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3425.Flajolet P. et al. - Mellin transforms and asymptotics- harmonic sums (2004).pdf

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Corrigenda to
“Mellin Transforms and Asymptotics: Harmonic Sums”,
by P. Flajolet,. X. Gourdon, and P. Dumas,
Theoretical Computer Science 144 (1995), pp. 3–58.
P. 11, Figure 1; P. 12, first diplay. [ca 2000, due to Julien Clément.]
The Mellin transform of f (1/x) is f ? (−s) [this corrects the third entry of
Fig. 1, P. 11],
f ? (s)
f (x)
f (1/x) f ? (−s)
Also [this corrects the first display on P. 12],
1
M f
; s = f ? (−s).
x
(The sign in the original is wrong.)
P. 20, statement of Theorem 4 and Proof. [2004-12-16, due to Manavendra
Nath Mahato] Replace the three occurrences of (log x)k by (log x)k−1 . (Figure 4
stands as it is.) Globally, the right residue calculation, in accordance with the rest
of the paper, is
(−1)k−1 −ξ
x−s
x (log x)k−1 .
=
Res
k
(s − ξ)
(k − 1)!
P. 48, Example 19. [2004-10-17, due to Brigitte Vallée]
Define
∞
X
X
r(k)
1
ρ(s) =
=
.
s
k
(m2 + n2 )s
m,n≥1
k=1
P∞
−m2 x2
Let Θ(x) =
. The Mellin transform of Θ(x1/2 ) is ζ(2s)Γ(s), and
m=1 e
accordingly [this corrects Eq. (61) of the original; the last display on P. 29 on
which this is based is correct]
√
π 1
1
1/2
√ − + R(x),
Θ(x ) =
2
x 2
where R(x) is exponentially small. By squaring,
√
π
π 1
1
1/2
√ + + R2 (x),
Θ(x ) =
−
4x
2
x 4
with again R2 (x) exponentially small. On the other hand, the Mellin transform of
Θ(x1/2 )2 is [this corrects the display before Eq. (61)]
M(Θ(x1/2 )2 , s) = ρ(s)Γ(s).
(Equivalently, the transform of Θ(x)2 is 21 ρ(s/2)Γ(s/2).) Comparing the singular
expansion of M(Θ(x1/2 )2 , s) induced by the asymptotic form of Θ(x1/2 )2 as x → 0
to the exact form ρ(s)Γ(s) of the Mellin transform shows that ρ(s) is meromorphic
1
2
in the whole of C, with simple poles at s = 1, 12 only and singular expansion [this
corrects the last display of page 48]
1 1
1
π
+ −
+
+ [0]s=−1 + [0]s=−2 + · · · .
ρ(s) 4(s − 1) s=1
2 s − 12 s=1/2
4 s=0
(Due to a confusion in notations, the expansion computed in the paper was relative
to ρ(2s) rather than to ρ(s); furthermore, a pole has been erroneously introduced
at s = 0 in the last display of P. 48 In fact ρ(0) = 14 , as stated above.)
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