The Riemann hypothesis: a million dollar problem
The Riemann hypothesis concerns the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ...Ubiquitous and fundamental in mathematics as they are, it is important and interesting to know as much as possible about these numbers. Simple questions would be: how are the prime numbers distributed among the positive integers? What is the number of prime numbers of 100 digits? Of 1,000 digits? These questions were the starting point of a groundbreaking paper by Bernhard Riemann written…
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- Full title :The Riemann hypothesis : a million dollar problem / Roland van der Veen, Leiden University, and Jan van de Craats, University of Amsterdam.
Summary
The Riemann hypothesis concerns the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ...Ubiquitous and fundamental in mathematics as they are, it is important and interesting to know as much as possible about these numbers. Simple questions would be: how are the prime numbers distributed among the positive integers? What is the number of prime numbers of 100 digits? Of 1,000 digits? These questions were the starting point of a groundbreaking paper by Bernhard Riemann written in 1859. As an aside in his article, Riemann formulated his now famous hypothesis that so far no one has come close to proving: All nontrivial zeroes of the zeta function lie on the critical line. Hidden behind this at first mysterious phrase lies a whole mathematical universe of prime numbers, infinite sequences, infinite products, and complex functions. The present book is a first exploration of this fascinating, unknown world. It originated from an online course for mathematically talented secondary school students organized by the authors of this book at the University of Amsterdam. Its aim was to bring the students into contact with challenging university level mathematics and show them what the Riemann Hypothesis is all about and why it is such an important problem in mathematics.
Read More ↓Edition Statements
- Publication Date:©2015
Publication Statements
- Publisher(s):Mathematical Association of America
- Place of Publication:Washington
- Language:eng
- Publication Date:©2015
Notes
Original title: De Riemann-hypothese : Een miljoenenprobleem, Epsilon Uitgaven, Utrecht, 2011.
Translated by the authors.
The Physical Object
- Description:xi, 144 pages : : illustrations ; : 23 cm.
Edition Identifiers
- ISBN:9780883856505, 0883856506
- oclc:ocn934248680
- LC Classification:QA246
- Dewey Decimal:512.73
Table of Contents
Prime numbers. Primes as elementary building blocks ; Counting primes ; Using the logarithm to count powers ; Approximations for ?(x) ; The prime number theorem ; Counting prime powers logarithmically ; The Riemann hypothesis : a look ahead ; Additional exercises -- The zeta function. Infinite sums ; Series for well-known functions ; Computation of ?(2) ; Euler's product formula ; Looking back and a glimpse of what is to come ; Additional exercises -- The Riemann hypothesis. Euler's discovery of the product formula ; Extending the domain of the zeta function ; A crash course on complex numbers ; Complex functions and powers ; The complex zeta function ; The zeroes of the zeta function ; The hunt for zeta zeroes ; Additional exercises -- Primes and the Riemann hypothesis. Riemann's functional equation ; The zeroes of the zeta function ; The explicit formula for ?(x) ; Pairing up the non-trivial zeroes ; The prime number theorem ; A proof of the prime number theorem ; The music of the primes ; Looking back ; Additional exercises -- Appendix A : Why big primes are useful -- Appendix B : Computer support -- Appendix C : Further reading and Internet surfing -- Appendix D : Solutions to…
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