  We present further results on a class of sums which involve complex powers of
the distance to points in a two-dimensional square lattice and trigonometric
functions of their angle, supplementing those in a previous paper (McPhedran
{\em et al}, 2008). We give a general expression which permits numerical
evaluation of members of the class of sums to arbitrary order. We use this to
illustrate numerically the properties of trajectories along which the real and
imaginary parts of the sums are zero, and we show results for the first two of
a particular set of angular sums (denoted ${\cal C}(1,4 m;s)$) which indicate
their density of zeros on the critical line of the complex exponent is the same
as that for the product (denoted ${\cal C}(0,1;s)$) of the Riemann zeta
function and the Catalan beta function. We then introduce a function which is
the quotient of the angular lattice sums ${\cal C}(1,4 m;s)$ with ${\cal
C}(0,1;s)$, and use its properties to prove that ${\cal C}(1,4 m;s)$ obeys the
Riemann hypothesis for any $m$ if and only if ${\cal C}(0,1;s)$ obeys the
Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds,
then ${\cal C}(1,4 m;s)$ and ${\cal C}(0,1;s)$ have the same distribution of
zeros on the critical line (in a sense made precise in the proof).
