scholarly journals The Piltz divisor problem in number fields: An improved lower bound by Soundararajan's method

2005 ◽  
Vol 117 (2) ◽  
pp. 187-206 ◽  
Author(s):  
K. Girstmair ◽  
M. Kühleitner ◽  
W. Müller ◽  
W. G. Nowak
Keyword(s):  
Lower Bound ◽  
Number Fields ◽  
Divisor Problem

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*.

Integral points on elliptic curves over number fields

1997 ◽  
Vol 122 (1) ◽  
pp. 9-16 ◽  
Author(s):  
N. P. SMART ◽  
N. M. STEPHENS
Keyword(s):  
Lower Bound ◽  
Elliptic Curves ◽  
Number Fields ◽  
Integral Points ◽  
Linear Forms ◽  
Special Cases ◽  
Explicit Lower Bound

In recent years there has been an interest in using elliptic logarithms to find integral points on elliptic curves defined over the rationals, see [23], [17], [6] and [12]. This has been partly due to work of David [5], who gave an explicit lower bound for linear forms in elliptic logarithms. Previously, integral points on elliptic curves had been found by Siegel's method; that is, a reduction to a set of Thue equations which could be solved, in principle, by the methods in [19]. For examples of this method see [3], [7], [16], [18], [21], [22] and [8]. Other techniques can be used to find all integral points in some special cases, see, for instance, [14].

Dirichlet divisor problem in number fields

1988 ◽  
Vol 44 (4) ◽  
pp. 750-757 ◽  
Author(s):  
E. I. Panteleeva
Keyword(s):  
Number Fields ◽  
Divisor Problem ◽  
Dirichlet Divisor Problem
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