Squaring the Riemann Hypothesis

Prof. Ross McPhedran,
School of Physics,
University of Sydney

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One of the great unsolved problems in mathematics is the Riemann
hypothesis, concerning the distribution of  the zeros of the Riemann
zeta function, of  importance for example in the distribution of prime
numbers. A problem of almost the same era concerned the evaluation of
lattice sums in solid state physics- for example, the Madelung sums
which gave estimates for the strength of the electrostatic forces
holding ionic crystals together. Surprisingly, the Riemann hypothesis
and the evaluation of lattice sums are linked, and it may be that the
methods which form the basis of solid state physics may shed new light
on the Riemann hypothesis.
I will briefly review the background to the Riemann hypothesis and the
evaluation of lattice sums, before discussing the behaviour of
two-dimensional lattice sums incorporating a phase or Bloch vector. I
will show that the dependence of the sums of this vector enables
interesting connections  between sums to be established. I will comment
on Hardy's expression for lattice sums as a product of the Riemann zeta
function and the Catalan beta function, and will present numerical
evidence that both have similar behaviour in relation to their
distribution of zeros and their asymptotic behaviour. I will also
discuss an angular lattice sum, exhibit a duplication formula for it,
and show that it too is a candidate for having a set of non-trivial
zeros lying on the critical line of the zeta function. I will discuss an
identity due to Kober, which relates three functions, two of which seem
to obey the Riemann hypothesis, while their difference
does not.