On integrals of Eisenstein series and derivatives of L-series

(1)

2004, No. 6

On Integrals of Eisenstein Series

and Derivatives of L-Series

Yifan Yang

1 Introduction

In[1], inspired by a formula of Ramanujan (see [3, page 207])

q1/9 (1 − q) 1 − q441 − q77· · · 1 − q22_{1 − q}55_{1 − q}88_{· · ·} = exp − C −1 9 1 q f(−t)9 f− t33 dt t , (1.1)

where f(−q) =∞_n₌₁(1 − qn_{) and C is a multiple of the value of certain Dirichlet L-series}

evaluated at 2, Ahlgren, Berndt, Yee, and Zaharescu established the following result that connects Eisenstein series, special values of Dirichlet L-series, and infinite products of certain form.(See [1] for a historical background of the above formula.)

Theorem 1.1(Ahlgren, Berndt, Yee, and Zaharescu). Suppose that α is real, k ≥ 2 is an

integer, and χ is a nontrivial Dirichlet character that satisfies χ(−1) = (−1)k_{. Then}_{, for}

0 < q < 1, qα ∞ n=1 1 − qnχ(n)n k−2 = exp − C − ₁ q α − ∞ n=1 d|n χ(d)dk−1tn dt t , (1.2) where C = L(2 − k, χ). (1.3) Received 14 August 2003.

at National Chiao Tung University Library on April 30, 2014

http://imrn.oxfordjournals.org/

(2)

This result is surprising in the sense that it connects three apparently disparate mathematical objects, and it is natural to ask whether there are similar identities for functions other than the Dirichlet characters. In this paper, we will show that this is indeed the case and thatTheorem 1.1is in fact a simple corollary of a general result. Theorem 1.2. Let a(n) be an arithmetic function such that

(i) a(n) nλ_{for some real number λ,}

(ii) the Dirichlet series A(s) =∞_n₌₁a(n)n−s_{can be analytically continued to the}

half-plane{s : Re s ≥ −} and satisfies |A(s)| e(π/2−)| Im s|_{in the region}

for some positive numbers and , (iii) A(0) = 0. Then, for 0 < q < 1, qα ∞ n=1 1 − qna(n)= exp − C − ₁ q α − ∞ n=1 d|n a(d)dtn dt t , (1.4) where C = A(0). (1.5) If we set a(n) = χ(n)nk−2_{, where χ(n) is a nonprincipal character modulo N with}

χ(−1) = (−1)k_{, then the function a(n) clearly satisfies the conditions in}_{Theorem 1.2}_(see

[2]). Therefore, identity (1.2) follows.

Our line of approach is diﬀerent from that in [1], in which the main ingredient is the representation L(2 − k, χ) = lim q→ 1− ∞ n=1 d|n χ(d)dk−1q n n (1.6)

for integers k ≥ 2 and nonprincipal Dirichlet character χ with χ(−1) = (−1)k_{, and}

Theorem 1.1follows immediately from this assertion. To prove(1.6), they started out by writing the sum on the right-hand side as a Riemann sum for some integral in a clever way. Thus, evaluating the limit of the sum is equivalent to evaluating a certain integral, which is done by contour integration and the residue theorem.

Here, we provide a simpler and more natural approach to this problem. Assume that χ is a character modulo N. We first observe that the integrand on the right-hand side of(1.2), with a suitable choice of α, is the Fourier expansion of the Eisenstein series with the character χ of weight k associated with the cusp ∞ on Γ0(N). Thus, it is very

(3)

natural to express the integrand as a Mellin integral. In particular, if we write t as e−u_,

then the Mellin transform of the integrand with respect to the variable u contains the L-series L(s, χ). This explains how special values of L-L-series come into the identity. To prove Theorem 1.1(and, implicitly, (1.6)), we only need to evaluate the integral of the Mellin integral in a straightforward manner. In fact, upon closer scrutiny, we can see that our argument actually works for any functions a(n) satisfying the conditions inTheorem 1.2. Finally, we remark that one may wonder what will happen if we replace the in-tegrand on the right-hand side of(1.2) by Eisenstein series associated with cusps other than∞. In that case, we can still obtain an expression for the right-hand side, but it is not as elegant as that on the left-hand side of(1.2).

2 Proof ofTheorem 1.2

Let a(n) be an arithmetic function satisfying the assumptions inTheorem 1.2and let F(q) be defined by F(q) = α − ∞ n=1 d|n a(d)dqn. (2.1)

By the well-known formula

e−z= 1 2πi

c+i∞ c−i∞ Γ (s)z

−s_ds, _(2.2)

which holds for all complex numbers z with Re z > 0 and all real numbers c > 0, we have, for all u > 0, Fe−u= α − ∞ d=1 a(d)d ∞ n=1 e−ndu = α − 1 2πi ∞ d=1 a(d)d ∞ n=1 _λ₊₃₊_i∞ λ+3−i∞ Γ (s)(ndu)−sds = α − 1 2πi _λ₊₃₊_i∞

λ+3−i∞ Γ (s)ζ(s)A(s − 1)u

−s_ds,

(2.3)

where A(s) denotes the Dirichlet series∞_n₌₁a(n)n−s. Let c be a positive number less than 1, and denote log(q−1_{) and log(c}−1_{) by x and δ, respectively. The last expression}

(4)

yields c qF(t) dt t = x δF e−udu = α(x − δ) − 1 2πi _λ₊₃₊_i∞ λ+3−i∞ Γ (s)ζ(s)A(s − 1) x1−s_{− δ}1−s 1 − s ds. (2.4)

Noting that Γ (s) =(s − 1)Γ(s − 1) and changing the variable from s to s + 1, we see that _c q F(t)dt t = α(x − δ) + 1 2πi _λ₊₂₊_i∞ λ+2−i∞ Γ (s)ζ(s + 1)A(s)x−s− δ−sds. (2.5) We now consider the integrals involving x and δ separately.

For the integral involving x, we observe that

ζ(s + 1)A(s) = ∞ d=1 ∞ n=1 a(d) n (nd) −s_. _(2.6)

Therefore, by (2.2) again, we have

1 2πi _λ₊₂₊_i∞ λ+2−i∞ Γ (s)ζ(s + 1)A(s)x−sds = ∞ d=1 a(d) ∞ n=1 1 ne −n dx = − ∞ d=1 a(d) log1 − qd. (2.7)

For the integral involving δ we use the residue theorem. We move the line of inte-gration to the vertical line Re s = −. This is justified by assumption (ii) and the upper-bound|Γ(σ + it)| |t|σ−1/2_e−π|t|/2_{. Each of the functions Γ (s) and ζ(s) has a simple pole}

at s = 0. By assumption (iii), the function A(s) has a zero at s = 0. The Taylor expansions at 0 of these three functions are given by

Γ (s) = 1

s+· · · , ζ(s + 1) = 1

s +· · · , A(s) = sA

_{(0) + · · · .} _(2.8)

Denoting A(0) by C, we thus have 1 2πi _λ+2+i∞ λ+2−i∞ Γ (s)ζ(s + 1)A(s)δ−sds = C + 1 2πi −+i∞ −−i∞ Γ (s)ζ(s + 1)A(s)δ−sds = C + Oδ. (2.9)

(5)

Combining(2.4), (2.7), and (2.9), and letting δ → 0, we hence obtain ₁ q F(t)dt t = −α log q − C − ∞ d=1 a(d) log1 − qd, (2.10)

which is equivalent to(1.4). This completes the proof ofTheorem 1.2.

Acknowledgment

The author would like to thank Professors Berndt and Hildebrand from the University of Illinois for reading the earlier versions of the manuscript and providing valuable comments.

References

[1] S. Ahlgren, B. C. Berndt, A. J. Yee, and A. Zaharescu, Integrals of Eisenstein series and derivatives

of L-functions, Int. Math. Res. Not. 2002 (2002), no. 32, 1723–1738.

[2] H. Davenport_{, Multiplicative Number Theory, 2nd ed., Graduate Texts in Mathematics, vol. 74,}

Springer-Verlag, New York, 1980, revised by Hugh L. Montgomery.

[3] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer-Verlag, Berlin, 1988,

with an introduction by George E. Andrews.

Yifan Yang: Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30005,

Taiwan

E-mail address:yfyang@math.nctu.edu.tw