THE LAPLACE TRANSFORM OF THE
FOURTH MOMENT OF THE ZETA-FUNCTION
Aleksandar Ivić
Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 11(2000), 41 − 48.
Abstract. The Laplace transform of |ζ( 12 +ix)|4 is investigated, for which a precise
expression is obtained, valid in a certain region in the complex plane. The method
of proof is based on complex integration and spectral theory of the non-Euclidean
Laplacian.
1. INTRODUCTION
Laplace transforms play an important rôle in analytic number theory. Of special
interest in the theory of the Riemann zeta-function ζ(s) are the Laplace transforms
Z
(1.1)
Lk (s) :=
0
∞
|ζ( 21 + ix)|2k e−sx dx
(k ∈ N, <e s > 0).
E.C. Titchmarsh’s well-known monograph [20, Chapter 7] gives a discussion of
Lk (s) when s = σ is real and σ → 0+, especially detailed in the cases k = 1 and
k = 2. Indeed, a classical result of H. Kober [14] says that, as σ → 0+,
N
(1.2)
γ − log(4πσ) X
L1 (2σ) =
+
cn σ n + O(σ N +1 )
2 sin σ
n=0
for any given integer N ≥ 1, where the cn ’s are effectively computable constants
and γ = 0.577 . . . is Euler’s constant. For complex values of s the function L1 (s)
1991 Mathematics Subject Classification. 11M06, 11F72, 11F66.
Key words and phrases. Riemann zeta-function, Laplace transform, complex integration,
spectral theory.
Typeset by AMS-TEX
2
Aleksandar Ivić
was studied by F.V. Atkinson [1], and more recently by M. Jutila [13], who noted
that Atkinson’s argument gives
(1.3)
∞
X
1
π
− 12 is
is
2
d(n) exp(−2πine−is ) + λ1 (s)
L1 (s) = −ie (log(2π) − γ + ( − s)i) + 2πe
2
n=1
in the strip 0 < <e s < π, where the function λ1 (s) is holomorphic in the strip
|<e s| < π. Moreover, in any strip |<e s| ≤ θ with 0 < θ < π, we have
λ1 (s) ¿θ (|s| + 1)−1 .
In [12] M. Jutila gave a discussion on the application of Laplace transforms to the
evaluation of sums of coefficients of certain Dirichlet series.
F.V. Atkinson [2] obtained the asymptotic formula
(1.4)
L2 (σ) =
1
1
1
1
1
(A log4 + B log3 + C log2 + D log + E) + λ2 (σ),
σ
σ
σ
σ
σ
where σ → 0+,
A=
1
, B = π −2 (2 log(2π) − 6γ + 24ζ 0 (2)π −2 )
2
2π
and
µ ¶ 13
+ε
1 14
λ2 (σ) ¿ε
.
σ
(1.5)
He also indicated how, by the use of estimates for Kloosterman sums, one can
8
improve the exponent 13
14 in (1.5) to 9 . This is of historical interest, since it is
one of the first instances of application of Kloosterman sums to analytic number
theory. Atkinson in fact showed that (σ = <e s > 0)
(1.6)
L2 (s) = 4πe
− 12 s
∞
X
√
1
d4 (n)K0 (4πi ne− 2 s ) + φ(s),
n=1
where d4 (n) is the divisor function generated by ζ 4 (s), K0 is the Bessel function,
and the series in (1.6) as well as φ(s) are both analytic in the region |s| < π. When
s = σ → 0+ one can use the asymptotic formula
K0 (z) =
1
2
√
¡
¢
πz −1/2 e−z 1 − 8z −1 + O(|z|−2 )
(| arg z| < θ <
3π
, |z| ≥ 1)
2
Laplace transform of the fourth moment of
|ζ( 12 + ix)|
3
and then, by delicate analysis, one can deduce (1.4)–(1.5) from (1.6).
The author [5] gave explicit, albeit complicated expressions for the remaining
coefficients C, D and E in (1.4). More importantly, he applied a result on the
fourth moment of |ζ( 12 + it)|, obtained jointly with Y. Motohashi [9], [11] (see also
[4]), to establish that
(1.7)
λ2 (σ) ¿ σ −1/2
(σ → 0+),
and this is where the matter presently rests.
For k ≥ 3 not much is known about Lk (s), even when s = σ → 0+. This is not
surprising, since not much is known about upper bounds for
Z
T
Ik (T ) :=
0
|ζ( 21 + it)|2k dt
(k ≥ 3, k ∈ N).
For a discsussion on Ik (T ) the reader is referred to the author’s monographs [3]
and [4]. One trivially has
Z
(1.8)
Ik (T ) ≤ e
0
∞
1
|ζ( 12 + it)|2k e−t/T dt = eLk ( ).
T
Thus any nontrivial bound of the form
(1.9)
µ ¶ck +ε
1
Lk (σ) ¿ε
σ
(σ → 0+, ck ≥ 1)
gives, in view of (1.8) (σ = 1/T ), the bound
(1.10)
Ik (T ) ¿ε T ck +ε .
Conversely, if (1.10) holds, then we obtain (1.9) from the identity
1
1
Lk ( ) =
T
T
Z
∞
Ik (t)e−t/T dt,
0
which is easily established by integration by parts.
4
Aleksandar Ivić
2. SPECTRAL THEORY AND THE
LAPLACE TRANSFORM OF |ζ( 12 + ix)|4
The purpose of this paper is to consider L2 (s), where s is a complex variable,
and to prove a result analogous to (1.3), valid in a certain region in C. We shall not
use Atkinson’s method and try to elaborate on (1.6). Our main tools are powerful
methods from spectral theory, by which recently much advance has been made in
connection with I2 (T ). For a competent and extensive account of spectral theory
the reader is referred to Y. Motohashi’s monograph [19]. Some of the relevant
papers on I2 (T ) are [6]–[11], [14]–[18] and [21].
We begin by stating briefly the necessary notation involving the spectral theory
of the non-Euclidean Laplacian. As usual {λj = κ2j + 14 } ∪ {0} will denote the discrete spectrum of the non-Euclidean Laplacian acting on SL(2, Z) –automorphic
forms, and αj = |ρj (1)|2 (cosh πκj )−1 , where ρj (1) is the first Fourier coefficient of
the Maass wave form corresponding to the eigenvalue λj to which the Hecke series
Hj (s) is attached. We note that
X
(2.1)
αj Hj3 ( 12 ) ¿ K 2 logC K
(C > 0).
κj ≤K
Our result is the following
THEOREM. Let 0 ≤ φ <
have
π
2
be given. Then for 0 < |s| ≤ 1 and | arg s| ≤ φ we
1
1
1
1
1
L2 (s) = (A log4 + B log3 + C log2 + D log + E) + G2 (s)
s
s
s
s
s

∞
(2.2)

X
¡
¢
1
,
+ s− 2
αj Hj3 ( 12 ) s−iκj R(κj )Γ( 12 + iκj ) + siκj R(−κj )Γ( 21 − iκj )


j=1
where
(2.3)
r ³
3
π −iy Γ( 14 − 2i y) ´
Γ(2iy) cosh(πy)
R(y) :=
2
2
Γ( 14 + 2i y)
and in the above region G2 (s) is a regular function satisfying (C > 0 is a suitable
constant)
(2.4)
½
¾
C log(|s|−1 + 20)
−1/2
G2 (s) ¿ |s|
exp −
.
(log log(|s|−1 + 20))2/3 (log log log(|s|−1 + 20))1/3
Laplace transform of the fourth moment of
|ζ( 12 + ix)|
5
Remark 1. The constants A, B, C, D, E in (2.2) are the same ones as in (1.4).
Remark 2. From Stirling’s formula for the gamma-function it follows that
−1/2
R(κj ) ¿ κj
. In view of (2.1) this means that the series in (2.2) is absolutely
convergent and uniformly bounded in s when s = σ is real. Therefore, when
s = σ → 0+, (2.2) gives a refinement of (1.7).
Remark 3. From (1.4) and (1.7) it transpires that λ(σ) is an error term when
0 < σ < 1. For this reason we considered the values 0 < |s| ≤ 1 in (2.2), although
one could treat the case |s| > 1 as well.
Remark 4. From (2.2) and elementary properties of the Laplace transform
one can easily obtain the Laplace transform of
Z
T
E2 (T ) :=
0
|ζ( 12 + it)|4 dt − T P4 (log T ),
P4 (x) =
4
X
aj x j ,
j=0
where a4 = 1/(2π 2 ) (for the evaluation of the remaining aj ’s, see [5]).
3. PROOF OF THE THEOREM
Note first that in the integral defining L2 (s) it suffices to consider only the range
of integration [1, ∞ ] , since the range [0, 1] trivially contributes ¿ 1. We start
from the well-known integral
Z
1
−z
(3.1)
e =
Γ(s)z −s ds
(<e z > 0, c > 0),
2πi (c)
and the function
Z
∞
Z2 (w) :=
1
|ζ( 12 + it)|4 t−w dt
(<e w > 1),
introduced and studied by Y. Motohashi [17], [19]. Here as usual
Z
Z
σ+iT
= lim
(σ)
T →∞
.
σ−iT
Y. Motohashi showed that Z2 (s) has meromorphic continuation over C. In the
half-plane σ = <e s > 0 it has the q
following singularities: the pole s = 1 of order 5,
simple poles at s =
1
2
± iκj (κj =
λj − 14 ) and poles at s = 12 ρ, where ρ denotes
complex zeros of ζ(s). The residue of Z2 (s) at s =
1
2
+ iκh equals
r ³
3
X
π −iκh Γ( 14 − 2i κh ) ´
αj Hj3 ( 12 ),
R0 (κh ) =
Γ(2iκ
)
cosh(πκ
)
2
h
h
2
Γ( 14 + 2i κh )
κ =κ
j
h
6
Aleksandar Ivić
and the residue at s = 21 − iκh equals R0 (κh ).
From (3.1) we have, for c > 1, 0 < |s| ≤ 1 and | arg s| ≤ φ,
Z ∞
|ζ( 12 + ix)|4 e−sx dx
1
Ã
!
Z ∞
Z
1
=
|ζ( 12 + ix)|4
Γ(w)(sx)−w dw dx
(3.2)
2πi
1
(c)
Z
1
Γ(w)s−w Z2 (w) dw.
=
2πi (c)
In (3.2) we shift the line of integration to the contour L (w = u + iv) consisting
of the curves
(3.3)
u=
1
2
− C log−2/3 |v|(log log |v|)−1/3
(|v| ≥ v0 > 0, C > 0)
and the segment
u = u0 , u0 =
1
2
− C log−2/3 |v0 |(log log |v0 |)−1/3 , |v| ≤ v0 .
Namely ζ(s) 6= 0 (see e.g., [3, Chapter 6]) for
σ ≥ 1 − A(log t)−2/3 (log log t)−1/3
(s = σ + it, t ≥ t0 > 0, A > 0)
and a suitable constant A. The function Z2 (w) will be regular on L, since the
poles w = 12 ρ, ζ(ρ) = 0) lie to the left of L. For any given η > 0 one has
(3.4)
Z2 (w) ¿ eη|=m w|
(w ∈ L)
if C in (3.3) is taken sufficiently small. This follows easily from the proof of
Lemma 1 of [7], which shows that the order of Z2 (w) is of the order given by (3.4)
if <e w > 0 and w stays away from the poles of Z2 (w). In the forthcoming work
[8] it is even shown that in the above region Z2 (w) is of polynomial growth in |v|,
which is more than what is required for our present purpose. Thus by the residue
theorem we obtain from (3.2)
(3.5)
Z ∞
Z
X
1
4 −sx
−w
1
|ζ( 2 + ix)| e
dx =
Res Γ(w)s Z2 (w) +
Γ(w)s−w Z2 (w) dw,
2πi
1
L
and the last integral is regular for 0 < |s| ≤ 1 and | arg s| ≤ φ. There are residues
at w = 1 and at w = 12 ± iκj . The contribution from w = 1 is (see (1.4))
1
1
1
1
1
(A log4 + B log3 + C log2 + D log + E),
s
s
s
s
s
7
Laplace transform of the fourth moment of |ζ( 12 + ix)|
while the residues at w =
s−1/2

∞
X

j=1
1
2
± iκj yield


¡
¢
3 1
−iκj
iκj
1
1
αj Hj ( 2 ) s
R(κj )Γ( 2 + iκj ) + s R(−κj )Γ( 2 − iκj )
,

where R(y) is defined by (2.3).
Write
Z
Γ(w)s−w Z2 (w) dw = I1 + I2 ,
L
say, where in I1 we have |v| ≤ V , and in I2 we have |v| > V , where V (À 1) is a
parameter to be chosen. Since
1
|s−w | = |s|−u e|v arg s|
π
(| arg s| ≤ φ), Γ(w) ¿ |v|u− 2 e− 2 |v|
(v À 1),
then setting δ(x) = C log−2/3 x(log log x)1/3 we obtain
Z
I1 ¿ |s|
−u0
V
+
1
π
1
|s|−( 2 −δ(v)) e−( 2 −φ)v dv ¿ |s|−u0 + |s|−( 2 −δ(V )) .
v0
Similarly we have
Z
I2 ¿ |s|
− 12
π
∞
e
−( π
2 −φ)v
− 12
dv = |s|
V
e−( 2 −φ)V
.
( π2 − φ)
Finally choosing
V = C1 log(|s|−1 + 20)
with suitable C1 > 0 we obtain (C2 > 0)
Z
Γ(w)s−w Z2 (w) dw
¾
½
C2 log(|s|−1 + 20)
−1/2
,
¿ |s|
exp −
(log log(|s|−1 + 20))2/3 (log log log(|s|−1 + 20))1/3
L
which in view of (3.5) completes the proof of the Theorem.
8
Aleksandar Ivić
References
[1] F.V. Atkinson, The mean value of the zeta-function on the critical line, Quart. J. Math.
Oxford 10(1939), 122-128.
[2] F.V. Atkinson, The mean value of the zeta-function on the critical line, Proc. London Math.
Soc. 47(1941), 174-200.
[3] A. Ivić, The Riemann zeta-function, John Wiley and Sons, New York, 1985.
[4] A. Ivić, Mean values of the Riemann zeta-function, LN’s 82, Tata Institute of Fundamental
Research, Bombay, 1991 (distr. by Springer Verlag, Berlin etc.).
[5] A. Ivić, On the fourth moment of the Riemann zeta-function, Publs. Inst. Math. (Belgrade)
57(71)(1995), 101-110.
[6] A. Ivić, The Mellin transform and the Riemann zeta-function, in “Proceedings of the Conference on Elementary and Analytic Number Theory (Vienna, July 18-20, 1996)”, Universität
Wien & Universität für Bodenkultur, Eds. W.G. Nowak and J. Schoißengeier, Vienna 1996,
112-127.
[7] A. Ivić, On the error term for the fourth moment of the Riemann zeta-function, J. London
Math. Soc. 60(2)(1999), 21-32.
[] A. Ivić, The Laplace transform of the fourth moment of the zeta-function, Univ. Beograd.
Publ. Elektrotehn. Fak. Ser. Mat. 11(2000), 41-48.
[8] A. Ivić, M. Jutila and Y. Motohashi, The Mellin transform of power moments of the zetafunction, Acta Arithmetica 95(2000), 305-342.
[9] A. Ivić and Y. Motohashi, A note on the mean value of the zeta and L-functions VII, Proc.
Japan Acad. Ser. A 66(1990), 150-152.
[10] A. Ivić and Y. Motohashi, The mean square of the error term for the fourth moment of the
zeta-function, Proc. London Math. Soc. (3)66(1994), 309-329.
[11] A. Ivić and Y. Motohashi, The fourth moment of the Riemann zeta-function, J. Number
Theory 51(1995), 16-45.
[12] M. Jutila, Mean values of Dirichlet series via Laplace transforms, in “Analytic Number Theory” (ed. Y. Motohashi), London Math. Soc. LNS 247, Cambridge University Press, Cambridge, 1997, 169-207.
[13] M. Jutila, Atkinson’s formula revisited, in “Voronoi’s Impact on Modern Science”, Book 1,
Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 1998, 139-154.
[14] H. Kober, Eine Mittelwertformel der Riemannschen Zetafunktion, Compositio Math. 3(1936),
174-189.
[15] Y. Motohashi, The fourth power mean of the Riemann zeta-function, in “Proceedings of the
Amalfi Conference on Analytic Number Theory 1989”, eds. E. Bombieri et al., Università di
Salerno, Salerno, 1992, 325-344.
[16] Y. Motohashi, An explicit formula for the fourth power mean of the Riemann zeta-function,
Acta Math. 170(1993), 181-220.
[17] Y. Motohashi, A relation between the Riemann zeta-function and the hyperbolic Laplacian,
Annali Scuola Norm. Sup. Pisa, Cl. Sci. IV ser. 22(1995), 299-313.
[18] Y. Motohashi, The Riemann zeta-function and the non-Euclidean Laplacian, Sugaku Expositions, AMS 8(1995), 59-87.
[19] Y. Motohashi, Spectral theory of the Riemann zeta-function, Cambridge University Press,
Cambridge, 1997.
[20] E.C. Titchmarsh, The theory of the Riemann zeta-function (2nd ed.), Clarendon Press, Oxford, 1986.
[21] N.I. Zavorotnyi, On the fourth moment of the Riemann zeta-function (in Russian), in “Automorphic functions and number theory I”, Coll. Sci. Works, Vladivostok, 1989, 69-125.
Laplace transform of the fourth moment of
Aleksandar Ivić
Katedra Matematike RGF-a
- ušina 7
Universitet u Beogradu, D
11000 Beograd, Serbia (Yugoslavia)
[email protected], [email protected]
|ζ( 12 + ix)|
9