Bugeaud and Laurent: values of Hecke–Mahler series

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Y. Bugeaud and M. Laurent, Transcendence and continued fraction expansion of values of Hecke-Mahler series, arXiv:2203.12901v1, submitted 24 March 2022, 29 pages.

Theorem 1.1 and the defining series were inspected in the arXiv version. For irrational $0<\theta<1$, $0\le\rho<1$, and nonzero algebraic $\alpha,\beta$ with $|\beta|<1$ and $|\beta\alpha^\theta|<1$, it gives transcendence of the Hecke–Mahler value $F_{\theta,\rho}(\beta,\alpha)$. The source proof has not been audited here.

The arXiv record, retrieved 2026-10-14, specifies arXiv's non-exclusive distribution license, with other rights reserved. No file of this source is held. No full transcription is provided under these terms.

Bears on. Problem 269: for distinct primes $p<q$, the choices $\theta=\log p/\log q$, $\rho=0$, $\beta=1/p$ and $\alpha=1/q$ meet these hypotheses. The two-prime running-lcm deduction expresses each factor as a rational affine function of the Hecke–Mahler value and the full sum as a nonconstant rational quadratic in that value. It retains the earlier Fan attribution and has author-recorded standing; no conclusion for three or more primes follows.