A Primer of Analytic Number Theory: From Pythagoras to Riemann
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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Abel's Theorem according algebra amicable pairs analogous antiderivative arithmetic Big Oh Big Oh notation calculus Chapter class number coefficients complex numbers compute congruent number conjecture constant converges absolutely define definition derivative Diophantus discriminant divides divisors elliptic curve equivalent error Euler product example Exercise exp(jr exp(x fact factor Fermat Figure finite Fundamental Theorem Geometric series gives goes to infinity graph Harmonic numbers inequality infinite series integer Interlude lattice points Lemma log(jt log(l log(n log(p log(x logarithms mathematician mathematics Mersenne numbers Mersenne prime modulo multiply Nicomachus notation number theory odd prime partial sums Pell's equation perfect number polynomial Prime Number Theorem proof prove quadratic Re(s real number rectangles relatively prime Riemann Hypothesis Riemann zeta function Section sequence series converges series expansion solutions Sophie Germain primes square modulo symmetry Taylor series term triangular numbers true values variables write zeros zeta function