A Hundred Years of Prime Numbers

This year marks the hundredth anniversary of the proof of the Prime Number Theorem (PNT), one of the most celebrated results in mathematics. The theorem is an asymptotic formula for the counting function of primes π(x) := #{p≤x:pprime} asserting that\[\pi (x)\sim \frac{x}{\log x}.\quad (\text{PNT})\...

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Bibliographic Details
Published inWho Gave You the Epsilon? p. 328
Main Authors PAUL T. BATEMAN, HAROLD G. DIAMOND
Format Book Chapter
LanguageEnglish
Published Mathematical Association of America 01.06.2009
Edition1
Subjects
Algebra
Applied mathematics
Applied statistics
Arithmetic
Counting numbers
Discrete mathematics
Estimation methods
Inferential statistics
Integers
Mathematical expressions
Mathematical functions
Mathematical theorems
Mathematics
Number theory
Numbers
Prime numbers
Pure mathematics
Rational numbers
Real numbers
Statistical estimation
Statistics
Zero
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Summary:This year marks the hundredth anniversary of the proof of the Prime Number Theorem (PNT), one of the most celebrated results in mathematics. The theorem is an asymptotic formula for the counting function of primes π(x) := #{p≤x:pprime} asserting that\[\pi (x)\sim \frac{x}{\log x}.\quad (\text{PNT})\] The twiddle notation is shorthand for the statement\[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\pi (x)}{(x/\log x)}=1.\] Here we shall survey early work on the distribution of primes, the proof of the PNT, and some later developments. Since the time of Euclid, the primes, 2, 3, 5, 7, 11, 13, …, have been known to be infinite in number. They appear to
ISBN:0883855690
9780883855690