# Supersymmetric 1-loop vacuum amplitude and Special values of L-functions

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The  zeta functions and L-functions of number theory are directly the arithmetic analogs of analytic regularization of traces of Feynman propagators (via the definition of zeta function of an elliptic differential operator). In particular the number theorist's special values of L-functions therefore correspond to 1-loop vacuum amplitudes and related quantities (vacuum energy, functional determinant, etc.) All ingredients of these analogies are well-known in the literature, though maybe the full statement is not advertized as widely as it ought to be. I have created a summary table here.

Now what is striking is that with these analogies, then vanishing of special values of zeta functions corresponds to vanishing of 1-loop vacuum amplitudes.

But vanishing of special values of zeta functions is the content of the generalized Riemann hypothesis and the vanishing of 1-loop vacuum amplitudes is of key interest in the study of supersymmetric (string) vacua. There seems to be an intimate relation between the two.

My question is: where is this relation being investigated?

One article in this direction that I am aware of is

• Carlo Angelantonj, Matteo Cardella, Shmuel Elitzur, Eliezer Rabinovici, Vacuum stability, string density of states and the Riemann zeta function,JHEP 1102:024,2011 (arXiv:1012.5091)

Is there more?

I should say that I am of course aware of a large and growing body of literature which tries to connect the Riemann hypothesis to physics in some way. But, unless I am missing something, then what I see in most of these discussions does not really fit the table. Notably some authors try to identify the Riemann zeta function with the partition function of some quantum mechanical system. But following through the function field analogy etc. shows clearly that the Riemann zeta function is not analogous to a partition function, but to the Mellin transform of a partition function (integrating out what in physics is the Schwinger parameter) and hence really is analogous to the regularized 1-loop vacuum amplitude. Most of the literature which one finds when looking for "Riemann hypothesis and physics" seems to be oblivious of this relation. My question is specifically about literature which does make the relation between vanishing of special values of zeta functions/L-functions and vanishing of 1-loop vacuum amplitudes in physics.

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