This is not really an answer, but somewhat long comment. Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects. Let me suggest some steps towards physical point of view (which are standard):

1. use Feynman-Kac formula: Tr (e^{tH} ) = path integral_{ over all closed paths} EXP( lenght of path)

2. Left hand side is obviosuly $sum_{eigenvalues} e^{t \lambda_i} - the "spectral part of the Selberg's formula (at least particalar case of it)

3. by the Stationary phase approximation we can expect that the principal contribution comes from the extremums of the action - in this case this would be geodesics. (This corresponds to limit from quantum mechanics to classical mechanics)

Remark: Steps 1-3 are morally true for any Laplacian on any Riemannian manifold

4. By some kind of trick one must show for this particular case of Riemann surfaces with hyperbolic metric, THE ONLY contribution comes from extremums i.e. geodesics. (Such things happens in many other situations - e.g. Duistermaat Heckman formula, supersymmetric quantum mechanics etc...; however for the particular case of the Selberg formula I do not know the reference where this should be exposed in the manner which I would love, but may be I just did not search enough).

Concerning yours particular question about the term $ \int_{0}^{E}dptanh(\pi p) p $, unfortunately I cannot derive it, but just some comments.

This term corresponds to closed paths of zero length i.e. just points on the manifold (for each point you have trivial path which starts and immediately stops in this point). So contribution of such paths reduces to $\int_{manifold} something$.

When you consider the so-called super-symmetric quantum this "something" is exactly Todd class and you get Atiyah-Singer theorem for Dirac operator (or "d-bar" operator is almost the same).

But Selberg's formula is more subtle - it is NOT super-symmetric quantum mechanics, but usual quantum mechanics. So my feeling is that you can do something similar and write down some explicit integral over the Riemann surface (i.e. 2d integral) - than you can take explicitly integral in one of the variables and the rest of integral will give you desired term $ \int_{0}^{E}dptanh(\pi p) p $.

I guess that should be written somewhere, but I do not know the reference.

This is not really an answer, but somewhat long comment. Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects. Let me suggest some steps towards physical point of view (which are standard):

1. use Feynman-Kac formula: Tr (e^{tH} ) = path integral_{ over all closed paths} EXP( lenght of path)

Here H = is Laplacian

2. Left hand side is obviosuly $sum_{eigenvalues} e^{t \lambda_i} - the "spectral part" of the Selberg's formula.

To compare with Wikipedia, take function "h(u)" (in Wikipedia) to be equal to exp(t u).

3. by the Stationary phase approximation we can expect that the principal contribution comes from the minimums of the action - in this case these would be closed geodesics. (This corresponds to limit from quantum mechanics to classical mechanics)

Remark: Steps 1-3 are morally true for any Laplacian on any Riemannian manifold

4. By some kind of trick one must show for this particular case of Riemann surfaces with hyperbolic metric, THE ONLY contribution comes from extremums i.e. geodesics. (Such things happens in many other situations - e.g. Duistermaat Heckman formula, supersymmetric quantum mechanics etc...; however for the particular case of the Selberg formula I do not know the reference where this should be exposed in the manner which I would love, but may be I just did not search enough).

Concerning yours particular question about the term $ \int_{0}^{E}dptanh(\pi p) p $, unfortunately I cannot derive it, but just some comments.

This term corresponds to closed paths of zero length i.e. just points on the manifold (for each point you have trivial path which starts and immediately stops in this point). So contribution of such paths reduces to $\int_{manifold} something$.

When you consider the so-called super-symmetric quantum this "something" is exactly Todd class and you get Atiyah-Singer theorem for Dirac operator (or "d-bar" operator is almost the same).

But Selberg's formula is more subtle - it is NOT super-symmetric quantum mechanics, but usual quantum mechanics. So my feeling is that you can do something similar and write down some explicit integral over the Riemann surface (i.e. 2d integral) - than you can take explicitly integral in one of the variables and the rest of integral will give you desired term $ \int_{0}^{E}dptanh(\pi p) p $.

It might be that "tanh" in this formula has the same origin as "sinh" in Atiyah-Singer theorems - which physically have a clear explanation as a regularized determinant of the harmonic oscillator.

I guess that should be written somewhere, but I do not know the reference.

This is not really an answer, but somewhat long comment. Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects. Let me suggest some steps towards physical point of view (which are standard):

1. use Feynman-Kac formula: Tr (e^{tH} ) = path integral_{ over all closed paths} EXP( lenght of path)

2. Left hand side is obviosuly $sum_{eigenvalues} e^{t \lambda_i} - the "spectral part of the Selberg's formula (at least particalar case of it)

3. by the Stationary phase approximation we can expect that the principal contribution comes from the extremums of the action - in this case this would be geodesics. (This corresponds to limit from quantum mechanics to classical mechanics)

Remark: Steps 1-3 are morally true for any Laplacian on any Riemannian manifold

4. By some kind of trick one must show for this particular case of Riemann surfaces with hyperbolic metric, THE ONLY contribution comes from extremums i.e. geodesics. (Such things happens in many other situations - e.g. Duistermaat Heckman formula, supersymmetric quantum mechanics etc...; however for the particular case of the Selberg formula I do not know the reference where this should be exposed in the manner which I would love, but may be I just did not search enough).

This is not really an answer, but somewhat long comment. Selberg's trace formula is, of course, crying to be interpreted by means techniques coming from physics, however, personally I did not see some text which would satisfy me in all respects. Let me suggest some steps towards physical point of view (which are standard):

1. use Feynman-Kac formula: Tr (e^{tH} ) = path integral_{ over all closed paths} EXP( lenght of path)

2. Left hand side is obviosuly $sum_{eigenvalues} e^{t \lambda_i} - the "spectral part of the Selberg's formula (at least particalar case of it)

3. by the Stationary phase approximation we can expect that the principal contribution comes from the extremums of the action - in this case this would be geodesics. (This corresponds to limit from quantum mechanics to classical mechanics)

Remark: Steps 1-3 are morally true for any Laplacian on any Riemannian manifold

4. By some kind of trick one must show for this particular case of Riemann surfaces with hyperbolic metric, THE ONLY contribution comes from extremums i.e. geodesics. (Such things happens in many other situations - e.g. Duistermaat Heckman formula, supersymmetric quantum mechanics etc...; however for the particular case of the Selberg formula I do not know the reference where this should be exposed in the manner which I would love, but may be I just did not search enough).

Concerning yours particular question about the term $ \int_{0}^{E}dptanh(\pi p) p $, unfortunately I cannot derive it, but just some comments.

This term corresponds to closed paths of zero length i.e. just points on the manifold (for each point you have trivial path which starts and immediately stops in this point). So contribution of such paths reduces to $\int_{manifold} something$.

When you consider the so-called super-symmetric quantum this "something" is exactly Todd class and you get Atiyah-Singer theorem for Dirac operator (or "d-bar" operator is almost the same).

But Selberg's formula is more subtle - it is NOT super-symmetric quantum mechanics, but usual quantum mechanics. So my feeling is that you can do something similar and write down some explicit integral over the Riemann surface (i.e. 2d integral) - than you can take explicitly integral in one of the variables and the rest of integral will give you desired term $ \int_{0}^{E}dptanh(\pi p) p $.

I guess that should be written somewhere, but I do not know the reference.