In 1986 Csordas, et al. (1) used some aspects of the theory of sign-regularity of kernels and matrices (see Karlin (2)) to prove the second order Turan inequalities, which are known to be a necessary condition for the truth of the Riemann Hypothesis. The key object in the proof is the function \(\Phi(u)\), which is a Fourier cosine transform of Riemann's \(\Xi\) function. By simplifying and then extending the Turan inequalities to all orders, we present a number of conjectures, which, if true, would result in a proof of the Riemann Hypothesis. Some of the conjectures have been proved, while some of the remainder are supported by extensive numerical computations. Crucial to the approach is the use of the \(m^\rm{th}\) cumulant of \(\Phi(u)\), denoted by \(\Psi(m,u)\). There appears to be 'simple' function which specifies \(m(r)\), the lowest value of \(m\) for which \(\Psi(m,u)\) is sign-regular of type \(\Bbb R_r\) for order \(r\). This may be an indication that a proof of sign-regularity is feasible.
We have also extended the Karlin technique of proving the required positivity of some of the infinite set of determinants that arise in the method, but some cases remain to be treated.
References:
John Nuttall