Two zeta functions contained in the Poincaré series(ポアンカレ級数に含有される 2 種のゼータ関数)
Takumi Noda ∗† （野田 工 日本大学・工学部）
Department of Mathematics, College of Engineering, Nihon University,Kôriyama, Fukushima 963–8642, JapanAbstract
Two zeta-functions associated with the classical Poincaré series attached to modular group are introduced. Integral representations, transformation formulas andsome functional properties are given. As an application, we obtain two new proofsof the Fourier series expansion of the Poincaré series attached to SL(2, Z).
This manuscript is a summarized version of [No1] and the forthcoming paper [No2].

1 Exponential type generating functions
Let H = {z ∈ C | Im(z) > 0} be the complex upper half plane. Let Γ (s) be the Gamma
function, F(α ; γ ; z) be Kummer’s confluent hypergeometric function of the first kind (cf.
[Er1, 6.5.(1)]), Jν (z) be the Bessel function of the first kind, and Iν (z) and Kν (z) be modified Bessel functions (cf. [Er2, 7.2.1 (2), 7.2.2. (12), (13)]). Throughout this manuscript,∫ (0+)
ζ (s) and ζ (s, α ) denote the Riemann and the Hurwitz zeta-function respectively, −∞denotes integration over a Hankel contour, starting at negative infinity on the real axis,encircling the origin with a small radius in the positive direction, and returning to the∫ (0+)
starting point. We write ∞eiθ an integration taken along a rotated Hankel contour, starting at ∞ei(−2π +θ ) , encircling the origin in the positive direction, and returning to the point∞eiθ .
First, we introduce following Dirichlet series:∞exp (−2π iλ /(n + z)).
(n + z)sn=0ζexp .II (s; λ ; z) := ∑(1)∗ takumi@ge.ce.nihon-u.ac.jp† The author was supported by Grants-in-Aid for Scientific Research (No. 16K05078), Japan Society for
the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technologyof Japan.

Here, let z ∈ C \ R, Re(2π iλ /z) > 0 and 0 < Re(z) ≤ 1 in (1). Then the power series
expansion involving the Hurwitz zeta-function∞(−2π iλ )mζexp .II (s; λ ; z) = ∑ζ (s + m, z)m!
m=0(2)

holds for s ∈ C \ {1, 0, −1, −2, ...}, and we have
Theorem 1 ([No2]) Let z ∈ H, 0 < Re(z) ≤ 1 and λ > 0, and assume π < θ < 3π /2.
Then an integral representation

ζexp .II (s; λ ; z) = (2π iλ )1−s(2π iλ )−sΓ (s − 1) +πi∫ (0+) (s−1)/2 zu/(2π iλ )ue∞eiθ

1 − eu/(2π iλ )√Ks−1 (2 u)du(3)holds for s ∈ C \ {1, 0, −1, −2, . . . }, which provides a holomorphic continuation to thewhole s-plane except on s ∈ {1, 0, −1, −2, . . . }. Above integral representation gives thefunctional relation

ζexp .II (s; λ ; z) + (−1)s ζexp .II (s; −λ ; 1 − z)∞√
= 2π ie−π is ∑ (n/λ )(s−1)/2 e2π izn Is−1 (4π i nλ ),(4)n=1for s ∈ C \ {1, 0, −1, −2, . . . }.

2 J-Bessel zeta-function
Let θ > 0, ν ∈ C and s be a complex variable. Next, we define J-Bessel zeta-function oforder ν − 1 as follows:√∞Jν −1 (2 θ n).
(5)Jν −1 (s; θ ) := ∑ν +1
ns + 2n=1
By the estimates of the J-Bessel function, the Dirichlet series above converges absolutelyin the region Re(s) > 0 when Re(ν ) > 1/2, and converges absolutely in the region Re(s) >⌊3/2 − Re(ν )⌋/2 − 1 when Re(ν ) ≤ 1/2. For the specific case, ν is an integer, Jν −1 (s; θ )converges absolutely in the region Re(s) > (1 − ν )/2.
Theorem 2 ([No1], Theorem 1.1) Let ν ∈ C and θ > 0. The J-Bessel zeta-function has
an integral representation
ν +1∫

θ s+ 2 Γ (−s) (0+) us eθ uF(−s; ν ; −u−1 )du,Jν −1 (s; θ ) =θu
2π iΓ (ν )−∞ 1 − e(6)

which provides a meromorphic continuation to the whole s-plane. Further, the transformation formula()ν −1
θ 2 Γ (−s) ∞−θsJν −1 (s; θ ) =(7)∑ (2π in) F −s; ν ; 2π inΓ (ν ) n=−∞,n̸=0holds for Re(s) < −1, and the power series expansion
ν −1∞

θ 2Jν −1 (s; θ ) = ∑ζ (s + 1 − m)(−θ )m ,Γ(ν+m)
m!
m=0(8)

holds for s ∈ C \ {0, 1, 2, ...}. The J-Bessel zeta-function also satisfies the following recurrence formula:νJν −1 (s; θ ) + Jν +1 (s − 1; θ ) = √ Jν (s; θ ).
(9)θRemark 1. The power series expressions (2) and (8) are one of the generalizations ofRamanujan’s formula [Ra] (the binomial type power series):∞Γ (s + m)
ζ (s + m)(−x)m ,
Γ(s)
m!
m=0

ζ (s, 1 + x) = ∑(10)

for |x| < 1 and s ∈ C\{1}. An exponential type series was initially studied by Chowla andHawkins [CH], and Gauss’ hypergeometric type and Kummer’s confluent hypergeometric
type series were introduced by Katsurada [Kt]. For related results and generalizations ofRamanujan’s formula (10), refer to [SC].
Remark 2. More general zeta functions twisted by hypergeometric or Bessel functions
are treated by Kaczorowski and Perelli [KP4]. They derived meromorphic continuationsof these zeta-functions via the properties of the nonlinear twists obtained in [KP1]–[KP3].

3 Relation to the Poincaré series
Let m ∈ Z>0 and H = {z ∈ C | Im(z)( a b )> 0} be the complex upper half-plane. We denoteγ (z) = (az + b)/(cz + d) for γ = c d ∈ SL2 (Z), and use the notation e(z) = exp(2π iz).
Let k ≥ 4 be an integer, and define the m-th Poincaré series attached to SL2 (Z) of weight
k bye(mγ (z)).
(11)
Pmk (z) := (−1)k ∑k(cz+d){c,d}(∗ ∗)Here{( ∗ ∗ )the summation} is taken over γ = 2c d , a complete system of representation of∼0 ∗ ∈ SL2 (Z) \SL2 (Z) = {(c, d) ∈ Z | gcd(c, d) = 1, c > 0 or c = 0, d = 1}. The
functional relation of ζexp .II (s; λ ; z) or transformation formula of Jν −1 (s; θ ) induce thefollowing proposition which leads to the Fourier expansion of Pmk (z).

Proposition 1 Let µ > 0. The equality
2π (−1)

k2∞ (∑n=1nµ) k−1∞2√e (−µ /(z + n))
Jk−1 (4π µ n)e(nz) = (−1)k ∑(z + n)kn=−∞(12)holds for positive integer k.
Remark 3. It is easy to see the functional relation (4) is equivalent to (12). To show(12) via the transformation formula (7) is rather complicated (see [No1, Proposition 4.1]). As is well-known, the equality (12), from right side to left side, can be shown by using
Fourier transform. Our procedure in the proofs of Proposition 1 differs from these existing
methods.
By the definition of Pmk (z), we seee(mγ (z)).
k(cz+d)2(c, d) ∈ Z , c > 0Pmk (z) = (−1)k e(mz) + (−1)k∑(13)

gcd(c, d) = 1According to ordinary steps, we rearrange the d-sum into n-sum with finite d-sum modulo
c, and apply Proposition 1 with taking µ = m/c2 and replacing z by z + d/c. Thus, weachieve the following;
Theorem 3 (Fourier series expansion of the Poincaré series)∞ (n ) k−1
2mn=1()∞
4π √1∑ Kc(m, n)Jk−1 c mn e(nz).
c=1 cPmk (z) = (−1)k e(mz) + (−1) 2 2π ∑kHere, the Kloosterman sum is defined as follows:( ¯)
md + ndKc (m, n) =,∑ ecd mod c¯ ≡ 1 mod c).
(ddgcd(c, d) = 1

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[Er2] Ibid. Vol. II, McGraw-Hill, New York, 1953.

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