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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 prove an identity involving generalised Euler-Briggs constants, Euler's constant, and linear forms in logarithms of algebraic numbers. This generalises and gives an alternative proof of an identity of Lehmer (1975). Further, this identity facilitates the investigation of the (conjectural) transcendental nature of generalised Euler-Briggs constants. Earlier investigations of similar type by the present authors involved the interplay between additive and multiplicative characters. This in turn...
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 study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.
Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products
(i=1,...,m) or (i=1,...,m)
to be algebraically dependent, where are non-zero integers and and are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....
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