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As the title says, is it known whether or not the Glaisher constant is a transcendental number?

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    Then there is the conjecture that any naturally generated constant that is not obviously rational or algebraic (by its construction, such as the square root o$f$ two) is transcendental. I don't think puttin$g$ a bounty on this will help.2012-08-29

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Math Reviews returns 10 papers when one searches for Glaisher and Kinkelin. Not one of the ten reviews has anything to say about irrationality or transcendence. Absence of evidence is not evidence of absence, but this suggests to me that nothing is known about the question.