数理解析研究所講究録
第 1659 巻 2009 年 106-115

106

Some asymptotic expansionsof the Eisenstein series
Takumi Noda * 野田
College of Engineering Nihon University

1

工

日本大学

工学部

Contents
1. Definition of the Eisenstein series for2. y-aspect of the Eisenstein series
3. Definition of the Airy function
4. Main Theorem and Corollaries
5. t-aspect of the Eisenstein series
6. Jutila’s formula
7. Other asymptotic expansions
8. Outline of the proof of the main theorem$SL_{Q}(\mathbb{Z})$2 Definition of the Eisenstein series
Let $i=\sqrt{-1},$ $s=\sigma+it\in \mathbb{C}$ and $H$ be the upper half plane. The nonholomorphic Eisenstein series forwith weight is$0$$SL_{2}(\mathbb{Z})$$E(z,s)=Y \sum_{\{c,d\}}|cz+d|^{-Zs}$.

(1)

Here $z=x+iy\in H$ , and the summation is taken over
, a complete system ofrepresentation of. The Fourier expansion is as follows:$(\begin{array}{l}**cd\end{array})$$\{(_{0*}^{**})\in SL_{2}(\mathbb{Z})\}\backslash SL_{2}(\mathbb{Z})$$\zeta(2s)E(z,s)$$=\zeta(2s)\gamma+\sqrt{\pi}\zeta(2s-1)arrow^{1}\Gamma sy^{1-s}\Gamma(s-)$$+ \frac{4}{\Gamma}\pi S’\sqrt{y}\sum_{n=1}^{\infty}n^{s^{1}}-z\sigma_{1-b}(n)K_{s_{2}^{1}}(2\pi ny)\cos(2\pi n\kappa)$

,

(2)

where
is the modified Bessel function andis the sum of s-th powersof positive divisors of . We call the first two terms of (2) are the constant term of$K_{v}(\tau)$$\sigma_{s}(n)$$n$$E(z,s)$.

’Supported by Grant-in-Aid for Scientific Research (C).

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3

$y\cdot aspect$of the Eisenstein series

It is well-known that the constant term represents the -aspect of $E(z,s)$ as
Because the Bessel function in (2) decays exponentially. Therefore there
exist positive constants
anddepending only on such that (except on the$yarrow\infty$.

$y$$A_{1}$$A_{2}$$s$poles)$|E(z,s)|\leqq A_{1}y^{{\rm Re}(s)}+A_{2}y^{1-{\rm Re}(s)}$$(yarrow\infty)$.

(3)

The invariance of $E(z,s)$ under the action ofgives the asymptotic behavior$yarrow0$
when. For every $y>0$ , except on the poles,$SL_{2}(\mathbb{Z})$(4)$|E(z,s)|\leqq\{\begin{array}{ll}A_{1}(y^{-{\rm Re}(s)}+y^{{\rm Re}(s)}) ({\rm Re}(s)>12)A_{2}(y^{-1+{\rm Re}(s)}+y^{1-{\rm Re}(s)}) ({\rm Re}(s)\leqq 1z).\end{array}$

The t-aspect of $E(z,s)$ is not simple. The non-constant terms in (2) are notnegligible when
Empirically, the behavior of $E(z,s)$ respect to ${\rm Im}(s)=t$ is similar to the be,havior of$tarrow\infty$.

$\zeta(s)^{2}$$E(z,s)$$\Leftarrow?\Rightarrow$$\zeta(s)^{2}$.

Problem.
Investigate the asymptotic behavior $E(z,s)$ with respect to ${\rm Im}(s)=t$.

4 Defimition of the Airy Function
The modified Bessel function$K_{v}(\tau)(v, \tau\in \mathbb{C})$is defined by the integral$K_{v}( \tau)=\frac{1}{2}\int_{0}^{\infty}u^{v-1}\exp(-\frac{1}{2}\tau(u+\frac{1}{u}))du$,

which satisfies the modified Bessel equation:$\frac{d^{2}w}{d\tau^{2}}+\frac{1}{\tau}\frac{dw}{d\tau}-(1+\frac{v^{2}}{\tau^{2}})w=0$

For$\tau\in \mathbb{R}$.

, the Airy function is defined by$Ai(\tau)=\frac{1}{\pi}\int_{0}^{\infty}\cos(\frac{1}{3}u^{3}+\tau u)du=\frac{1}{\sqrt{3}\pi}\tau^{z}K_{1}13(\frac{2}{3}\tau^{3}2)$,

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which satisfies the differential equation$\frac{d^{2}w}{d\tau^{2}}=\tau w$

The representation of Ai

Ai$(\tau)$for.

$\tau\in \mathbb{C}(|\arg\tau|<\pi)$is.

$( \tau)=\frac{\exp(-2^{3}3\tau z)}{2\pi}\int_{0}^{\infty}\exp(-\tau^{1}zu)\cos(\frac{1}{3}u^{3}2)u^{-z}du1$

In which ffactional powers take their principal values.

The modified Bessel function
decays exponentially for large . The
asymptotic expansion of
for the cases
or $\tau-t=o(\tau^{\iota/3})$ are obtainable by using saddle-point method.
However, in the transitional regions, namely
is nearly equal to 1 while$|\tau-t|$ is large, the investigation becomes much more involved.
As an another approach, the theoiy of asymptotic solutions of differential equations are employed,
where the Airy function plays a fundamental role (cf. [4], 7.4.3, [14]).
$K_{it}(\tau)$$\tau$$\tau/t\oint 1$$K_{it}(\tau)$$\tau/t$

5

Main Theorem and Corollaries

Theorem 1 Let $z=x+iy\in H$ and $t>289.Assume$ thatfor any positive$t-(4\logt)^{21}3fJ\leqq
2\pi yN<t$$N\geqq
0$
constant , andis any integer satisfying. Define$y<t^{1}s^{-\delta}$$\delta$$\frac{2}{3}\tau_{n}=t\log\frac{t+\{t^{2}-(2\pi y)^{2}\}^{1}\tau}{2\pi y}-\{t^{2}-(2\pi ny)^{2}\}^{1}z$Then for every$E(z_{2}^{1}+it)$$\epsilon>0$

.

,

Ai

$= \frac{4\sqrt{2}\pi^{11}2^{+i\iota_{\mathcal{Y}}}2}{\zeta(1+2it)}\sum_{n=1}^{N}n^{-it}\sigma_{2it}(n)\{t^{2}-(2\pi ny)^{2}\}^{-\iota};\cos(2\pi n\kappa)\tau_{n}^{i}$$(-\tau_{n}^{2}3)$$+y^{1}y^{1}e^{i\theta}+O(31\#+\epsilon^{1}3^{+\epsilon}$

Here

.

. The implied O-constant depends at most$e^{i\theta}=\pi^{-2\dot{u}}\zeta(2it)\Gamma(it)/\overline{\zeta(2it)\Gamma(it)}$on and .
$\epsilon$$\delta$

Corollary 1 Suppose $t-t^{11}z+r\leqq 2\pi yM\leqq t$ . For every$E(z_{2}^{1}+it)$$\epsilon>0$,$= \frac{4\sqrt{2}\pi^{it}y\}}{\zeta(1+2\dot{u})}\sum_{n=1}^{M}n^{-\dot{u}}\infty_{it}(n)\{t^{2}-(2\pi ny)^{2}\}^{-}l\cos(2\pi n\kappa)\cos(\tau_{n}-)1$$+y^{\iota\iota}z^{+\dot{u}}+yz^{-it}e^{i\theta}+O(y^{-}3\not\supset tz(\log t)^{1}+ylz^{+\epsilon}\log(t/\mathcal{Y}))$.

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Corollary 2 Let $z=x+iy\in H$. Assume that $c0<y<fJ^{-c_{1}}1$ for some positive
constants
and . Then for every,$c_{0}$$\epsilon>0$$c_{1}$as$E(z_{2}^{1}+it)=O(y^{-}21t^{z+\epsilon}1)$$tarrow\infty$.

Remark. Corollary 2 is a convexity bound for the Eisenstein series which includes the y-factor.

6$t$.aspect of the Eisenstein series

Known fact 1. A convexity bound is known (see [16] (p. 258)), which is aconsequence of the Phragm\’en-Lindel\"of convexity principle;$E(z_{z+it)=O_{y}(t^{z+\epsilon})}^{1^{1}}$.

Known fact 2. The spectral theoiy of automorphic foms gives the following
estimate:$\sum_{0<t_{j}<T}|u_{j}(z)|^{2}+\frac{1}{2\pi}|E(z_{2}^{1}\acute{0^{T}},+it)|^{2}dt=O(T^{2}+Ty)$

is an orthonormal system of cusp forms for. (See for example [6], (13.1).)
A convexity bound for the Riemann zeta-function isHere$\{u_{j}\}$$SL_{2}(\mathbb{Z})$

.

with

$\Delta u_{j}=(_{F}^{1}+$$t_{j}^{2})u_{j}$$\zeta(2^{+it)=O(t^{F^{+\epsilon}})}1^{1}$.

Further, the sub-convexity bound, the classical result for the Riemann zeta-function due to Hardy-Littlewood is$\zeta(\iota^{\iota_{+\epsilon}}z+it)=O(t^{6})$.

The mean value theorem of the Riemann zeta-function on the critical line is$\int_{0}^{T}|\zeta(z1+it)|^{2}dt=T\log T+$(error).

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From the standpoint of the similarity between $E(z,s)$ andfollowing$c_{0i}\dot{u}ecture$$\epsilon$.

Assume $y\geqq c0$ for a positive constant$c_{0}$$\zeta(s)^{2}$. For any positive,$E(z_{2}^{1}+it)=O(t^{\epsilon}+y^{2})1$

as, we set up the$tarrow\infty$.

Here the y-factor comes ffom the constant term.

7 Jutila’s formula
Theorem (Jutila 1984) Let $t\geqq 12\pi,$and$\delta$

be a positive number,$t^{\delta}\leqq N\leqq t/12\pi$,$N’=t/2\pi+N/2-(N^{2}/4+Nt/2\pi)z1$ .

Define$f(t,n)=2t$ arcsinh $\sqrt{\pi n/2t}+(\pi^{2}n^{2}+2\pi nt)^{1}z+\pi/4$.

Then,$|\zeta(z1+it)|^{2}$$=2^{\iota 1}2 \sum_{n=1}^{N}(-1)^{n}d(n)n^{-}z(z1^{1}+\frac{t}{2\pi n})^{-r}\cos(f(t,n))$$+2 \sum_{n=1}^{N^{l}}d(n)n^{-z}cos1(t\log(t/2\pi n)-r_{7}^{\pi}-)+O(N^{11}rt^{-}\tau(\log t)^{2}+\log t)$

Remark 1. Jutila’s formula is a differentiated version of Atkinson’s formula. In
the proofs of these formulas, Voronoi’s summation formula and the saddle-point
method are used as the main instruments.
Remark 2. There are some differences between Theorem 1 and the formulas on
the square of the Riemann zeta-function. Atkinson type formulas usually have
two summations, whereas Theorem 1 and Corollary 1 consist of one summation. This differenc is explained by each approximate functional equations;$\sum_{1\leqq n\leqq N}$$e$$\zeta^{2}(s)=\sum_{n=1}^{N}d(n)n^{-s}+\pi^{2s-1}\frac{\Gamma^{2}(-)}{\Gamma^{2}(S2)}\sum_{n=1}^{N’}d(n)n^{s-1}+O(N^{1}z^{-\sigma}\log t)$,$NN’=(t/2\pi)^{2},$ $N\geqq 1,$ $N’\geqq 1$ .
where
For the case of $E(z,s)$ , the Fourier expansion (2) itself may be regarded as one
self dual (approximate) functional equation except the constant term. Originally
Voronoi’s summation formula consists of one summation.
$0\leqq\sigma\leqq 1,$

.

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8

Other asymptotic expansions
Define the holomorphic Eisenstein series foras$SL_{Q}(\mathbb{Z})$$E_{s}( z)=\sum_{(m,n)\in \mathbb{Z}^{2}\backslash (0,0)}(m+nz)^{-s}$.

Theorem 2 (K. Matsumoto $f9]$) Assume $0<|\arg(z)|<\pi$ and ${\rm Re}(s)>-N+1$
for any positive integer $N$, then for$|z|\geq 1$,$E_{s}(z)=(1+e^{\pi is})\zeta(s)+O(|z|^{-{\rm Re}(s)-N})$.

For$|z|\leq 1$$E_{s}(z)$,$=(1+z^{-s})(1+e^{\pi is}) \zeta(s)+(e^{\pi is}-e^{-\pi is})\frac{\zeta(s-1)}{s-1}z^{-1}-(1+\frac{e^{\pi is}+e^{arrow\pi is}}{2})\zeta(s)$$+ \sum_{1\leq k\leq N-1,k:odd}(e^{\pi is}-e^{-\pi is})(\begin{array}{l}-sk\end{array})\zeta(s+k)\zeta(-k)z^{k}+O(|z|^{N})$Define the non-holomorphic Eisenstein series of weight attached to$k$$E_{k}(z,s)= \frac{1}{2}\sum_{c,d=-\infty}^{\infty}.(cz+d)^{-k}|cz+d|^{-2s}$.

$SL_{2}(\mathbb{Z})$

,

as
(5)

$(c,d)=1$

and define Ramanujan’s$\Phi$-function$\Phi_{s_{1},s_{2}}(e(z))=\sum_{l_{1},l_{2}=1}^{\infty}l_{1}^{s_{1}}l_{2}^{s_{2}}e(l_{1}l_{2}z)=\sum_{l=1}^{\infty}\sigma_{s_{1}-s_{2}}(l)l^{s_{2}}e(lz)$.

Theorem 3 (M. Katsurada [8]) For any $z\in H$ and any integer $N\geq 0$ the following fomula holds $in-N<{\rm Re}(s)<1+N$ except at $s=1$ .
$E_{0}(z,s)=1+ \frac{\sqrt{\pi}\Gamma(s-1/2)\zeta(2s-1)}{\Gamma(s)\zeta(2s)}y^{1-2s}+\frac{(2\pi)^{2s}}{\Gamma(s)\zeta(2s)}\{S_{N}(s;z)+R_{N}(s;z)\}$Here$S_{N}(s;z)= \sum_{n=0}^{N-1}\frac{(-1)^{n}(s)_{n}(1-s)_{n}}{n!}\Phi_{s-n-1,-s-n}^{*}(e(z))(4\pi y)^{-s-n}$

with$\Phi_{s_{1^{S}2}}^{*},(e(z))=\Phi_{s_{1},s_{2}}(e(z))+\Phi_{s_{1},s_{2}}(\overline{e(z)})$.

The remainder $temR_{N}(s;z)$ is estimated as$R_{N}(s;z)=O\{(|t|+1)^{2N}e^{-2\pi y}y^{-{\rm Re}(s)-N}\}$

.

.

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Remark. Theorem 3 yields various known results on $E_{0}(z,s)$ , including its functional properties and its asymptotic aspects as $zarrow 0$ . Especially the MellinBarnes integral transformation shows that the functional equation of $E_{0}(z,s)$ reduces eventually into the simple property$\Phi_{s_{1},s_{2}}(e(z))=\Phi_{s_{2},s_{1}}(e(z))$.

Theorem 4 (M. Katsurada, T Noda (to appear))$E_{k}(z,s)$$=1+(-1)^{k/2}2 \pi\frac{\Gamma(b+k-1)}{\Gamma(s)\Gamma(s+k)}\frac{\zeta(2+k-1)}{\zeta(2s+k)}(2y)^{1-2s-k}$$+ \frac{(-1)^{k/2}(2\pi)^{2s+k}}{\zeta(1s+k)\Gamma(s+k)}\{S_{N+k/2}(s,2s+k;z)+R_{N+k/2}(s,2s+k;z)\}$$+ \frac{(-1)^{k/2}(2\pi)^{Z+k}}{\zeta(2s+k)\Gamma(s)}\{S_{N-k/2}(s+k,2s+k;-\overline{z})+R_{N-k/2}(s+k,2s+k;-\overline{z})\}$

holds in the $region-N-k/2<{\rm Re}(s)<N-k/2+1$ except at the complex zeros of$\zeta(2s+k)$ and at the real poles
of $E_{k}(s;z)$ . Here the remainder tems are estimated$as$$R_{N+k/2}(s;2s+k;z)=O\{(|t|+1)^{2N+k}e^{-2\pi y}y^{-\sigma-N-k/2}\}$

and.

$R_{N-k/2}(s+k;2s+k;-\overline{z})=O\{(|t|+1)^{2N-k}e^{-2\pi y}y^{-\sigma-N-k/2}\}$

Remark 1. The asymptotic expansion of $E_{k}(s;z)$ established by transferringfrom the derived asymptotic expansion of $E_{0}(s;z)$ (Theorem 3) to that of $E_{k}(s;z)$through successive use of Maass’ weight change operators.
Remark 2. Theorem 4 also gives a new altemative proof of the Fourier expansion of $E_{k}(z,s)$ , consequently gives new proofs of various results on $E_{k}(z,s)$ , forexample, functional equation, special values, the Kronecker limit formula, theeigenfunction equation for the non-Euclidean Laplacian and so on.

9 Outline of the proof of the main theorem
Balogh [3] gave one uniform asymptotic expansion of the modified Bessel
function by using Airy functions. Balogh’s result is based on Olver’s works. The
following proposition (Olver [14], Chap.11, p. 425) is the uniform asymptotic
expansion of the modified Bessel function of imaginary order, which is crucial inthis report.
$m\geqq 0$ and
Proposition 1 ([3], Olver $f14]$ p.425) For
with$t\in \mathbb{R}_{>0},$$u\in \mathbb{C}$$|\arg(u)|<\pi$,$K_{it}(tu)=\urcorner^{-\exp}t^{3}\pi(-E^{t)}\pi(A)^{1};\{$

Ai$(-t^{2}s \xi)\sum_{k=0}^{m}A\Delta\varphi t$$+t^{-} zAi’(-t^{2}4s\xi)\sum_{k=0}^{m-1_{B_{t}}}\ovalbox{\tt\small REJECT}+\epsilon_{2m+1}(t,\xi)\}$

.

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.

(with oscillation) as $\tauarrow-\infty$

decays rapidly as
and decays slowly. More precisely, we have following

Proposition 2 (I) For

with $|\arg\tau|<\pi$, we have

The Airy function Ai$(\tau)(\tau\in \mathbb{R})$$\tau\in \mathbb{C}$$\tauarrow\infty$$Ai(\tau)=\frac{\exp(-52_{T^{2)}}^{3}}{2\pi^{11}z\tau\tau}\sum_{l=0}^{n-1}a_{l}(-2^{3}3T^{2})^{-l}+(error)$, we have(II) For

Ai.

$\tau\in \mathbb{R}_{>0}$$(- \tau)=\frac{1}{\pi^{1}\tau^{1}}\sum_{l=0}^{n-1}a_{l}\cos(2^{3}1\pi z^{\tau^{z}-}\tau^{\pi-}\tau^{l)}(_{3}^{2_{T^{2}}^{3}})^{-l}+(error)$.

In (I) and (II), fractional powers of take their principal values.
$\tau$On the critical line

$s= \frac{1}{2}+it$, we divide the summation of (2) into five seg-ments:$\zeta(1+2it)E(z_{2}^{1}+it)=S_{0}(z,t)+S_{1}(z,t)+S_{2}(z,t)+S_{3}(z,t)+S_{\infty}(z,t)$.

Here$S_{0}(z,t)= \zeta(1+2it)_{\mathcal{Y}^{2}}^{\iota_{+it}}+\sqrt{\pi}\zeta(2it)\frac{\Gamma(it)}{r(1z+it)}y^{1}z^{-it}$.

For $j=1,2,3$ ,,$S_{j}(Z,t)z+\dot{u}1$and$s_{\infty}(Z,t) z+it1+it)^{-1}\sqrt{y}\sum_{n=N_{3}}^{\infty}n^{it}\sigma_{-2\dot{u}}(n)K_{it}(2\pi ny)\cos(2\pi nx)$.

Applying the estimations Proposition 1, Proposition 2 and$\zeta(1+it)^{-1}=O((\log t)^{2}3($log log$t)^{1}3)$$(t\geqq 2)$

to $S_{j}(z,t)$ , we obtain the the proof of Theorem 1.

References[1] F. V. Atkinson, The mean-value of the Riemann zeta-fiunction, Acta Math.

81, 1949, 353-376.
[2] C. B. Balogh, Unifonn asymptotic expansions of the modiffid Besselfunctionof the third kind of large imaginary order, Bull.Amer. Math. Soc., 72, 1966,
40-43.

114[3]

C. B. Balogh, Asymptotic expansions of the modified Besselfunction of thethird kind of imaginary order, SIAM-J.-Appl.-Math. 15, 1967, 1315-1323.

[4] A. Erd\’elyi, et al. Higher Transcendental Functions, Vol.

$\Pi$

, McGraw-Hill,

New York, 1953.
[5] A. Ivi6, The Riemann zeta-function, A Wiley-Intersince Publication, 1985.
[6] H. Iwaniec, Spectral Methods of Automorphic Forms, 2nd ed., Graduate

Studies in Mathematics vol. 53, A.M.S., Revista Matem\’atica Iberoamericana, 1995.
[7] M. Jutila, Transfomation fomulae for Dirichlet polynomials, J. Number

Theory, 18, 1984, 135-156.
[8] M. Katsurada,

Complete asymptotic expansions associated with Epsteinzeta-frnctions, Ramanujan J. (2007), 14, 249-275[9] K. Matsumoto, Asymptotic expansions of double zeta-fmctions of Bames, of

Shintani, and Eisenstein series, Nagoya Math. J. 172, 2003, 59-102[10] F. W. J. Olver, The asymptotic solutions of linear differential equations of

the second order for large values of a parameter, Philos. Trans. Roy. Soc.
London, Ser. A, 247, 1954, 307-327.
[11] F. W. J. Olver, The asymptotic expansions ofBesselfunctions of large order,

Ibid., 247, 1954, 328-368.
[12] F. W. J. Olver, Error bounds for first approximations in tuming-point prob-lems, SIAM-J.-Appl.-Math. 11, 1963, 748-772.
[13] F. W. J. Olver, Error bounds for asymptotic expansions in tuming-point

problems, Ibid., 12, 1964, 2m-2l4.
[14] F. W. J. Olver, Asymptotics and special functions, A K Peters, 1997. (Aca-demic Press, 1974.)[15] P. Shiu, A Bmn-Ttchmarsh theorem for multiplicative functions, J. Reine

Angew. Math., 313, 1980, 161-170.
[16] A. Terras, Hamonic analysis on symmetric spaces and applications I,

Springer-Verlag 1985.
[17] E. C. Titchmarsh and D. R. Heath-Brown, The Theory of the Riemann zeta-

fimction (2nd edition), Oxford science publications, 1986.

115
[18]

G. F. Voronoi, Sur une fonction transcendante et ses applications \‘a la som]e Normale, 21, (3), 1904, 207-268,
mation de quelques s\’eries, Ann.
$\acute{E}co$459-534.