# What is the mathematical meaning of this symbol? [closed]

Someone asked me, and I told them I would try to find out... what is the meaning of this symbol:

B'L     or     BL'

(I'm not sure if the tick comes before or after the L. It was found on a "nerd clock". The value of this symbol, by the way, is 1.

## closed as off topic by Felipe Voloch, Alain Valette, Gerald Edgar, j.c., Andy PutmanFeb 19 '13 at 22:19

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Apparently, it is supposed to be Legendre's constant, also known as $1$.

• This is the clock: photodump.com/wp-content/uploads/2009/09/geek-clock.jpg – Dan Piponi Apr 22 '10 at 22:24
• That makes me flash on a dialogue from my youth: Professor (slow Texas drawl): That's almost as useful as knowing Bieberbach's constant. Student (the stooge): What's Bieberbach's constant? Professor: One-fourth. – Carl Weisman Apr 22 '10 at 23:26
• Isn't there a similar story (with Edmund Landau) concerning the "Caratheodory constant"? – Yemon Choi Apr 22 '10 at 23:33
• Can someone please post a link where we can buy the clock? – Joel David Hamkins Apr 23 '10 at 0:43
• A clock set in Computer Modern is somewhat cute :) – Mariano Suárez-Álvarez Apr 23 '10 at 1:02

Hi,

This was supposed to be a comment to Mariano's answer, but it seems too long for a comment.

Somebody gave me that clock for Christmas and all along I thought $B_L'=1$ was some silly Physics constant... but it seems Mariano is right, and this refers to Legendre's constant (at least, that's the case according to this other site).

I was terribly curious, though, to find out where this notation came from and decided to go right to the source, "Essai sur la Theorie des Nombres", by Legendre. Amazingly, our friends at Google have scanned the whole book, and the whole volume is freely available here. It is a large volume, however! So it was not easy to locate the exact place where Legendre talks about the prime counting function. A nice paper by Goldstein in the American Math Monthly, "A history of the prime number theorem"" was very helpful to locate the exact reference: p. 394-398 in the second edition of the "Essai sur..." by Legendre.

In p. 395, Legendre explains that $\pi(x)$, the number of primes $\leq x$, seems to grow like $$\frac{x}{A\log x + B}$$ and conjectures that $A=1$ and $B=-1.08366\ldots$ (now a famous mistake, since later on the proofs of the prime number theorem would show that $B=1$). But, in any case, Legendre himself called this constant $B$ and I suppose somebody added the subscript $L$, to $B_L$, to remind us of Legendre's name.

However, I am still puzzled by the apostrophe, $B_L'$.

• It seems that according to this answer: mathoverflow.net/a/273337/41139, the prime number theorem shows that $B = -1$ rather that 1, rendering Legendre's guess a lot more accurate. – Vincent Jun 30 '17 at 8:21
• Great answer by the way, thanks for all the detective work! – Vincent Jun 30 '17 at 8:21