On Minakshisundaram-Pleijel zeta functions of spheres
HTML articles powered by AMS MathViewer
- by E. Carletti and G. Monti Bragadin PDF
- Proc. Amer. Math. Soc. 122 (1994), 993-1001 Request permission
Abstract:
The aim of this paper is to show that the Minakshisundaram-Pleijel zeta function ${Z_k}(s)$ of k-dimensional sphere ${\mathbb {S}^k},k \geq 2$ (defined in $\Re e(s) > \frac {k}{2}$ by \[ {Z_k}(s) = \sum \limits _{n = 1}^\infty {\frac {{{P_k}(n)}}{{{{[n(n + k - 1)]}^s}}}} \] with $(k - 1)!{P_k}(n) = \mathcal {R}(n + 1,k - 2)(2n + k - 1)$ where the "rising factorial" $\mathcal {R}(x,n) = x(x + 1) \cdots (x + n - 1)$ is defined for real number x and n nonnegative integer) can be put in the form \[ (k - 1)!{Z_k}(s) = {\sum \limits _{l = 0}^\infty {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} ^{2l}}\left ( {\begin {array}{*{20}{c}} { - s} \\ l \\ \end {array} } \right )\sum \limits _{j = 0}^{k - 1} {{B_{k,}}_j\zeta (2s + 2l - j,\frac {{k + 1}}{2})} \] where ${B_{k,j}}$ are explicitly computed. The above formula allows us to find explicitly the residue of ${Z_k}(s)$ at the pole $s = \frac {k}{2} - n,n \in \mathbb {N}$, \[ \frac {1}{{(k - 1)!}}\sum \limits _{h = 0}^{\frac {k}{2} - 1} {{{\sum \limits _{\begin {array}{*{20}{c}} {l + h = n} \\ {l \geq 0} \\ \end {array} } {{{( - 1)}^l}\left ( {\frac {{k - 1}}{2}} \right )} }^{2l}}\left ( {\begin {array}{*{20}{c}} {n - \frac {k}{2}} \\ l \\ \end {array} } \right )} {B_{k,k - 2h - 1}}.\] In passing, we also obtain apparently new relations among the Stirling numbers.References
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- E. Carletti and G. Monti Bragadin, On Dirichlet series associated with polynomials, Proc. Amer. Math. Soc. 121 (1994), no. 1, 33–37. MR 1179586, DOI 10.1090/S0002-9939-1994-1179586-5
- Peter B. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, vol. 11, Publish or Perish, Inc., Wilmington, DE, 1984. MR 783634
- Charles Jordan, Calculus of finite differences, 3rd ed., Chelsea Publishing Co., New York, 1965. Introduction by Harry C. Carver. MR 0183987
- S. Minakshisundaram and Å. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242–256. MR 31145, DOI 10.4153/cjm-1949-021-5
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 993-1001
- MSC: Primary 58G26; Secondary 11M36
- DOI: https://doi.org/10.1090/S0002-9939-1994-1249872-9
- MathSciNet review: 1249872