about Schmidt's problem [Electron. J. Combin. 11 (2004), #R22], which has been solved in a previous paper [Acta Arith. 127 (2007), 17--31]. The new approach is based on a  $q$-analogue of our recent result in  [J. Number Theory 132 (2012), 1731--1740] derived from $q$-Pfaff-Saalschütz identity.

Microcanonical condensate fluctuations in one-dimensional weakly-interacting Bose gases

2020 ◽  
Vol 34 (35) ◽  
pp. 2050410
Author(s):  
Ji-Xuan Hou
Keyword(s):  
Number Theory ◽  
Low Temperature ◽  
Ground State ◽  
Particle Number ◽  
Microcanonical Ensemble ◽  
Bose Gases ◽  
New Approach ◽  
One Dimensional ◽  
Weakly Interacting ◽  
Counting Statistics

Weakly interacting Bose gases confined in a one-dimensional harmonic trap are studied using microcanonical ensemble approaches. Combining number theory methods, I present a new approach to calculate the particle number counting statistics of the ground state occupation. The results show that the repulsive interatomic interactions increase the ground state fraction and suppresses the fluctuation of ground state at low temperature.

Idempotents, group membership and their applications

2018 ◽  
Vol 68 (6) ◽  
pp. 1231-1312
Author(s):  
Štefan Porubský
Keyword(s):  
Number Theory ◽  
Group Membership ◽  
Ring Theory ◽  
Maximal Subgroups ◽  
Structural Description ◽  
Structural Elements ◽  
New Approach ◽  
Czechoslovak Math ◽  
Semigroup Approach ◽  
Insight Into

AbstractŠ. Schwarz in his paper [SCHWARZ, Š.:Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.:Contribution to the theory of torsion semigroups, Czechoslovak Math. J.3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.

On a problem of Erdös related to common factor differences

2019 ◽  
Vol 15 (05) ◽  
pp. 1059-1068
Author(s):  
Andrew Bremner
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Common Factor ◽  
Difference Set ◽  
Acta Arith ◽  
Factor Difference

Let [Formula: see text] be a positive integer. The factor-difference set [Formula: see text] of [Formula: see text] is the set of absolute values [Formula: see text] of the differences between the factors of any factorization of [Formula: see text] as a product of two integers. Erdős and Rosenfeld [The factor–difference set of integers, Acta Arith. 79(4) (1997) 353–359] ask whether for every positive integer [Formula: see text] there exist integers [Formula: see text] such that [Formula: see text], and prove this is true when [Formula: see text]. Urroz [A note on a conjecture of Erdős and Rosenfeld, J. Number Theory 78(1) (1999) 140–143] shows the result true for [Formula: see text]. The ideas of this paper can be extended, and here, we show the result true for [Formula: see text] by proving there are infinitely many sets of four integers with four common factor differences.

scholarly journals On the order of a modulo n, on average

2016 ◽  
Vol 12 (08) ◽  
pp. 2073-2080 ◽  
Author(s):  
Sungjin Kim
Keyword(s):  
Number Theory ◽  
Positive Constant ◽  
Acta Arith ◽  
Multiplicative Order ◽  
Orders Of Elements

Let [Formula: see text] be an integer. Denote by [Formula: see text] the multiplicative order of [Formula: see text] modulo integer [Formula: see text]. We prove that there is a positive constant [Formula: see text] such that if [Formula: see text], then [Formula: see text] where [Formula: see text] It was known for [Formula: see text] in [P. Kurlberg and C. Pomerance, On a problem of Arnold: The average multiplicative order of a given integer, Algebra Number Theory 7 (2013) 981–999] in which they refer to [F. Luca and I. E. Shparlinski, Average multiplicative orders of elements modulo [Formula: see text], Acta Arith. 109(4) (2003) 387–411.

scholarly journals AN EXTENSION OF A RESULT OF ERDŐS AND ZAREMBA

2020 ◽  
Vol 63 (1) ◽  
pp. 193-222
Author(s):  
MICHEL JEAN GEORGES WEBER
Keyword(s):  
Recent Result ◽  
Finite Sequence ◽  
New Approach ◽  
Euler’S Constant ◽  
Euler's Constant

AbstractErdös and Zaremba showed that $ \limsup_{n\to \infty} \frac{\Phi(n)}{(\log\log n)^2}=e^\gamma$ , γ being Euler’s constant, where $\Phi(n)=\sum_{d|n} \frac{\log d}{d}$ .We extend this result to the function $\Psi(n)= \sum_{d|n} \frac{(\log d )(\log\log d)}{d}$ and some other functions. We show that $ \limsup_{n\to \infty}\, \frac{\Psi(n)}{(\log\log n)^2(\log\log\log n)}\,=\, e^\gamma$ . The proof requires a new approach. As an application, we prove that for any $\eta>1$ , any finite sequence of reals $\{c_k, k\in K\}$ , $\sum_{k,\ell\in K} c_kc_\ell \, \frac{\gcd(k,\ell)^{2}}{k\ell} \le C(\eta) \sum_{\nu\in K} c_\nu^2(\log\log\log \nu)^\eta \Psi(\nu)$ , where C(η) depends on η only. This improves a recent result obtained by the author.

scholarly journals Une nouvelle approche dans la théorie des entiers friables

2017 ◽  
Vol 153 (3) ◽  
pp. 453-473
Author(s):  
Régis de la Bretèche ◽  
Gérald Tenenbaum
Keyword(s):  
Number Theory ◽  
Counting Function ◽  
New Approach ◽  
Probabilistic Number Theory ◽  
Probabilistic Number ◽  
Improved Accuracy

Using a new approach starting with a residue computation, we sharpen some of the known estimates for the counting function of friable integers. The improved accuracy turns out to be crucial for various applications, some of which concern fundamental questions in probabilistic number theory.

scholarly journals On sums of coefficients of polynomials related to the Borwein conjectures

2021 ◽  
Author(s):  
Ankush Goswami ◽  
Venkata Raghu Tej Pantangi
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Asymptotic Formula ◽  
Recent Result ◽  
Asymptotic Estimate ◽  
Partial Sums ◽  
Absolute Value ◽  
Partial Sum ◽  
Special Cases ◽  
Error Terms

AbstractRecently, Li (Int J Number Theory, 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci China Math, 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $$p>3$$ p > 3 . In this work, we extend Li’s method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $$p=3, 5$$ p = 3 , 5 , the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J, 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $$p=2, 3, 5, 7, 11, 13$$ p = 2 , 3 , 5 , 7 , 11 , 13 and for $$p>15$$ p > 15 , we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large n.

scholarly journals Isometric Reflection Vectors and Characterizations of Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-3
Author(s):  
Donghai Ji ◽  
Senlin Wu
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Number Theory ◽  
Recent Result ◽  
Hilbert Spaces ◽  
Unit Sphere ◽  
Substantial Reduction ◽  
Nonempty Interior

A known characterization of Hilbert spaces via isometric reflection vectors is based on the following implication: if the set of isometric reflection vectors in the unit sphereSXof a Banach spaceXhas nonempty interior inSX, thenXis a Hilbert space. Applying a recent result based on well-known theorem of Kronecker from number theory, we improve this by substantial reduction of the set of isometric reflection vectors needed in the hypothesis.

scholarly journals Haar’ s Measure using Triangular Fuzzy Finite Topological Group

2019 ◽  
Vol 8 (11) ◽  
pp. 1581-1583
Keyword(s):  
Number Theory ◽  
Topological Group ◽  
Invariant Measure ◽  
Fuzzy Number ◽  
Measure Theory ◽  
Triangular Fuzzy Number ◽  
New Approach

In this paper, A new approach is used to apply Haar’s measure theory to triangular fuzzy number theory for comprehending and generalizing the uniqueness of invariant measure when there are uncertainty and risk. If T ~ is a triangular fuzzy finite Topological group and X ~ is its subgroup, X ~ also being a triangular fuzzy number, then )

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