  In 1961, Solomon gave upper and lower bounds for the sum of all the entries
in the character table of a finite group in terms of elementary properties of
the group. In a different direction, we consider the ratio of the character
table sum to the sum of the entries in the first column, also known as the
character degree sum, in this work. First, we propose that this ratio is at
most two for many natural groups. Secondly, we extend a conjecture of Fields to
postulate that this ratio is at least one with equality if and only if the
group is abelian. We establish the validity of this property and conjecture for
all finite irreducible Coxeter groups. In addition, we prove the conjecture for
generalized symmetric groups. The main tool we use is that the sum of a column
in the character table of an irreducible Coxeter group (resp. generalized
symmetric group) is given by the number of square roots (resp. absolute square
roots) of the corresponding conjugacy class representative.
  As a byproduct of our results, we show that the asymptotics of character
table sums is the same as the number of involutions in symmetric,
hyperoctahedral and demihyperoctahedral groups. We also derive explicit
generating functions for the character table sums for these latter groups as
infinite products of continued fractions. In the same spirit, we prove similar
generating function formulas for the number of square roots and absolute square
roots in $n$ for the generalized symmetric groups $G(r,1,n)$.
