The Gregory function and its completed Gregory transform · arXivDeskAbstract
We study the entire interpolation G(z)=∫01(zx)dx
of the Gregory coefficients. Its completion satisfies the positive Markov-transform identity
sin(πz)πzG(z)=n=1∑∞n−zn∣Gn∣. Consequently, every zero is real and simple; the negative zeros are the integers
−1,−2,… , and one zero
lies in each
. We derive complete logarithmic asymptotics for
, determine the Cartwright growth and canonical products of
, and realize
spectrally. The resulting relative determinant yields
γ=n=1∑∞(n1−ρn1).