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EIGENVALUES ASYMPTOTICS 13
Here Res(s0) denotes the residue of ZA(s) at s0 and C (m,q) is a constant.
Then we have
d1 = 2 1−d/2 1 Γ(m/(2q))Γ((pm − qn)/(2q))
,
q(1 + q) Γ(m/2)Γ(pm/(2q))
1−d/2 1 Γ(m/(2q))
d2 = 2
pq Γ(m/2)Γ(n/2) ,
d3 = 2 1−d/2
1 Γ(m/(2q))ψ((m + mq + nq)/(2q))
Γ(n/2)
pqΓ(m/2)
(1 + q)Γ((m + mq)/(2q))
+ C (m,q)
−
p
γΓ(m/(2q))
q(1 + q)Γ(m/2)
+ Γ(m/(2q))
m + mq
ψ
−
pqΓ(m/2) 2q
p
1 + q ψ
n
m + mq + nq
− ψ
2
2q
−n/2 (1 + q)Γ((m + mq)/(2q))Γ(n(1 + p + q)/(2p)) n(1 + q)
d4 = 2 ZA
pΓ(n/2)Γ((m + mq + nq)/(2q)
2p
where ψ(z) = Γ ′ (z)/Γ(z) is the digamma function and γ is the Euler constant.
In the particular case where q = m = 1, we have s0 = 1, Res(s0) = 2 −1/2
and C (1,1) = 2 −1/2 (γ + log 2). Moreover, for Res > 1, we see that ZA(s) =
(2 s/2 − 2 −s/2 )R(s) where R(s) is the Riemann zeta function. Thus we have
−(1+n)/2 Γ((p − n)/2)
d1 = 2 ,
Γ(p/2)
d2 = 2 (1−n)/2 1
pΓ(n/2) ,
d3 = 2 (1−n)/2
1 2 − p + 2
Γ(n/2)
3/2
γ −
2p
p
2 ψ
n
+
2
21/2
log 2 ,
p
1−n/2 Γ(n(2 + p)/(2p))
d4 = 2
pΓ(n/2)Γ(1 + n/2) (2n/(2p) − 2 −n/(2p)
n
)R .
p
For a more precise argument, see [3].
References
[1] J. Aramaki, On an extension of the Ikehara Tauberian theorem, Pacific J. Math.,
133(1) (1988), 13-30.
[2] , Complex powers of vector valued operators and their applications to asymptotic
behavior of eigenvalues, J. Funct. Anal., 87(2) (1989), 294-320.
[3] , An extension of the Ikehara Tauberian theorem and its application, Acta
Math. Hungarica, 71(4) (1996), 297-326.
[4] K. Itô and H.P. Mckean, Jr., Diffusion Processes and Their Sample Paths, Springer,
Berlin, Heidelberg, New York, 1974.