In the Beginning, There Was... Euler's Formula!

I will start this blog the way Bernhard Riemann started his paper: with Euler's product formula that John Derbyshire called the golden key:

\zeta(s)=\sum_{n\ge1}n^{-s}=\prod_{p}(1-p^{-s})^{-1}

This holds for any complex number s with \Re s > 1. If you look up a proof in any modern textbook, you will find a number technical rearrangements that end up in an examination of the absolute convergence on both sides. But actually, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes. Let's start by writing out the sum on the left hand side:

\zeta(s)=1^{-s}+2^{-s}+3^{-s}+4^{-s}+5^{-s}+6^{-s}+7^{-s}+8^{-s}+9^{-s}+10^{-s}+\ldots

Now, let's sift out all even terms by multiplying the equation by 2^{-s}

2^{-s}\zeta(s)=2^{-s}+4^{-s}+6^{-s}+8^{-s}+10^{-s}+12^{-s}+14^{-s}+16^{-s}+18^{-s}+\ldots

and subtracting the second from the first equation:

(1-2^{-s})\zeta(s)=1^{-s}+3^{-s}+5^{-s}+7^{-s}+9^{-s}+11^{-s}+13^{-s}+15^{-s}+\ldots

OK, all even terms are gone, now let's eliminate all remaining multiples of 3. We multiply by 3^{-s}

3^{-s}(1-2^{-s})\zeta(s)=3^{-s}+9^{-s}+15^{-s}+21^{-s}+27^{-s}+33^{-s}+39^{-s}+45^{-s}+\ldots

and subtract again:

(1-3^{-s})(1-2^{-s})\zeta(s)=1^{-s}+5^{-s}+7^{-s}+9^{-s}+11^{-s}+13^{-s}+\ldots

I think by now the pattern is clear. We continue by multiplying all the primes 5^{-s}, 7^{-s}, 11^{-s}, ..., and continue subtracting from what we've got before, and arrive at

\cdots(1-17^{-s})(1-13^{-s})(1-11^{-s})(1-7^{-s})(1-5^{-s})(1-3^{-s})(1-2^{-s})\zeta(s)=1

Dividing by the factors on the left hand side, we arrive at our final result:

\zeta(s)=(1-2^{-s})^{-1}(1-3^{-s})^{-1}(1-5^{-s})^{-1}(1-7^{-s})^{-1}\cdots=\prod_{p}(1-p^{-s})^{-1}

Magic!

9 thoughts on “In the Beginning, There Was... Euler's Formula!”

  1. Near the beginning of his 1859 paper Riemann incorrectly assumes that the complex variable s =(1/2) + ti is a zeta power. Riemann fails to recognise that an expression containing an imaginary number such as (1/2) + ti cannot be a power unless the base is a log base such as e. The best known example of this is Cotes's formula (not mentioned by Riemann), cosu + isinu equals e^(iu), where it is not possible for e to be meaningfully replaced by other values, also e^(1/2) X e^(iu) equals e^[(1/2) +iu]. This means that Riemann is badly wrong in applying as a power s = (1/2) +ti and in failing to recognise t as an angle. It also means that practically all the arguments in his 1859 paper are fallacious.

    1. Nice to read a comment here, though I fail to follow through as well as to see the connection to this particular post, particularly since every statement of Riemann's paper has been thoroughly scrutinised and (save the RH) rigorously proved afterwards. Can you elaborate?

  2. Unlike the individuals referred to in Marcus Schepke's comment, I am not and never have been a professional academic. This means that I can reach conclusions based on mathematical rules such as Cotes's formula which the professionals have overlooked and certainly not scrutinised. I have done something similar to Fermat's Last Theorem but this takes just over 400 words, and should be published In April 2014.

    1. I'd be indeed interested to read this - I hope you will keep me posted? Particularly interesting would be to hear about Cotes's formula since I couldn't find any reference to this whatsoever. Is there any chance you can give us a preview on this?

  3. Cotes's fomula can be constructed by bringing together the infinite series for sines, the infinite series for cosines and the infinite series for antilog e discovered by Isaac Newton in his epistola posterior 1676. The inability of e to be replaced by any other value when it has a complex power is my own invention. As also is the fact that the coefficient of i has to be an angle. The same also applies to -1 equals e^[(PI). i]. In the literature on the Riemann hypothesis the only slightly erroneous reference to Cotes's formula I can find is on page 124 of H.M. Edwards's book.

  4. Further to my comment of 11 December 2013, I ought to clarify my observation about the inability of e being replaced by any other value. I now consider that it is possible for e to be replaced by other values but only because the general formula for any variable base n and power u should be the
    constant n^(u/logn). When n equals e, log n will be clearly 1. The fact that n^(t/logn) has to be a constant particularly when applied to i that is n^(ti/logn) undermines virtually all the arguments in Riemann's 1859 paper.

  5. A simpler explanation of my comment of 17 December 2013 is that antilog e can be expressed as n^(1/logn) whatever the value of n.

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