On 3/13/2013 6:01 PM, Simon Plouffe wrote:

I have a friend in Moncton (NB, canada) a math professor, Paul Deguire, browsing around with the history of math and came accros this formula (well known ?) of pi :

http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80

see the center of the page with prime numbers, also this one http://mathworld.wolfram.com/PiFormulas.html formulas 60 and 61.

It deals with prime numbers , an infinite product and pi.

Is there someone that knows a reference for this formula, when it was found by Euler ?

There are actually two different formulas in these references; the first Wikipedia formula is pi/4 = 3/4 * 5/4 * 7/8 * 11/12 * 13/12 * 17/16 * 19/20 * ..., where the numerators are the odd primes and the denominators are the nearest multiples of four. The MathWorld formulas amount to pi/2 = 3/2 * 5/6 * 7/6 * 11/10 * 13/14 * 17/18 * 19/18 * ...; this time the denominator is the nearest double of an odd integer. According to a 1924 submission by New Zealand mathematician Alexander Aitken to Edinburgh Mathematical Notes, which can be read online at http://journals.cambridge.org/abstract_S1757748900001754 , the first of these formulas appeared without proof in a letter of Euler to James Stirling dated 27 July 1738 (Euler was in Russia, so that's an Old Style date). Aitken suggests that Euler obtained this "satis notatu dignam" ("quite noteworthy") result by factoring the Gregory-Leibniz formula for pi/4 = arctan 1, i.e., 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ... = (1 - 1/3 + 1/9 - ...)(1 + 1/5 + 1/25 + ...)(1 - 1/7 + 1/49 - ...) ... = 1/(1 + 1/3) * 1/(1 - 1/5) * 1/(1 + 1/7) * ... = 3/4 * 5/4 * 7/8 * ... . I can't find the full text of the letter online, so I don't know if the similar-looking pi/2 formula is in it as well or not. By the way, the second prime-related formula in the Wikipedia article can be obtained from the pi/2 formula by reversing the technique above (i.e., turning the Euler product into a sum). -- Fred W. Helenius fredh@ix.netcom.com