Secant zeta functions

We study the series ψ s ( z ) ≔ ∑ n = 1 ∞ sec ( n π z ) n − s and prove that it converges under mild restrictions on z and s . The function possesses a modular transformation property, which allows us to evaluate ψ s ( z ) explicitly at certain quadratic irrational values of z . This supports our conjecture that π − k ψ k ( j ) ∈ Q whenever k and j are positive integers with k even. We conclude with some speculations on the Bernoulli numbers.