共查询到20条相似文献,搜索用时 15 毫秒
1.
Let ψ(x) denote the digamma function. We study the linear independence of ψ(x) at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series. 相似文献
2.
We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number x with 0<x<1, the number logΓ(x)+logΓ(1−x) is transcendental with at most one possible exception. Assuming Schanuel's conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ-function as well as the transcendence of certain series of the form , where P(x) and Q(x) are polynomials with algebraic coefficients. 相似文献
3.
Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbers
4.
Chester Weatherby 《Journal of Number Theory》2011,131(4):705-715
We consider the transcendence of the multi-indexed series
5.
Le Maohua 《Monatshefte für Mathematik》1993,116(3-4):283-285
LetD be a positive square free integer, and leth(–D) denote the class number of
. Furthermore letp be an odd prime with
. In this note we prove that ifp {5, 7} orp>3·106, then the equation
, has no positive integer solution (x, y). 相似文献
6.
Yohei Tachiya 《Journal of Number Theory》2007,125(1):182-200
We prove the transcendence results for the infinite product , where Ek(x), Fk(x) are polynomials, α is an algebraic number, and r?2 is an integer. As applications, we give necessary and sufficient conditions for transcendence of and , where Fn and Ln are Fibonacci numbers and Lucas numbers respectively, and {ak}k?0 is a sequence of algebraic numbers with log‖ak‖=o(rk). 相似文献
7.
Taka-aki Tanaka 《Journal of Number Theory》2004,105(1):38-48
It is proved that the function , which can be expressed as a certain continued fraction, takes algebraically independent values at any distinct nonzero algebraic numbers inside the unit circle if the sequence {Rk}k?0 is the generalized Fibonacci numbers. 相似文献
8.
Kumiko Nishioka 《Monatshefte für Mathematik》1997,123(2):135-148
Algebraic independence of the numbers
, where{R
n
}
n
0 is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler's method. 相似文献
9.
Takeshi Kurosawa 《Journal of Number Theory》2007,123(1):35-58
Duverney and Nishioka [D. Duverney, Ku. Nishioka, An inductive method for proving the transcendence of certain series, Acta Arith. 110 (4) (2003) 305-330] studied the transcendence of , where Ek(z), Fk(z) are polynomials, α is an algebraic number, and r is an integer greater than 1, using an inductive method. We extend their inductive method to the case of several variables. This enables us to prove the transcendence of , where Rn is a binary linear recurrence and {ak} is a sequence of algebraic numbers. 相似文献
10.
Éric Gaudron 《Journal of Number Theory》2007,127(2):220-261
We establish new measures of linear independence of logarithms on commutative algebraic groups in the so-called rational case. More precisely, let k be a number field and v0 be an arbitrary place of k. Let G be a commutative algebraic group defined over k and H be a connected algebraic subgroup of G. Denote by Lie(H) its Lie algebra at the origin. Let u∈Lie(G(Cv0)) a logarithm of a point p∈G(k). Assuming (essentially) that p is not a torsion point modulo proper connected algebraic subgroups of G, we obtain lower bounds for the distance from u to Lie(H)k⊗Cv0. For the most part, they generalize the measures already known when G is a linear group. The main feature of these results is to provide a better dependence in the height loga of p, removing a polynomial term in logloga. The proof relies on sharp estimates of sizes of formal subschemes associated to H (in the sense of Bost) obtained from a lemma by Raynaud as well as an absolute Siegel lemma and, in the ultrametric case, a recent interpolation lemma by Roy. 相似文献
11.
The Euler–Lehmer constants γ(a,q) are defined as the limits We show that at most one number in the infinite list is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If f:Z/qZ→Q is such that f(a)=±1 and f(q)=0, then Erdös conjectured that If , we show that the Erdös conjecture is true. 相似文献
12.
13.
Pingzhi Yuan 《Journal of Number Theory》2003,102(1):1-10
Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities
|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N 相似文献
14.
Xavier Le Breton 《Discrete Mathematics》2006,306(15):1776-1780
We extend a result of J.-P. Allouche and O. Salon on linear independence of formal power series associated to polynomial extractions of quasistrongly p-additive sequences. The original result was on the Fp-linear independence and we extend it to the Fp[X]-linear independence. 相似文献
15.
16.
Tomohiro Yamada 《Journal of Number Theory》2010,130(9):1889-1897
We shall make a slight improvement to a result of p-adic logarithms, which gives a nontrivial upper bound for the exponent of p dividing the Fermat quotient xp−1−1. 相似文献
17.
We study the algebraic independence of two inductively defined sets. Under the hypothesis of Schanuel's conjecture we prove that the exponential power tower E and its related logarithmic tower L are linearly disjoint. 相似文献
18.
Jhon J. Bravo 《Quaestiones Mathematicae》2016,39(3):391-400
In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where Fk is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation. 相似文献
19.
We sharpen work of Bugeaud to show that the equation of the title has, for t = 1 or 2, no solutions in positive integers x, y, z and k with z > 1 and k > 3. The proof utilizes a variety of techniques, including the hypergeometric method of Thue and Siegel, as well as an assortment of gap principles. 相似文献
20.
The equation is studied for periodic algebraically-valued functions f and, in particular, a well known problem of Chowla in this context is resolved. The work depends on an application of a theorem of the first author concerning linear forms in the logarithms of algebraic numbers. 相似文献