Deninger'sapproach to the RH and a talk of his which "gave...the impression that a proof of the Riemann hypothesis is just around the corner..."
his "proof" of the Riemann Hypothesis"...explains the mathematical motivation forhis Riemann Hypothesis proof and reveals that he proved the Bieberbach conjecture sothat he could get funding to work on the Riemann Hypothesis."In June 2004, Louis de Branges of the .
The media coverage therefore appears to be much ado about nothing."
["How the Riemann Hypothesis was not proved" (Excerpts from two letters from P.
to a Fourier transform (more exactly to Cosine Fourier transform). If the Riemann hypothesis is true then we have to prove that all zeros of the function occur for.
Müller's comments on why this 'proof' is flawed.]
Xian-Shun Luo, (12/2008) [abstract:] "In this paper we will give a simple proof of Riemann Hypothesis, considered to be one of the greatest unsolved problem in mathematics, related to inverse scattering problem and radom matrices.
First we shortly represent the transition from the Riemann zeta function of complex variable to the xi function introduced already by Riemann and derive for it by means of the Poisson summation formula a representation which is convergent in the whole complex plane (Section 2 with main formal part in Appendix
To the Xi function in mentioned integral transform we apply the second mean-value theorem of real analysis first on the imaginary axes and discuss then its extension from the imaginary axis to the whole complex plane. For this purpose we derive in Appendix B in operator form general relations which allow to extend a holomorphic function from the values on the imaginary axis (or also real axis) to the whole complex plane which are equivalents in integral form to the Cauchy-Riemann equations in differential form and apply this in specific form to the Xi function and, more precisely, to the mean-value function on the imaginary axis (Sections 3 and 4).
Yuan-You Fu-Rui Cheng, (10/2008)[abstract:] "The Riemann zeta function is a meromorphic function on the whole complex plane.
Then in Section 5 we accomplish the proof with the discussion and solution of the two most important equations (10) and (11) for the last as decisive stage of the proof. These two equations are derived in preparation before this last stage of the proof. From these equations it is seen that the obtained two real equations admit zeros of the Xi function only on the imaginary axis. This proves the Riemann hypothesis by the equivalence of the Riemann zeta function to the Xi function and embeds it into a whole class of functions with similar properties and positions of their zeros.
right that means to the critical line and call this Xi function with. We derive some representations for it among them novel ones and discuss its properties, including its derivatives, its specialization to the critical line and some other features. We make an approach to this function via the second mean value theorem of analysis (Gauss-Bonnet theorem, e.g.,   ) and then we apply an operator identity for analytic functions which is derived in Appendix B and which is equivalent to a somehow integrated form of the Cauchy-Riemann equations. This among other not so successful trials (e.g., via moments of function) led us finally to a proof of the Riemann hypothesis embedded into a proof for a more general class of functions.
The Sections 6-7 serve for illustrations and graphical representations of the specific parameters (e.g., mean-value parameters) for the Xi function to the Riemann hy- pothesis and for other functions which in our proof by the second mean-value problem are included for the existence of zeros only on the imaginary axis. This is, in particular,
In this Section we represent the known transition from the Riemann zeta function to a function and finally to a function with displaced complex
Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic.
As already expressed in the Introduction, the most promising way for a proof of the Riemann hypothesis as it seems to us is the way via a certain integral representation of the related xi function. We sketch here the transition from the Riemann zeta function to the related xi function in a short way because, in principle, it is known and we delegate some aspects of the derivations to Appendix A.