Transcendental infinite sums

Transcendental values of the digamma function

“…We hasten to point out that theorems similar to Theorems 9 and 10 have been proved in [1]. In a related context, Erdős (see [8]) conjectured that there is no function defined on the integers…”

“…Thus, the theorem resolves Chowla's question. In 2001, Adhikari, Saradha, Shorey and Tijdeman [1] noted that the theory of linear forms in logarithms can be used to show that in fact, the sum in the above theorem is transcendental whenever it converges. In particular, the special value L(1, χ) of Dirichlet's L-function attached to a non-trivial character mod q is transcendental.…”

Infinite sums as linear combinations of polygamma functions

“…Since the infinite sum in the latter expression of T does not vanish, by [1,Theorem 3] we conclude that T is transcendental.…”

“…Obviously by (1) and (4), the numbers ψ (k) (1)/ζ(k + 1), g (k) (1)/ζ(k + 1), ψ (k) (1/2)/ζ(k + 1) are rational (here ζ(s) = ∞ n=1 1/n s is the Riemann zeta function) and therefore from (2), (5) we get the following inclusions: (7) ψ (2k−1) (m), g (2k−1) (m), ψ (2k−1) (m + 1/2) ∈ Q × · π 2k + Q, m ∈ N.…”

“…. , α m of Q(x) are distinct rational numbers was considered in [1], where by Baker's theory it was proved that each of the numbers (8) is either a computable algebraic number or is transcendental. In particular, if Q(x) is a reduced polynomial, i.e., if α 1 , .…”