Values of the Legendre chi and Hurwitz zeta functions at rational arguments

Abstract

We show that the Hurwitz zeta function, ζ ( ν , a ) \zeta (\nu ,a) , and the Legendre chi function, χ ν ( z ) \chi _\nu (z) , defined by \[ ζ ( ν , a ) = ∑ k = 0 ∞ 1 ( k + a ) ν , 0 > a ≤ 1 , Re ν > 1 , \zeta (\nu ,a)=\sum _{k=0}^\infty \frac {1}{(k+a)^\nu },\quad 0>a\le 1,\operatorname {Re}\,\nu >1, \] and \[ χ ν ( z ) = ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) ν , | z | ≤ 1 , Re ν > 1 with ν = 2 , 3 , 4 , … , \chi _\nu (z)=\sum _{k=0}^\infty \frac {z^{2k+1}}{(2k+1)^\nu },\quad |z|\le 1,\operatorname {Re}\,\nu >1 \text {with} \nu =2,3,4,\dotsc , \] respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained as a corollary to this result. Among them is the further simplification of the summation formulae from our earlier work on closed form summation of some trigonometric series for rational arguments. Also, these transform relations make it likely that other results can be easily recovered and unified in a more general context.