Monthly Archives: January 2012

The Riemann hypothesis: Properties of the zeta function

In spite of the important relation between ζ(s) and the sequence of prime numbers, Riemann was not especially interested in number theory. His 8-page paper of 1859, entitled On the Number of Primes Less Than a Given Magnitude was, in … Continue reading

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The Riemann hypothesis: The distribution of primes

It was known in antiquity that there is no “largest” prime number, and hence that there must be infinitely many primes. Euclid proved in in Book IX of his Elements. For suppose the set S of primes is finite. Consider … Continue reading

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The Riemann hypothesis: The product formula

The so-called “fundamental theorem of arithmetic” states than any integer is a product of powers of prime numbers in a unique way. Euler was (apparently) the first to realize that this fact could be expressed as an identity between an … Continue reading

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The Riemann hypothesis: Preliminaries

The Riemann hypothesis is the statement that the zeros of a certain complex-valued function ζ(s) of a complex number s all have a certain special form. That is, if we look at ζ(s) = 0 as an equation to solve, … Continue reading

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The Riemann hypothesis: Introduction

The Riemann hypothesis is the statement that the zeros of a certain complex-valued function ζ(s) of a complex number s all have a certain special form. It’s probably the most famous currently unsolved problem in mathematics. I will be posting … Continue reading

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