The Computer as Crucible: An Introduction to Experimental by Keith Devlin, Jonathan Borwein

By Keith Devlin, Jonathan Borwein

Keith Devlin and Jonathan Borwein, famous mathematicians with services in numerous mathematical specialties yet with a standard curiosity in experimentation in arithmetic, have joined forces to create this creation to experimental arithmetic. They conceal various subject matters and examples to offer the reader a great feel of the present nation of play within the quickly starting to be new box of experimental arithmetic. The writing is obvious and the reasons are greater via appropriate old proof and tales of mathematicians and their encounters with the sphere through the years.

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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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