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Introduction and Dedication
viii
Such an attitude might prove an effective antidote to a generally
declining interest in pure mathematics. But it is not so much as incen-
tive that we proffer what might best be called “the fun and games”
approach to mathematics, but as a revelation of its true nature. The
insistence on simplicity asserts a mathematics that is both “magi-
cal” and coherent. The solution that strives to master these qualities
restores to mathematics that element of adventure that has always
supplied its peculiar excitement. That adventure is intrinsic to even
the most elementary description of analytic number theory.
The initial step in the investigation of a number theoretic item
is the formulation of “the generating function”. This formulation
inevitably moves us away from the designated subject to a consider-
ation of complex variables. Having wandered away from our subject,
it becomes necessary to effect a return. Toward this end “The Cauchy
Integral” provesto be an indispensabletool.Yetitleads us,inevitably,
further afield from all the intricacies of contour integration and they,
in turn entail the familiar processes, the deformation and estimation
of these contour integrals.
Retracing our steps we find that we have gone from number theory
to function theory, and back again. The journey seems circuitous, yet
in its wake a pattern is revealed that implies a mathematics deeply
inter-connected and cohesive.