  We describe a numerical algorithm for evaluating the numbers of roots minus
the number of poles contained in a region based on the argument principle with
the function of interest being written as a Mellin transformation of a usually
simpler function. Because the function to be transformed may be simpler than
its Mellin transform whose roots are to be sought we express the final
integrals in terms of the former accepting higher dimensional integrals.
Nonlinear terms are expressed as convolutions approximating reciprocal values
by exponential sums. As an example the final expression is applied to the
Riemann Zeta function. The procedure is very inefficient numerically. However,
depending on the function to be investigated it may be possible to find
analytical estimates of the resulting integrals.
