Richard's Approach

This 11 page proof is developed from a well-known starting point and only uses established, unchallenging mathematics (See MATHEMATICS USED) - although there are a number of major steps where the theory moves into a new area.

Georg Friedrich Bernhard Riemann

Georg Friedrich Bernhard Riemann

The original form of the Zeta-function was as a series, with very simple complex number terms, that was seen to have far reaching implications for areas of mathematics such as prime number theory. Riemann produced a modified entire Xi-function that eliminated a set of ‘trivial zeros’ and two poles on the real axis and defined an integral to generate this revised form, which retained all of the original ‘non-trivial’ zeros. The locations of these have important implications and Riemann’s Hypothesis was that they all lie on the critical line half way across the critical strip that flanks and lies to the right of the imaginary axis. Since that time it has been proved that they all lie within the critical strip and that an infinite number of them lie on the critical line but the final requirement that they all lie on the critical line has previously eluded proof.

The required proof involved two major steps. The first of these, proved in Section 2, established that a particular configuration of simple poles can only produce finite location zeros lying on a straight line. The second step, covered in Section 3, showed that Riemann's function can be expressed in the required configuration with the straight line coinciding with the Critical Line, which proves that his Hypothesis is correct.

Section 2 proves the collinearity of the finite location zeros generated by a set of simple poles lying on a straight line having alternating sign real residues of magnitudes such that their running total also alternates in sign and ultimately equals nought, by:-

2.1     Performing an integration to derive a function having the correspondingly weighted set of logarithmic singularities.

2.2     Showing that on each side of the line the imaginary part of this function has constant values between the singularities which change sign as each one is passed and are antisymmetric between the two sides of the line, becoming zero outside the occupied span.

2.3     Deducing that the non-zero level lines of this imaginary part that emerge from each singularity are constrained to arc between neighbouring ones, leaving the zero value level lines as the only ones that are available to produce their self-intersection that has to occur at the zeros of the original function.

2.4     Recognising that the only place where these zero value level lines can self intersect without contravening requirements applying to analytic functions is on the straight line outside the span occupied by the singularities, which means that this must be where any finite location zeros are situated.

Section 3 shows that Riemann's Xi function, which has the same zeros as the Zeta function in the Critical Strip, can be expressed in terms of simple poles that lie on the Critical Line and comply with the requirements of Section 2, by:-

3.1     Applying a simple affine transformation to Riemann's integral for the Xi function, in terms of his Phi function, to improve its symmetry.

3.2  Integrating this equation by parts to generate an integral in terms of the derivative of Phi.

3.3  Re-expressing this derivative in a form that permits the integral for Xi to be carried out to give an expression in terms of simple poles.

3.4  Combining these poles in pairs to transform them into a form which is expressed in terms of simple poles in a new variable in which they all lie on the positive real axis and comply with the requirements of Section 2, while  the Critical Line is now located on its negative half, which proves Riemann’s Hypothesis.

The proof can be downloaded here. (See DOWNLOAD PDF)