Newman D.J. Analytic Number Theory

Graduate texts in Mathematics
Springer-Verlag, New York, 1998, 81 pagesThe 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 consideration
of complex variables. Having wandered away from our subject,
it becomes necessary to effect a return. Toward this end The Cauchy
Integral proves to be an indispensable tool.Yet it leads 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.I. The Idea of Analytic Number Theory
II. The Partition Function
III. The Erdös–Fuchs Theorem
IV. Sequences without Arithmetic Progressions
V. The Waring Problem
VI. A Natural Proof of the Nonvanishing of L-Series
VII. Simple Analytic Proof of the Prime Number Theorem