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{{harvs|txt|author-link=Stephen Hawking|last=Hawking|first=Stephen|year=1977}} showed that in flat space, in which the eigenvalues of Laplacians are known, the [[zeta function (operator)|zeta function]] corresponding to the [[partition function (quantum field theory)|partition function]] can be computed explicitly. Consider a scalar field ''φ'' contained in a large box of volume ''V'' in flat spacetime at the temperature ''T'' = ''β''<sup>−1</sup>. The partition function is defined by a [[path integral formulation|path integral]] over all fields ''φ'' on the Euclidean space obtained by putting ''τ'' = ''it'' which are zero on the walls of the box and which are periodic in ''τ'' with period ''β''. In this situation from the partition function he computes energy, entropy and pressure of the radiation of the field ''φ''. In case of flat spaces the eigenvalues appearing in the physical quantities are generally known, while in case of curved space they are not known: in this case asymptotic methods are needed.
Another method defines the possibly divergent infinite product ''a''<sub>1</sub>''a''<sub>2</sub>.... to be exp(−ζ′<sub>''A''</sub>(0)). {{
|author2-link=Isadore Singer}} used this to define the [[determinant]] of a positive [[self-adjoint operator]] ''A'' (the [[Laplacian operator|Laplacian]] of a [[Riemannian manifold]] in their application) with [[eigenvalue]]s ''a''<sub>1</sub>, ''a''<sub>2</sub>, ...., and in this case the zeta function is formally the trace of ''A''<sup>−''s''</sup>. {{ |author2-link=Åke Pleijel}} showed that if ''A'' is the Laplacian of a compact Riemannian manifold then the [[Minakshisundaram–Pleijel zeta function]] converges and has an analytic continuation as a meromorphic function to all complex numbers, and {{ }} extended this to [[Elliptic differential operator|elliptic pseudo-differential operator]]s ''A'' on compact Riemannian manifolds. So for such operators one can define the determinant using zeta function regularization. See "[[analytic torsion]]." {{harvtxt|Hawking|1977}} suggested using this idea to evaluate path integrals in curved spacetimes. He studied zeta function regularization in order to calculate the partition functions for thermal graviton and matter's quanta in curved background such as on the horizon of black holes and on de Sitter background using the relation by the inverse [[Mellin transform]]ation to the trace of the kernel of [[heat equation]]s.
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* {{note|Hard16}}G.H. Hardy and J.E. Littlewood, "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes", ''Acta Mathematica'', '''41'''(1916) pp. 119–196. ''(See, for example, theorem 2.12)''
* {{Citation | last1=Hawking | first1=S. W. | author1-link=Stephen Hawking | title=Zeta function regularization of path integrals in curved spacetime | doi=10.1007/BF01626516 | mr=0524257 | date=1977 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=55 | issue=2 | pages=133–148|bibcode = 1977CMaPh..55..133H | s2cid=121650064 | url=http://projecteuclid.org/euclid.cmp/1103900982 }}
* {{note|Mo97}} V. Moretti,
* {{Citation | last1=Minakshisundaram | first1=S. | last2=Pleijel | first2=Å. | title=Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds | doi=10.4153/CJM-1949-021-5 | doi-access=free | mr=0031145 | date=1949 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=1 | issue=3 | pages=242–256}}
* {{Citation | last1=Ray | first1=D. B. | last2=Singer | first2=I. M. | title=''R''-torsion and the Laplacian on Riemannian manifolds | doi=10.1016/0001-8708(71)90045-4 | mr=0295381 | date=1971 | journal=[[Advances in Mathematics]] | volume=7 | issue=2 | pages=145–210| doi-access=free }}
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