On the Number of Primes Less Than a Given Magnitude
August 3, 2009 11:58 AM Subscribe
This post was deleted for the following reason: This sort of thing could use just a skosh more context. -- cortex
I just finished Cryptonomicon. On the whole, I didn't like it. But I did feel very smart that I could follow along when Waterhouse and Turing discussed Riemann-Zeta functions
posted by tylerfulltilt at 12:05 PM on August 3, 2009
posted by tylerfulltilt at 12:05 PM on August 3, 2009
I don't usually complain about posts...but....a post comprised of a single link in German to a mathematics .pdf? Maybe we could get some background on why this is an important paper? Bueller...Bueller...
posted by The Light Fantastic at 12:06 PM on August 3, 2009
posted by The Light Fantastic at 12:06 PM on August 3, 2009
My God, people, the answer is obvious!
posted by Astro Zombie at 12:07 PM on August 3, 2009
posted by Astro Zombie at 12:07 PM on August 3, 2009
PDF title page
On the Number of Prime Numbers less than a Given Quantity.
(Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.)
Bernhard Riemann
[Monatsberichte der Berliner Akademie, November 1859.]
Translated by David R. Wilkins
Preliminary Version: December 1998
Riemann's paper, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of primes less than a given quantity), by Bernhard Riemann, was first published in the Monatsberichte der Berliner Akademie, in November 1859. This paper, just six manuscript pages in length, introduced radically new ideas to the study of prime numbers — ideas which led, in 1896, to independent proofs by Hadamard and de la Vallée Poussin of the prime number theorem. This theorem, first conjectured by Gauss when he was a young man, states that the number of primes less than x is asymptotic to x/log(x). Very roughly speaking, this means that the probability that a randomly chosen number of magnitude x is a prime is 1/log(x) (source).
From all this, my non-mathematically tuned mind thinks of two things: Du Du hast. Du hast mich... and It's Log, Log, it's big, it's heavy, it's wood. / It's Log, Log, it's better than bad, it's good!
posted by filthy light thief at 12:09 PM on August 3, 2009 [1 favorite]
On the Number of Prime Numbers less than a Given Quantity.
(Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse.)
Bernhard Riemann
[Monatsberichte der Berliner Akademie, November 1859.]
Translated by David R. Wilkins
Preliminary Version: December 1998
Riemann's paper, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse (On the number of primes less than a given quantity), by Bernhard Riemann, was first published in the Monatsberichte der Berliner Akademie, in November 1859. This paper, just six manuscript pages in length, introduced radically new ideas to the study of prime numbers — ideas which led, in 1896, to independent proofs by Hadamard and de la Vallée Poussin of the prime number theorem. This theorem, first conjectured by Gauss when he was a young man, states that the number of primes less than x is asymptotic to x/log(x). Very roughly speaking, this means that the probability that a randomly chosen number of magnitude x is a prime is 1/log(x) (source).
From all this, my non-mathematically tuned mind thinks of two things: Du Du hast. Du hast mich... and It's Log, Log, it's big, it's heavy, it's wood. / It's Log, Log, it's better than bad, it's good!
posted by filthy light thief at 12:09 PM on August 3, 2009 [1 favorite]
Ok, a paper from 1859, one of many from a prolific and influential mathematician who died young. No additional context. Did you mean to post this to MetaIdeasForCollaborativeFPPWriting?
posted by effbot at 12:09 PM on August 3, 2009
posted by effbot at 12:09 PM on August 3, 2009
Venn ist das nurnstuck git und Slotermeyer? Ya! Beigerhund das oder die Flipperwaldt gersput!
posted by fearfulsymmetry at 12:10 PM on August 3, 2009 [3 favorites]
posted by fearfulsymmetry at 12:10 PM on August 3, 2009 [3 favorites]
This is a prime example of something or other, but I'm not sure what.
posted by It's Raining Florence Henderson at 12:14 PM on August 3, 2009 [2 favorites]
posted by It's Raining Florence Henderson at 12:14 PM on August 3, 2009 [2 favorites]
Addendum: The translator, Dr. David R. Wilkins is knowledgable about Georg Friedrich Bernhard Riemann. And there is Reimann's page on wikipedia.
posted by filthy light thief at 12:14 PM on August 3, 2009
posted by filthy light thief at 12:14 PM on August 3, 2009
Fearfulsymmetry beat me to it. I will now die laughing.
posted by ooga_booga at 12:14 PM on August 3, 2009
posted by ooga_booga at 12:14 PM on August 3, 2009
In the first case the constant of integration is determined if one lets the
real part of β become infinitely negative; in the second case the integral from 0
to x takes on values separated by 2πi, depending on whether the integration
is taken through complex values with positive or negative argument, and
becomes infinitely small, for the former path, when the coefficient of i in the
value of β becomes infinitely positive, but for the latter, when this coefficient
becomes infinitely negative.
posted by kbanas at 12:18 PM on August 3, 2009
real part of β become infinitely negative; in the second case the integral from 0
to x takes on values separated by 2πi, depending on whether the integration
is taken through complex values with positive or negative argument, and
becomes infinitely small, for the former path, when the coefficient of i in the
value of β becomes infinitely positive, but for the latter, when this coefficient
becomes infinitely negative.
posted by kbanas at 12:18 PM on August 3, 2009
ffiffiffiffiffiffiffi ... Wow, hey, at least I got that out of all this. Those are some pretty great symbols.
posted by kbanas at 12:20 PM on August 3, 2009
posted by kbanas at 12:20 PM on August 3, 2009
I agree that the FPP doesn't give much context, but this is in fact one of the most important mathematical results of the past few centuries.
Briefly: the study of prime numbers is part of number theory, a very old branch of mathematics going back to the Greeks. Primes are in some sense the "atoms" from which all numbers are built. The problem Riemann addresses is then rather natural: how can we estimate the number of primes less than 10000? Less than 1,000,000? Less than 100,000,000? Nowadays we can answer these questions exactly via computer calculation, but there is always a limit -- if you want to know how many primes there are less than 10^1000, you'll have to use some kind of estimate.
Anyway, Riemann discovered that this problem is related to the properties of a certain function, now called the Riemann zeta function, which can be expressed as an integral involving complex numbers. In other words, to understand the most basic objects of classical mathematics, the prime numbers, which people had been chewing on for thousands of years, it was absolutely necessary to make reference to the comparatively novel and high-test techniques of calculus and complex numbers. This was a revolutionary insight and it's not an exaggeration to say that it affects the life of just about every working number theorist just about every day.
Maybe an analogy: it's a bit like the fact that modern genomic techniques can now be used to address questions about the history of life that biologists have wrestled with, inconclusively, for years. It's incredibly exciting when new tools open up new avenues of attack on old problems.
I think John Derbyshire's _Prime Obsession_, despite its dumb title, is the best book for non-mathematicians about all this stuff.
posted by escabeche at 12:23 PM on August 3, 2009 [2 favorites]
Briefly: the study of prime numbers is part of number theory, a very old branch of mathematics going back to the Greeks. Primes are in some sense the "atoms" from which all numbers are built. The problem Riemann addresses is then rather natural: how can we estimate the number of primes less than 10000? Less than 1,000,000? Less than 100,000,000? Nowadays we can answer these questions exactly via computer calculation, but there is always a limit -- if you want to know how many primes there are less than 10^1000, you'll have to use some kind of estimate.
Anyway, Riemann discovered that this problem is related to the properties of a certain function, now called the Riemann zeta function, which can be expressed as an integral involving complex numbers. In other words, to understand the most basic objects of classical mathematics, the prime numbers, which people had been chewing on for thousands of years, it was absolutely necessary to make reference to the comparatively novel and high-test techniques of calculus and complex numbers. This was a revolutionary insight and it's not an exaggeration to say that it affects the life of just about every working number theorist just about every day.
Maybe an analogy: it's a bit like the fact that modern genomic techniques can now be used to address questions about the history of life that biologists have wrestled with, inconclusively, for years. It's incredibly exciting when new tools open up new avenues of attack on old problems.
I think John Derbyshire's _Prime Obsession_, despite its dumb title, is the best book for non-mathematicians about all this stuff.
posted by escabeche at 12:23 PM on August 3, 2009 [2 favorites]
MarkovPost
Poor Andrey Markov, your name is associated with mathematical processes and elements, but that is all overshadowed by the Markov algorithm and Markov chain generators.
posted by filthy light thief at 12:25 PM on August 3, 2009
Poor Andrey Markov, your name is associated with mathematical processes and elements, but that is all overshadowed by the Markov algorithm and Markov chain generators.
posted by filthy light thief at 12:25 PM on August 3, 2009
« Older "I live in hope I can jump before I am pushed." | It's the end of the world as we ... think it might... Newer »
This thread has been archived and is closed to new comments
posted by DecemberBoy at 12:00 PM on August 3, 2009