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What mathematicians call the Mellin transform relating a theta function to its (completed) zeta function
$\hat \zeta(s) = \int_0^\infty t^{s-1} \hat \theta(t) \, d t$is precisely what physicists call the Schwinger parameter-formulation which takes the partition function of the worldline formalism to the zeta-regulated Feynman propagator
$Tr H^{-s} = \int_0^\infty t^{s-1} Tr \exp(- t H) \, d t \,.$I have tried to briefly mention this relation in relevant entries and to cross-link a bit. But more should be done.
I just copied the above 2 formulas to the entry Mellin transform.
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