In this note, we show that if b > 1 is an integer, f(X) ∈ Q[X] is an integer valued quadratic polynomial and K > 0 is any constant, then the b-adic number ∑n≥0 (an / bf(n)), where an ∈ Z and 1 ≤ |an| ≤ K for all n ≥ 0, is neither rational nor quadratic.
In this paper, we give transcendental numbers φ and ψ such that (i) both φ and ψ have explicit g-adic expansions, and simultaneously, (ii) the vector has an explicit expression in the Jacobi-Perron algorithm (cf. Theorem 1). Our results can be regarded as a higher-dimensional version of some of the results in [1]-[5] (see also [6]-[8], [10], [11]). The numbers φ and ψ have some connection with algebraic numbers with minimal polynomials x³ - kx² - lx - 1 satisfying (1.1) k ≥ l ≥0, k + l ≥ 2 (k,l...
We generalize and unify the proofs of several results on algebraic independence of arithmetic functions and Dirichlet series by using a theorem of Ax on the differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.
We compute upper and lower bounds for the approximation of hyperbolic functions at points
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of and , as well as to explain Rivoal’s recent result on infiniteness of irrational numbers in the set of odd zeta values, and to prove that at least one of the four numbers , , , and is irrational.