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Topic: Prime numbers and pi
Replies: 33   Last Post: Dec 10, 2012 3:43 PM

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 Franz Gnaedinger Posts: 330 Registered: 4/30/07
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.