Showing papers in "Journal of Number Theory in 1985"

TL;DR: Apery as discussed by the authors proved congruences for the numbers an = Σ0n(kn)2(kn + k) and un = Ω(n) 2(kn+k) 2 in his irrationality proof for ζ(2) and ε(3).

TL;DR: In this paper, a better than Thue-Siegel-Roth-Schmidt theorem bounding the approximation of rational functions over valued differential fields was established, based on using an auxiliary polynomial to construct an auxiliary function.

TL;DR: In this paper, the authors generalize some results of Bloch [3] concerning class field theory for curves over local fields, and develop a theory including the bad reduction case.

TL;DR: In this paper, practical computational techniques are described to determine the Galois group of a polynomial over the rationals, and each transitive permutation group of degree 3 to 7 is realized as a Galois groups over rationals.

TL;DR: In this article, the Dirichlets L-series of the form Σk = 1∞(2k2k)−1 k−n was shown to be summable exactly in terms of Dirichlet L series for values of n up to and including 5.

TL;DR: In this article, it was proved that the generalized Voronoi algorithm computes the fundamental units of all fields of unit rank 2, i.e., of the totally real cubic fields.

TL;DR: In this paper, a conjecture on l-adic independence of algebraic numbers is presented, which generalizes the classical ones of Leopoldt and Gross, and asserts that the ladic rank of a G submodule of Kx depends only on the character of its Galois representation.

TL;DR: In this article, the authors focus on estimating the taille, dans chaque ensemble de mesure positive du tore, du sous-ensemble ou un tel polynome est grand.

TL;DR: It is known that (n2n) is not squarefree if n > 4 as discussed by the authors, but it is shown that n2n is squarefree when n > n 0.

TL;DR: In this paper it was shown that if l ≢= 2 mod 4 and n is the product of 4 distinct primes congruent to 1 mod l, then l2 (l, if l is odd) divides the class number hn+ of the nth cyclotomic field.

TL;DR: A formula expressing the Drinfeld discriminant as a product of cyclotomic polynomials was proved in this paper, where the authors also proved that the discriminant can be expressed as a function of the cyclotome.

TL;DR: In this article, it is shown that if f is a path corresponding to a limit point and if T n(f) denotes the subtree of T which branches off from f at height n, roughly speaking, T n (f) converges to a tree T ′(f), which is called the derived tree of T at f at f. When this derived tree is essentially finite for all f associated with a limit points of S, then all elements of S are effectively determined in a small neighbourhood of this limit point.

TL;DR: In particular, if A is an asymptotic basis of order h, and if F⊆A, card(F) = k, and A\F is a finite subset of A, then the exact order of A/F is O(hk+1) as mentioned in this paper.

TL;DR: In this paper, Masser and Wustholz showed that the product of log A1, log An in the lower bound is of exactly the same form as in the upper bound for linear forms in logarithms of algebraic numbers.

TL;DR: In this article, it was shown that if ν is replaced by an arbitrary, integer-valued function f with certain properties assumed about its average order, then the equation n+ f(n) = m + f(m) has infinitely many solutions with n ≠ m.

TL;DR: In this paper, the authors evaluate some unusual infinite products, of which the following is a typical example: 3 − 1 3 = 1 2 · 6 4 · 8 9 · 12 10 · 14 15 · 16 17 · 20 21 · 22 23 · 27 25.

TL;DR: In this paper, the arithmetics of crossed products are studied and the definition of a crossed product order is tightened and it is shown that such orders are one-headed, i.e., that the idealizer of radical embedding always leads to the same hereditary order.

TL;DR: In this paper, Mobius inversion over the lattice of subgroups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup.

TL;DR: In this article, the binomial coefficient was used as the greatest common divisor for the first time, and was modified on 2016-08-08 to 2.1.

TL;DR: In this article, it was shown that as x approaches ∞, #{n ≤ x: d(n) divides n} = ( x √ log x)( log log x) −1 + 0(1).

TL;DR: In this paper, it was proved that the ladder structure is invariant with order when the order is decreased from its permissible maximum value for the corresponding ladder. But this invariance property does not necessarily persist in reverse, as increasing the order need not produce a valid ladder with rational coefficients.

TL;DR: In this article, the Berry-Esseen theorem was used to prove that Ψ(x, y) is a weighted lower approximation to the number defined by the title of this paper, using a reinterpretation of this sum in probability terminology.

TL;DR: In this article, it was shown that the spectrum of nonzero minima of an indefinite quadratic form with real coefficients in n variables with nonzero determinant d is discrete.

TL;DR: In this article, a simple new expression and a simple proof of the congruences of Mirimanoff which appeared in his papers of 1910 and 1911 are given, under the hypothesis that p does not divide the second factor of the class number of the p-cyclotomic field.

TL;DR: In this paper, the Dirichlet divisor conjecture was shown to hold for all e ≥ 1 2 and r ≥ 2 with e = 0 (with xe replaced by log x in case a = 1 2 ).

TL;DR: In this article, the conjugacy problem of the modular group was discussed, and its solution was used to compute the class number of certain real quadratic number fields with respect to a theorem of Olga Taussky.

TL;DR: In this article, necessary and sufficient conditions for U(RG) not to contain a free subgroup of rank two when G is finite or the extension of a solvable torsion group by a Torsion-free nilpotent group are presented.

TL;DR: The explicit lower bound due to Furtwangler of Cn, the n-dimensional Diophantine approximation constant is C n ≥ 1 Δ n+1 1 2, where Δn+1 is the minimum absolute discriminant of a real number field of degree n + 1, where [x] is the integer part of the real number x as mentioned in this paper.

TL;DR: In this article, a short proof is given that the theta functional is invariant under the Weil representation, and the explicit determination of the eighth root of unity which arises is also shown.

TL;DR: In this article, an asymptotic formula for the number of positive integers n ≤ x such that (d, f ( n )) = 1, where f is an integer-valued multiplicative function such that f (p ) is a polynomial in p for p prime, and where d has no prime divisor from a certain finite exceptional set.