Mellin Transform of The Riesz Function
The Riesz function is related to the Riemann zeta function via its Mellin transform. If we take
we see that if then
converges, whereas from the growth condition we have that if then
converges. Putting this together, we see the Mellin transform of the Riesz function is defined on the strip . On this strip, we have
From the inverse Mellin transform, we now get an expression for the Riesz function, as
where c is between minus one and minus one-half. If the Riemann hypothesis is true, we can move the line of integration to any value less than minus one-fourth, and hence we get the equivalence between the fourth-root rate of growth for the Riesz function and the Riemann hypothesis.
G. H. Hardy gave the integral representation of using Borel resummation as
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