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Re: Prime numbers and pi
Posted:
Apr 18, 2009 8:42 AM


How did Euler solve the Basel problem
'1x1 '2x2 '3x3 '4x4 '5x5 '6x6 '7x7 '8x8 '9x9 ... = ???
Let us look at a simpler series whose value can easily be found:
'1x1 '2x2 '4x4 '8x8 '16x16 '32x32 ... = 1 '3 or 4/3
1 '3 = '1x1 '3 1 '3 = '1x1 '2x2 '12 1 '3 = '1x1 '2x2 '4x4 '48 1 '3 = '1x1 '2x2 '4x4 '8x8 '16x16 '192
Now let us transform the first series into a stairway
'1x1 ? '1x1 '2x2 ? '1x1 '2x2 '3x3 ? '1x1 '2x2 '3x3 '4x4 ? '1x1 '2x2 '3x3 '4x4 '5x5 ?
Are there final terms that fit logically into the pattern? Yes, the doubled inverses of the odd numbers:
"1 '1x1 "3 '1x1 '2x2 "5 '1x1 '2x2 '3x3 "7 '1x1 '2x2 '3x3 '4x4 "9 '1x1 '2x2 '3x3 '4x4 '5x5 "11
Euler was very good at calculating in his head and at interpreting a number, so he was well able to guess the approximated value pi x pi / 6 which can also be given as the infinite product involving all primes in the square
2x2/1x3 x 3x3/2x4 x 5x5/4x6 x 7x7/6x8 x 11x11/10x12 x 13x13/12x14 x 17x17/16x18 x 19x19/18x20 ...
The identity of the infinite series and the infinite product marks the begin of the zeta function, the basic link between pi and the primes.



