Riemann’s Hypothesis

 

[Editor: This is the second post in a series about the Millenium Problems. This post is about the Riemann Hypothesis, probably the single most fascinating unsolved problem in all of mathematics. This hypothesis was first proposed by the mathematician Bernhard Riemann. Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.]

The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011.

It all started with Leonard Euler who in 1740 considered the function zeta(s)  (the zeta function) given  by the series,

zeta(s)=1+frac{1}{2^s}+frac{1}{3^s}+ldots

For s=1,  the series diverges and for s data-recalc-dims=1" /> it converges to give a function called Euler’s zeta function. The importance of this function in number theory became apparent when Euler proved,

zeta(s)=((12^{-s})^{-1}(13^{-s})^{-1}(15^{-s})^{-1}ldots

where the product involved all the factors of the form  (1p^{-s})^{-1} for every prime p. This gives an interesting proof that there are infinite number of primes, because otherwise zeta(1) would have been finite.

Bernhard Riemann, 1863

The connection of zeta function to number theory became more significant when Riemann extended the definition of the function to the whole complex s-plane except s=1, where it has a simple pole,  by first using the fact that the series converges for Re(s) data-recalc-dims=1" />, giving an analytic function in this region and then by using analytic continuation. The extended function is called Riemann’s zeta function. It was found to have zeros at  – 2, -4, -6, …  and these were all the real zeros. Riemann also found that the function must have an infinite number of non-real zeros. Riemann conjectured that all these zeros are of the form frac{1}{2}+ bi i.e. that all these complex zeros lie on the line Re(z)=frac{1}{2} , called the critical line. In 1859 he used the zeta function to investigate the pattern of primes. He hoped that the proof of his conjecture would lead to the proof of a conjecture made by Gauss that the number of primes below any positive integer n is approximately n/ log n and that the approximation becomes better and better as n approaches infinity. However later Hadamard and de la Valle Poussin proved Gauss’s conjecture which came to be known as the Prime Number Theorem, by exploiting the connection between the primes and Riemann’s zeta function without the need of proving Riemann’s conjecture (which is now known as Riemann’s hypothesis). Still if Riemann’s hypothesis is proved it will lead to many other significant  results in analytic number theory.

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Malay Dutta

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14 Comments
  • Peter L. Griffiths
    Posted at 22:33h, 31 August Reply

    As regards the Riemann hypothesis, very few people seem to know that the non-trivial zeta evaluations discovered by Euler are all terms of the cotangent series. The first term 1/2 becomes Riemann's critical line which apparently converts the zeta evaluations into zero evaluations. It is not just the line which is critical, I confess to being highly critical of the whole process.

    • Gonit Sora
      Posted at 23:22h, 04 September Reply

      Thanks for the info, do keep visiting.

  • Peter L. Griffiths
    Posted at 22:22h, 21 September Reply

    Further to my comment of 31 August 2013, Riemann's 1859 paper is largely based on the more difficult paper by Dirichlet 'There are infinitely many prime numbers in all arithmetic progressions with first term and difference coprime' 1837. Dirichlet needs to be clarified before tackling Riemann.

  • Peter L. Griffiths
    Posted at 21:28h, 25 September Reply

    Dirichlet's paper seems to be proposing a method of generating primes from a simple formula a +nd, where a is 4 and d is 3, and n can be any integer from 1 to infinity. This results in a few composites and omits a few primes, but followers of Riemann seem to think it has possibilities in locating primes which have no roots and hence in locating zeros, which are certainly not the same as zetas.

  • Peter L. Griffiths
    Posted at 23:03h, 31 October Reply

    Further to my previous comments, I have come to the conclusion that Euler's product formula for primes forming the basis of the Riemann hypothesis is wrong. The sum of all numbers can be reconciled with the sum of all primes by systematically deducting the composites from the sum of all numbers, leaving the sum of all primes. The formula quoted by Riemann does not do this.

  • Peter L. Griffiths
    Posted at 22:51h, 02 November Reply

    Further my comment of 31 October 2013, Euler's product formula for primes can be corrected by dividing the Harmonic series by the composite fractions including 1 followed by 1/4 contained in the Harmonic series.

  • Peter L. Griffiths
    Posted at 22:40h, 03 November Reply

    Further to my comment of 2 November 2013 which I accept is difficult to understand. We have to distinguish finite amounts such as the composite fractions from their location which initially may be transfinite, but these composite fractions can be brought together into the finite location dividing the Harmonic series.

    • Gonit Sora
      Posted at 18:28h, 04 November Reply

      Thanks for your comments and interest in this article.

  • Peter L. Griffiths
    Posted at 23:00h, 05 November Reply

    Further to my recent comments, I have come to the conclusion that it is not only all the primes (P) but also all the composites (C) which are deducted from the Harmonic series (H) leaving
    1 =H.Product(P-1)/P which is Euler's formula, and not C =H.Product (P-1)/P which is wrong.

  • Peter L. Griffiths
    Posted at 22:48h, 25 November Reply

    This comment I hope delivers the fatal thrust. 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
    cosu +isinu equals e^(iu) where it is not possible for e to be meanfully replaced by other values.
    This means that Riemann is badly wrong in applying as a power s =1/2 +ti. It also means that all the arguments in his 1859 paper are practically fallacious.

  • Peter L. Griffiths
    Posted at 21:14h, 19 December Reply

    Further to my comment of 25 Novmber 2013, in which I say 'Riemann fails to recognise that an expresion containing an imaginary number such as 1/2 +ti cannot be a power unless the base is e'. I should add 'or its equivalent such as n^(1/logn) which is also always e even though n may be any value'. This makes no difference to my conclusion.

  • Peter L. Griffiths
    Posted at 21:11h, 23 August Reply

    A possible approach to identifying primes could be to apply Euler's discovery H(1-[1/2])(1-[2/3])(1-[4/5])(1-[6/7]) ....=1, where H is the harmonic series 1+(1/2) +(1/3) +(1/5) .... particularly where the series is not carried to infinity.

  • Peter L. Griffiths
    Posted at 19:54h, 30 August Reply

    A Mystery which needs to be solved, why is it that one half the Harmonic series (1/2) +(1/4) +(1/6) .... is not what remains being 1 +(1/3) +(1/5) +(1/7)..... when deducted from the full Harmonic series 1 + (1/2) + (1/3) +(1/4)....The positioning of the terms seems to be important, but the first term 1 needs to be retained to compute the full prime product series.

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