Determinants of the Laplacians on complex projective spaces \(\mathbf{P}^n(\mathbb{C})(n\geq 1)\) (Q1753819)

The purpose of this well-written article is to determine explicitly the determinants of the Laplacians on the complex projective space by studying the asymptotic expansion of their heat coefficients. These asymptotic expansions for the corresponding Dirichlet series are given by the quite general so-called Minakshisundaram-Pleijel formula. All important formulas distinguish between even and odd values of \(n\), for the heat coefficients \(a_k^n\). First, the spectral zeta function of \(M\), i.e., the Mellin transform of the heat trace, is introduced. The heat kernel on the complex projective \(n\)-space admits a spectral representation in terms of the spherical function, which can be expressed in terms of the normalised Jacobi polynomials. The first main theorem expresses the heat coefficients \(a_k^n\) in terms of Bernoulli numbers. The proof uses the fact that the asymptotic expansion the first theta function can be expressed as a sum with coefficients Bernoulli numbers. As examples, the heat coefficients for the special cases \(n = 1,2, 3, 4, 5\) are explicitly computed. Then, by using the Hurwitz zeta function, the heat coefficients are described by the residues of the spectral zeta functions. Finally, the determinant of the Laplacian is expressed via the spectral zeta function. These two formulas are quite complicated powers of products of exponentials etc. The last proof seems quite simple and uses the digamma function and Bernoulli polynomials. As before, various special cases of \(n\) are considered.