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Let me first apologize, I've been reading this Math book in hebrew and since my Math knowledge is pretty basic I'm not firmiliar with the terminology in the English lanauge, I promise to do my best tho.

The book I'm reading is called "The music of the prime numbers"(thats the hebrew name atleast, should be pretty popular one).

Maybe it is just me but this book has a tendency to sometimes show some stuff without really explaning them.

Anyway, I got to this part which I don't really understand. Oiler took the Zetha(?) function and used the fact that every number can be written as the product of prime numbers to basically write the Zetha function in a different way.

firstly the book shows an example on how you can write 1/60 using primes--> 1/2x1/2x1/3x1/5.

Than is more or less jumps to this:

Zehtha(x)= 1/1^x + 1/2^x + 1/3^x + .... + 1/n^x + ....

hope I wrote this correctly, basically its a sequence of 1/1 +1/2 + 1/3 each at the power of x.

So this is the Zetha function, and with Oilers' improvment they jump to this:

=(1+1/2^x + 1/4^x + ....)x(1 + 1/3^x + 1/9^x+...)x....x(1+1/p^x+1/(p^2)^x)+....)x...

I hope this is clearly written, mind the the Xs between the brackets are not the variable x but simply the times icon.

I have no clue if I was suppost to understand why this is true from the book or not, but I definitly don't understand this and would be happy to.

Thanks in advanced.