**History of the Formula:** In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of spinor/tensor fields on the compact Riemann surface $M$ (for simplicity in this question I'm assuming $n\geq0$). Their result reads:
$$\det(\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-c_n\mathcal{X}(M)}.$$

Where $\mathcal{Z}_{n-[n]}(s)$ are two Selberg zeta functions, and $c_n$ is a constant which is explicitly computed in the article above.

In their exposition there are for sure some trivial mistakes. One comes from an error in the reference paper "Fourier coefficients of the resolvent for a Fuchsian group" (1977) by Fay, and it has been considered in the review paper "Geometry of String Perturbation Theory" (1988) of D'Hoker and Phong. But it doesn't lead to any change in the above considered formula.

Another trivial mistake is a misuse of the volume formula for an hyperbolic Riemann surface, this has been first pointed out by Bolte and Steiner, in their (unpublished) paper: "Determinants of Laplace-like Operators on Riemann Surfaces" (1988). In this article the authors use a new way to compute the same quantity $\det(\Delta_n^+)$, but they arrive to a different result. Calling $\frac{1}{2}\Delta_n^+$ what D'Hoker and Phong called $\Delta_n^+$ (according to Bolte and Steiner), it reads:
$$\det(\frac{1}{2}\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-k_n\mathcal{X}(M)}.$$

Where the constant $k_n$ has been explicitly computed, and it turns out to be different from $c_n$.

Unfortunately the mistakes pointed out above are not enough to account for the difference between $c_n$ and $k_n$. Bolte and Steiner suspect that the error in D'Hoker and Phong could come from a missing factor $2$ in the definition of $\Delta_n^+$.

The end of the story seems to be the article "Notes on determinants of Laplace-type operators on Riemann surfaces" (1990) of Oshima. In this article the author recalls the correction of some mistakes in the original paper of D'Hoker and Phong, moreover he points out a conceptual mistake made by Bolte and Steiner. Finally, simply assembling previous results, he provides the following formula:
$$\det(\frac{1}{2}\Delta_n^+)=\mathcal{Z}_{n-[n]}(n+1)\cdot e^{-l_n\mathcal{X}(M)}.$$

Again $l_n$ is given explicitly, and it turns out to be different from both $c_n$ and $k_n$.

**Question:** Is it really the end of the story? It seems to me that the whole situations is quite messy, for example no one seems to point out new specific mistakes in the original article of D'Hoker and Phong (there must be some, according to Oshima). In specific, do the experts agree on the validity of the formula provided by Oshima? Are there other successive articles on the same topic?

**Note**: I added the tags "Arithmetic Geometry" and "Analytic Number Theory" because the laplacians $\Delta_n^+$ are conjugated to the Maass Laplacians $D_n$ acting on automorphic forms of weight $n$ on $M$. So I suspect an answer could come from people in arithmetic as well.

Thank you very much for reading all this!

This post imported from StackExchange MathOverflow at 2014-07-30 08:58 (UCT), posted by SE-user Giovanni De Gaetano