A History of Greek Mathematics, Volume 2: From Aristarchus by Thomas Heath

By Thomas Heath

Volume 2 of an authoritative two-volume set that covers the necessities of arithmetic and contours each landmark innovation and every vital determine, including Euclid, Apollonius, and others.

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69777465796400798203 The three marked with an asterisk are likely to be harder work. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 36 The Computer as Crucible 2. Name that sum. Identify ∞ ∑ r(n)7 (1 + 14n + 76n2 + 168n3 ) n =0 where 1 8 2n , 1 (2n − 1) 1 3 Γ n+ · ··· 2 2 r (n) := 2 2 = √ . n! π Γ ( n + 1) Here, ∞ Γ( x ) := 0 t( x−1) e−t dt (the Gamma function) is the unique function on the positive real numbers satisfying the functional equation xΓ( x ) = Γ( x + 1), Γ(1) = 1, whose logarithm is convex. In particular, this means Γ(n + 1) = n!

6) in some detail the experimental process that led to the discovery of a generating function for the even ζ-values. An ordinary generating function for a sequence { an } is a formal power n series ∑∞ n=0 an x . In nice cases, the sum can be evaluated in closed form and leads to a great deal of information about the sequence. For example, the generating function for the averaged harmonic sum n −1 ∞ 1 k ∑ ∑ n =1 k =1 xn n evaluates to 1 1 1 11 log(1 − x )2 = x2 + x3 + x4 + O x5 . 2 2 2 24 The main result discovered regarding the even ζ-values is the identity ∞ ∑ k =1 ∞ 1 = 3 ∑ 2 k2 − x2 k =1 k 2k k 1 1 − x2 k2 k −1 1 − 4x2 m2 ∏ 1 − x 2 m2 m =1 .

It is unusual for such a function to have closed forms at both 1/4 and 1. The values Calkin obtained were, wait for it: 1, − − 1 1 25 1 3 1 1 1 , , − , , − , , − , , 3 25 147 1089 20449 48841 312987 55190041 1 1 49 , , − , ... 14322675 100100025 32065374675 So now you know where that sequence in the previous chapter came from. Indeed it is known (and can be proved—see [Borwein and Bailey 08]) that, for each positive integer k, you have 3 F2 6k, − 2k, 2k + 1 1 4k + 1, 2k + 21 = 6k k 4k k . along with a similar formula for odd k.

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