Refinement of the Chowla–Erdős method and linear independence of certain Lambert series

2019 ◽  
Vol 31 (6) ◽  
pp. 1557-1566
Author(s):  
Daniel Duverney ◽  
Yohei Tachiya
Keyword(s):  
Infinite Series ◽  
Rational Number ◽  
Linear Independence ◽  
Arithmetic Function ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Lambert Series ◽  
Independence Results

AbstractIn this paper, we refine the method of Chowla and Erdős on the irrationality of Lambert series and study a necessary condition for the infinite series {\sum\theta(n)/q^{n}} to be a rational number, where q is an integer with {|q|>1} and θ is an arithmetic function with suitable divisibility and growth conditions. As applications of our main theorem, we give linear independence results for various kinds of Lambert series.

scholarly journals Linear independence of certain Lambert series

2014 ◽  
Vol 142 (10) ◽  
pp. 3411-3419 ◽  
Author(s):  
Florian Luca ◽  
Yohei Tachiya
Keyword(s):  
Linear Independence ◽  
Lambert Series

On Diophantine approximations of the solutions of q-functional equations

2002 ◽  
Vol 132 (3) ◽  
pp. 639-659 ◽  
Author(s):  
Tapani Matala-aho
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Linear Independence ◽  
Number Fields ◽  
Growth Conditions ◽  
Linear Forms ◽  
Dimension Estimate ◽  
Linearly Independent ◽  
Diophantine Approximations ◽  
Image Position

Given a sequence of linear forms in m ≥ 2 complex or p-adic numbers α1, …,αm ∈ Kv with appropriate growth conditions, Nesterenko proved a lower bound for the dimension d of the vector space Kα1 + ··· + Kαm over K, when K = Q and v is the infinite place. We shall generalize Nesterenko's dimension estimate over number fields K with appropriate places v, if the lower bound condition for |Rn| is replaced by the determinant condition. For the q-series approximations also a linear independence measure is given for the d linearly independent numbers. As an application we prove that the initial values F(t), F(qt), …, F(qm−1t) of the linear homogeneous q-functional equation where N = N(q, t), Pi = Pi(q, t) ∈ K[q, t] (i = 1, …, m), generate a vector space of dimension d ≥ 2 over K under some conditions for the coefficient polynomials, the solution F(t) and t, q ∈ K*.

scholarly journals A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Xinguang Zhang ◽  
Lishan Liu ◽  
Yonghong Wu ◽  
Yujun Cui
Keyword(s):  
Nonlinear Operator ◽  
Blow Up ◽  
Growth Conditions ◽  
Necessary Condition ◽  
Radial Solutions ◽  
Hessian Equation ◽  
Nonexistence And Existence ◽  
Sufficient And Necessary Condition

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained. 

scholarly journals On 'Best' Rational Approximations to $\pi$ and $\pi+e$

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Continued Fraction ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

scholarly journals On Rational Approximations to $\pi+e$: 920/157 Is the 'Best' Rational Approximation to $\pi+e$ with a Denominator of at Most Three Digits

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations ◽  
Best Rational Approximation

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. This is not so surprising when one considers the empirical computations around these two rational approximations to $\pi$. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, $920/157$ turns out to be the only rational number of this type.

The law of large numbers for additive arithmetic functions

1975 ◽  
Vol 78 (1) ◽  
pp. 33-71 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Law Of Large Numbers ◽  
Arithmetic Function ◽  
Sufficient Conditions ◽  
Growth Conditions ◽  
Arithmetic Functions ◽  
Additive Arithmetic ◽  
Large Numbers ◽  
Coprime Integers ◽  
Weak Growth ◽  
Image Position

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequenciesWe shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

scholarly journals Diophantine Approximations of Infinite Series and Products

2016 ◽  
Vol 24 (1) ◽  
pp. 71-82 ◽  
Author(s):  
Ondřej Kolouch ◽  
Lukáš Novotný
Keyword(s):  
Infinite Series ◽  
Linear Independence ◽  
Infinite Products ◽  
Survey Paper ◽  
Diophantine Approximations

Abstract This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos’ results on irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

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