Legendre's constant

This article utilizes technical mathematical notation for logarithms. All instances of log(x) without a subscript base should be interpreted as a natural logarithm, commonly notated as ln(x) or log_e(x).

Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function $\pi (x)$ . The value that corresponds precisely to its asymptotic behavior is now known to be 1.

Examination of available numerical data for known values of $\pi (x)$ led Legendre to an approximating formula.

Legendre constructed in 1808 the formula

\pi (x)\approx {\frac {x}{\log(x)-B}},

where $B=1.08366$ (OEIS: A228211), as giving an approximation of $\pi (x)$ with a "very satisfying precision".^[1]^[2]

Today, one defines the value of $B$ such that

\pi (x)\sim {\frac {x}{\log(x)-B}},

which is solved by putting

B=\lim _{n\to \infty }\left(\log(n)-{n \over \pi (n)}\right),

provided that this limit exists.

Not only is it now known that the limit exists, but also that its value is equal to $1,$ somewhat less than Legendre's $1.08366.$ Regardless of its exact value, the existence of the limit $B$ implies the prime number theorem.

Pafnuty Chebyshev proved in 1849^[3] that if the limit B exists, it must be equal to 1. An easier proof was given by Pintz in 1980.^[4]

It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term

\pi (x)=\operatorname {Li} (x)+O\left(xe^{-a{\sqrt {\log x}}}\right)\quad {\text{as }}x\to \infty

(for some positive constant a, where O(…) is the big O notation), as proved in 1899 by Charles de La Vallée Poussin,^[5] that B indeed is equal to 1. (The prime number theorem had been proved in 1896, independently by Jacques Hadamard^[6] and La Vallée Poussin,^[7] but without any estimate of the involved error term).

Being evaluated to such a simple number has made the term Legendre's constant mostly only of historical value, with it often (technically incorrectly) being used to refer to Legendre's first guess 1.08366... instead.

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JAMES GRIME: So I've got another prime number generating formula for you. So this one is more modern. It was 1947, a mathematician called Mills, and he found this formula. He said there exist numbers-- which we're going to call theta, the Greek letter theta-- where you raise it to the power 3 to the power n. And you actually then have to round it down. That gives you fractions, so you actually round it down to the nearest whole number, which you write like this. That's the symbol for rounding it down. There exists a number like this that will always give you primes for every value of n. So you might have n equals 1, n equals 2, n equals 3, and every value is a prime. The smallest value for theta where this will work is called Mills' Constant. I'll write it out for you. It's called theta, and Mills' Constant is 1.306377883863080690, something, something, something, something. So you can see it's not a whole number. So the value you get when you raise it, it's not going to be a whole number, and you have to round it down. Let me just try the first few to show you what I mean. Let's do n equals 1. Let's use my constant. You put it in. So then it's 3 to the power 1, so it's just cubed. So what I'm going to do is, for each time, I'm going to take this constant and then raise it to the power 3 to the power n. And I'll do that n equals 1, n equals 2, and so on. So, n equals 1. 3 to the power n is just 3. So that's going to be theta cubed. That's the first prime. And what is the answer? What is theta cubed? Theta cubed is 2.229, something, something, something. You round it down, so the prime is 2. Hey! We've got that is prime. In fact, every number should be prime. Let's try the next one. Let's do n equals 2. So this is going to be 3 squared. That's theta to the power 9. So take the constant to the power 9, and that is going to be 11.082, something, something. Round it down, you get 11. So you get gaps. It's not consecutive primes, but every one is a prime. You get 11, that's the next one. Let's do n equals 3. So I now need theta 3 to the power 3, 27. Theta to the power 27 is 1,361.000, something, something, something. 1,361 is a prime as well. Let's do n equals 4. Theta to the power 81. You get 252,100,887-- BRADY HARAN: Oh, yeah. That's clearly a prime. JAMES GRIME: Which is clearly a prime, which is a a prime-- Point something, something, something, but you round it down. But you're actually guaranteed to get a prime every time. BRADY HARAN: Do you know what my conclusion is? That number's awesome! JAMES GRIME: I know. I completely agree. That number is completely awesome. So the rock stars of math, so I have pi, and e, and these golden ratios, and things like that. It's not nearly as famous. It's a constant that gives you primes. Brilliant! I love it. BRADY HARAN: Is it proven? I mean, there's a lot of n's you could put into that equation. Is this a proven thing? JAMES GRIME: Yep. This is proven. This is proven. So you might think then, what's the big deal? We've got a formula to find guaranteed primes. How amazing is that? And the problem is, did you notice how big the powers were getting? The powers very quickly become so huge that even computers can't deal with the problem. So the next one, n equals 5, is theta to the power 243. And this is about 1.602 times 10 to the 28. Once you start getting bigger than that, you can't do it. The computers can't cope with that. The other problem is you need to know this constant to a great accuracy. Because, what I've written out for you-- let's say if I terminated it there-- isn't accurate enough to tell me what the next prime is. The next prime actually is something huge. We'll put it on the screen, I think, instead of writing it out. But this was not accurate enough to work it out. That's a bit of a problem. You need to know theta very, very accurately. BRADY HARAN: Is theta rational? Does theta have an end? JAMES GRIME: We don't know. We don't know. How amazing is that? We don't know if theta is irrational or not. That's a cool question as well. We don't know. One of the other problems with theta is, at the moment, we don't know any way to really work it out, apart from taking one of the Mill primes-- You take one of the Mill primes, cube root it, and theta is approximately that. So I'm afraid it's a bit of a circular argument. We haven't got a good way of working out theta. You have to know the primes in the first place to work out theta. BRADY HARAN: Oh, well I'm not so impressed by it anymore. JAMES GRIME: I know. So you can see why it's not so practical. One of the things that we might be able to do is, if Riemann's hypothesis is true-- Riemann's hypothesis is a very important hypothesis in mathematics that hasn't been proven yet, that is a Millennium prize, and it's related to crimes and how they're distributed. If that is true-- (WHISPERING) it probably is true-- If it's true, then we have another way to find Mill primes. Using that method, we can now calculate what theta is probably going to be. It's been calculated up to about 7,000 digits, but it is relying on if the Riemann hypothesis is true, which hasn't been proven. (WHISPERING) It's probably true. We have worked out larger Mill primes, and we have worked out theta to a large number of places. BRADY HARAN: Our thanks to audible.com for supporting this video. They have a huge range of audio books you can listen to. Great for putting on your handheld device or listening to in the car. And you can download a free book at audible.com/numberphile. This is the part where we get to recommend a book, which I always enjoy. And today I'd like to recommend "Foundation" by Isaac Asimov. The first in a brilliant series of books, probably my favorite series of books ever. So go to audible.com/numberphile, get a free book, and why not check out anything by Isaac Asimov, but especially "Foundation," or any of its sequels and prequels.

References

^ Legendre, A.-M. (1808). Essai sur la théorie des nombres. Courcier. p. 394.
^ Ribenboim, Paulo (2004). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 188. ISBN 0-387-20169-6.
^ Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, page 17. Third (corrected) edition, two volumes in one, 1974, Chelsea 1974
^ Pintz, Janos (1980). "On Legendre's Prime Number Formula". The American Mathematical Monthly. 87 (9): 733–735. doi:10.2307/2321863. ISSN 0002-9890. JSTOR 2321863.
^ La Vallée Poussin, C. Mém. Couronnés Acad. Roy. Belgique 59, 1–74, 1899
^ Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques, Bulletin de la Société Mathématique de France, Vol. 24, 1896, pp. 199–220 Online Archived 2012-07-17 at the Wayback Machine
^ « Recherches analytiques sur la théorie des nombres premiers », Annales de la société scientifique de Bruxelles, vol. 20, 1896, pp. 183–256 et 281–361

External links

Weisstein, Eric W. "Legendre's constant". MathWorld.

v t e Prime number conjectures
Hardy–Littlewood 1st 2nd Agoh–Giuga Andrica's Artin's Bateman–Horn Brocard's Bunyakovsky Chinese hypothesis Cramér's Dickson's Elliott–Halberstam Firoozbakht's Gilbreath's Grimm's Landau's problems Goldbach's weak Legendre's Twin prime Legendre's constant Lemoine's Mersenne Oppermann's Polignac's Pólya Schinzel's hypothesis H Waring's prime number

This page was last edited on 23 December 2023, at 15:02

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