(08/14/2009, 04:50 PM)jaydfox Wrote: But Bo, these zeroes are a trivial result of the "change of base" concept.
Well if you only look at the case \( a=e \), \( b=\eta \); it may be trivial (though singularities not zeros) because everything stays on the real axis.
However I was considering general real bases \( a \) and \( b \) and the possibly non-real singularities in the region \( B_n \).
As you say yourself its difficult to determine whether the non-real possible singularities (which I completely specified) indeed exist or cancel out by appropriate choices of the logarithm. I showed that \( f_3 \) indeed has complex singularities that dont cancel out. And also that there are complex singularities for other bases that dont cancel out.
The example of \( f_3 \) shows also that your suggested path-continuation does not work. If you have branch points then different paths to the same point may result in different values of \( f_3 \). To keep it continuous you have to specify cuts.