Transcendence of the log gamma function and some discrete periods

Abstract

We study transcendental values of the logarithm of the gamma function. For instance, we show that for any rational number xtranscendental with at most one possible exception. Assuming Schanuel's conjecture, this possible exception can be ruled out. Further, we derive a variety of results on the Γ -function as well as the transcendence of certain series of the form ∞ n=1 P (n)/Q (n), where P (x) and Q (x) are polynomials with algebraic coefficients.

Linear and algebraic independence of generalized Euler–Briggs constants

“…This forces that d a = 0 for all a since ξ a 's are multiplicatively independent. Again going back to (5) and following the same argument, we get e b = 0, f c = 0 for all b, c. This completes the proof.…”

Linear independence of digamma function and a variant of a conjecture of Rohrlich

“…It has been established in [10] that if we assume the conjecture of Schanuel, log π is not a Baker period. Thus, we see that in the proof of Theorem 3, the term q/2 a=1 (a,p)=1 c a log π = q/2 a=1 (a,p)=1 c a log sin πa q is necessarily equal to zero when the coefficients c a are algebraic numbers.…”

“…The notion of periods has been introduced by Kontsevich and Zagier [11] and these Baker periods are examples of transcendental periods. As mentioned by the authors, the co-primality condition cannot be dispensed with in their conjecture as illustrated by the following example:…”

Transcendental Sums Related to the Zeros of Zeta Functions

“…We shall need the following consequence of the above conjecture which is not difficult to deduce. This was done in an earlier work of ours (see [8] for details).…”