Laplace transforms and the Riemann zeta function



Publication Details

Hill, J. M. (2007). Laplace transforms and the Riemann zeta function. Integral Transforms and Special Functions, 18 (3), 193-205.


In this article we examine certain formulae applying to the Riemann zeta function in the critical strip 0 < Re(s) < 1, and by means of the Laplace transform we demonstrate new relations between existing formulae. Müntz has proposed a formula for the Riemann zeta function in the critical strip, which involves an arbitrary function satisfying certain conditions. From this formula it is clear that a resolution of the Riemann hypothesis may hinge on a successful method for dealing with the sum-integral difference. The EulerMacLaurin summation formula is one such device. Here, we generate new expressions for certain Laplace transforms involving the sum-integral difference. Subsequently, the formulae thus far established are generalized using an arbitrary positive number m, which makes apparent the dependence of classical analysis on formulae and results associated with the exponential function which is characterized by the particular case m = 1.

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