A Primer of Analytic Number Theory: From Pythagoras to Riemann

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Cambridge University Press, Jun 23, 2003 - Mathematics - 383 pages
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This 2003 undergraduate introduction to analytic number theory develops analytic skills in the course of studying ancient questions on polygonal numbers, perfect numbers and amicable pairs. The question of how the primes are distributed amongst all the integers is central in analytic number theory. This distribution is determined by the Riemann zeta function, and Riemann's work shows how it is connected to the zeroes of his function, and the significance of the Riemann Hypothesis. Starting from a traditional calculus course and assuming no complex analysis, the author develops the basic ideas of elementary number theory. The text is supplemented by series of exercises to further develop the concepts, and includes brief sketches of more advanced ideas, to present contemporary research problems at a level suitable for undergraduates. In addition to proofs, both rigorous and heuristic, the book includes extensive graphics and tables to make analytic concepts as concrete as possible.
 

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Contents

Sums and Differences
1
Products and Divisibility
24
Order of Magnitude
43
Averages
64
Calculus
83
Primes
96
Series
111
Basel Problem
146
Symmetry
216
Explicit Formula
229
Modular Arithmetic
254
Pells Equation
260
Elliptic Curves
274
Analytic Theory of Algebraic Numbers
295
Solutions
327
Bibliography
375

Eulers Product
159
Complex Numbers
187
The Riemann Zeta Function
193

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