A dominated convergence theorem for Eisenstein series

Based on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp τ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau = 0$$\end{document}. As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.


Introduction
In this paper we prove a dominated convergence theorem for Eisenstein series. Roughly speaking, it states an upper bound for specific partial sums of Eisenstein series at purely imaginary arguments near the cusp τ = 0. This can be applied to questions involving Lfunctions assigned to products of Eisenstein series, since we obtain better control of the corresponding Mellin integral. One of the main ingredients we use is an alternative elementary approach to modular forms [6]. It relies on a class of very simple functions which we will call weak functions. A weak function ω is a 1-periodic meromorphic function in the entire plane, which has the following properties: B Johann Franke johnnyfranke@web.de 1 Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany (i) All poles of ω are simple and lie in Q.
(ii) The function ω tends to 0 rapidly as the absolute value of the imaginary part increases, so for all M > 0 as |y| → ∞.
By Liouville's theorem one quickly sees that each weak ω is essentially just a rational function R ∈ C(X ) with (only simple) poles only in roots of unity, such that R(0) = R(∞) = 0.
Here we put ω(z) := R(e(z)), where e(z) := e 2πiz . One defines W N to be the space of weak functions with the property, that ω(z/N ) only has poles in Z. We associate to ω a periodic divisor function β ω (x) := −2πires z=x ω(z). Now one can show the following construction theorem for modular forms for the congruence subgroup Theorem 0.1 (cf. Theorem 4.7 in [6]) Let k ≥ 3 and N 1 , N 2 > 1 be integers. There is a homomorphism In the case that k = 1 and k = 2 the map stays well-defined under the restriction that the function z → z k−1 η(z)ω(zτ ) is removable in z = 0.
The main tools for the proof are Weil's converse theorem and the following observation.
In this paper, we continue the study of this new perspective to modular forms and apply it to Dirichlet series. We first want to investigate the space ϑ k (W N 1 ⊗ W N 2 ) and it will turn out, that it is generated by Eisenstein series.
The cases k = 1 and k = 2 can be treated similarly. We want to apply the series representations of ϑ k in terms of rational functions to Dirichlet series. To every modular form f (τ ) = n≥0 a(n)q n/N of weight k for some congruence subgroup ⊂ SL 2 (Z) we can associate an L-function L( f , s) given by In the case that f is a cuspidal Hecke eigenform its L-function is entire, has an Euler product expansion and encodes deep arithmetic information.
We give a proof for a dominated convergence theorem for Eisenstein series arising from rational functions. In order to formulate it, we need the concept of the height of a N -periodic function β. The height of such a β is defined to be the largest integer d such that for all integers α ≥ 0: T → ∞.
Theorem 0.4 (cf. Theorem 2.14) Let ω ⊗ η ∈ W N 1 ⊗ W N 2 be a pair of weak functions such that ω is removable in z = 0 and κ N 2 β η has height d. Then for all α ∈ N 0 there is a constant C β,ω,α > 0 such that uniformly for all T ∈ N and y ∈ [0, 1] An application of this theorem is a new, in some sense more natural, representation of L-functions associated to products of Eisenstein series in terms of a generalized Dirichlet series. Modifying the sum a bit leads to convergence in a much wider region. In particular, we have the following theorem.
Note that this representation of the L-function of the considered product is more natural since it is a direct generalization of the formula in the case l = 1, where the series directly splits into a product of two Dirichlet L-functions. An important question, which is still unsolved in the very general case, is that which modular forms can be written as sums of products of Eisenstein series. But there is a lot of progress in this field. Dickson and Neururer have shown in [5], that, if k ≥ 4, N = p a q b N where p a , q b are powers of primes and N is square free, the space M k ( 0 (N )) is generated by E k ( 0 (N ), χ 0,N ) and a subspace containing products of two Eisenstein series. A similar result for M k ( 0 ( p)) and k ≥ 4, where p is prime, is due to Imamoḡlu and Kohnen [7] (for p = 2) and Kohnen and Martin [8] for p > 2. Recently, Raum and Xia proved in [10] that essentially all modular forms of weight 2 can be represented by products of Eisenstein series. For a correspondence between values of L-functions for products of pairs of different Eisenstein series see [3].
The paper is organized as follows. In the first section we identify generators for the space of modular forms that arise from rational functions. In the second section we prove a Dominated convergence theorem for Eisenstein series, which provides an upper bound for several partial sums of the series involving weak functions for modular forms near the cusp τ = 0. In the last section we apply this theorem to L-functions associated to products of Eisenstein series.
Notation. We use the introduced notations W N for the vector space of weak functions of level d|N , and W ± N for the odd and even function part. As usually, N is the set of positive integers and N 0 = N ∪ {0}. Throughout is a positive integer. We briefly define k = (k 1 , . . . , k ) ∈ N to be a vector of positive integers. We write |k| = k 1 + · · · + k .
For any set L we define L C 0 to be the space of all functions f : L → C, that are zero everywhere (except finitely many x ∈ L). The subspace Especially when going over to Fourier series it will be useful to identify We will identify functions f ∈ F C 0 N with N -periodic functions f : For any Dirichlet character ψ modulo N we define the Gauss sum G(ψ) := N −1 n=0 ψ(n)e 2πin/N . For the generalized Gauss sum it will be more convenient to use the more general notion of a discrete Fourier transform Note that we have an inverse transformation We use the same notation for functions f ∈ F C 0 1 N and have κ N F N f = F N κ N f . For d|N we also use the trivial injection where the left hand side is d-periodic and can be seen as a function of F d .
For the complex variable z = x + iy we write e(z) := e 2πiz and for the complex variable τ we define q := e 2πiτ . We also use the notation ζ M := e 2πi M for roots of unity. We denote by C N the group of all characters modulo N . Also we write C N for the set of all characters modulo d, where d divides N . We write χ 0,d for the principal character modulo d. In particular, χ 0,1 denotes the trivial character.

The space of weak modular forms
For Dirichlet characters χ and ψ modulo positive integers M and N , respectively, and some integer k ≥ 3 one defines the corresponding Eisenstein series for τ ∈ H (= upper half plane) via This series converges absolutely and uniformly on compact subsets of the upper half plane and defines a holomorphic function in that region. One can show that (1.1) leads to a non-zero function if and only if χ(−1)ψ(−1) = (−1) k and that the E k are modular forms of weight k for the congruence subgroups with Nebentypus character χψ of 0 (M, N ). The cases k = 1, 2 are treated differently, see also [9] on p. 274 ff. or [4]. Every Eisenstein series admits a Fourier series. The coefficients are well-known and given by where as usual q := e 2πiτ and L(ψ, s) is the Dirichlet L-function. Note that in the case that ψ is primitive one has (F N ψ)(a) = ψ(a)(F N ψ) (1) and one obtains the simpler expression d|n d k−1 ψ(d)χ (n/d) for the coefficients up to a constant. It is clear that every ϑ k (ω ⊗ η; τ ) admits a Fourier expansion. Since we only focus on the non-trivial cases we assume ω ⊗ η ∈ (W M ⊗ W N ) ± if (−1) k = ±1. It is given by According to (1.2) we conclude for non-principal characters (1.4) In particular, if χ and ψ are primitive and hence conjugate up to a constant under the Fourier transform, this simplifies to Already here the connection between Eisenstein series and weak functions is intuitively clear.
In this section we want to find generators for the space ϑ k (W N 1 ⊗ W N 2 ). We call their elements weak modular forms. In other words, the vector space V k ( 1 (N 1 , N 2 )) of all weak modular forms is the image of the linear map It is an easy exercise to verify that the discrete Fourier transform defines an isomorphism The next theorem provides generators for the space V k ( 1 (N 1 , N 2 )).
where χ and ψ run over all non-trivial characters modulo d 1 |N 1 and Proof It is clear that the Fourier transform preserves the subspaces of odd and even functions. Hence, for characters χ(−1)ψ(−1) = (−1) k , we have the Fourier expansion This proves the theorem.
For our investigations we are especially interested in a subspace of V k which we will denote by U k and which contains all weak modular forms which arise from weak functions that are removable in z = 0. In the following we shall give generators for U k . Let H N i ⊂ W N i be the subspace of weak functions that are removable in z = 0. Then we have In other words, the space H N is given by weak elements ω(z) such that β ω (0) = 0. On the periodic function side, we define the subspace of these coefficients by (F 1 is generated by the elements E k (χ, ψ; N 1 d 2 N 2 d 1 τ ) and the linear combinations where 1 < d i < N i and χ, ψ are non-principal characters modulo d 1 and d 2 , respectively, such that sgn(χψ) = (−1) k .
Proof Since all considered weak functions are removable in z = 0, we can apply the theorem to all positive weights k = 1, 2, . . .. The proof works similarly to the one of Theorem 1.1 and we omit it. Theorem 1. 3 We have the following.
(i) The space of weak modular forms of weight k = 1 is given by In particular, it is generated by the elements given in Theorem 1.2 for k = 1. (ii) The space of weak modular forms of weight k = 2 is given by In particular, it is generated by the elements in Theorem 1.2 for k = 2 and E 2 ( , where χ and ψ are non-principal characters modulo d 1 |N 1 and d 2 |N 2 , respectively.
In the last section we would like to investigate L-functions of products of weak functions. To formalize this, we give the following final definition.
to be the vector space of all modular forms that can be written as a sum j c j f 1, j · · · f , j , where each f r , j is an element of V k r ( 1 (N 1 , N 2 )). Analogously, we define the subspace N 2 )). We will call the modular forms in U k ( 1 (N 1 , N 2 )) higher weak modular forms.
investigate finite sums of the form on the upper half plane in detail, where α ≥ 0 is an integer, β is some N -periodic function (N ∈ N >1 ) and ω(z) is some weak function of level M with a removable singularity in z = 0. By Theorems 1.1, 1.2 and 1.3, expression (2.1) will converge to a linear combination of Eisenstein series as T tends to infinity, if β = β η comes from a weak function. The purpose of the Dominated convergence theorem is now to give a condition providing a non-trivial upper bound for the sum (2.1). In general, there will be no non-trivial "small" upper bound of (2.1) in terms of T , τ and α. However, when replacing T by N T and τ by iy, where 1 ≥ y > 0, it is possible, but quite technical, to give a "small" uniform upper-bound in the sense that it is independent of the choice of T . This upper bound is of the form Cy w with some integer w. This is summarized in Theorem 2.14. Before going into the proofs, we sketch the idea why dominated convergence of Eisenstein series is useful. When considering L-functions of modular forms (vanishing in the cusps τ ∈ {0, i∞}), we first look at the Mellin transform While convergence of integral and sum is no problem on the interval [1, ∞], the situation looks different for (0, 1]. A priori, we will only be allowed to switch integral and sum in the obvious region of absolute convergence. In this "trivial region" it is well-known that we end up with the ordinary Dirichlet series for the L-function. But if we can rearrange the Fourier series to a series of Lambert type and give "small" upper bounds for the partial sums (2.1), we may use Lebesgue's dominated convergence theorem to switch integral and sum also in non-trivial regions. As a result, we obtain a generalized form of Dirichlet series that also converges in a wider region to L( f ; s). All of this will be explained in Sect. 3. We will start this section with a classical result.

Theorem 2.1 (Faulhaber's formula)
We have for all α ∈ N 0 and T ∈ N: Here, the B k denote the Bernoulli numbers.
It is a trivial but very important observation for us that the left sum defines a unique polynomial in T by interpolation, which is given on the right hand side. We will not prove Theorem 2.1. It can be verified, for example, by using Euler-MacLaurin summation. For more details on this topic, the reader is advised to consult [2] on p. 21-31.

Definition 2.2
Let N be a positive integer and β : Z → C a function. We say that β has height d (with respect to N ), if for all α ∈ N 0 and T ∈ N: Here, the complex numbers γ α,β (u) only depend on α, β and u. The height of the zero function is always defined to be ∞. We denote by [N , d] the vector space of functions with height (with respect to N ) at least d.
Like in Theorem 2.1, the key property of functions in Definition 2.2 is that the left side defines a polynomial. We easily see that the constant sequence β( j) = 1 and more generally, β( j) = j d will have heights −1 and −d − 1, respectively, where d ≥ 0 is some integer. But while here the negative height causes an increase in the growth of the considered sums, we are rather interested in the opposite phenomenon of a non-negative height. In this case we obtain a decrease in the growth. Periodic functions with this feature play the key role when looking for "small" upper bounds of partial sums (2.1). Of course, not all functions β do have a height.
We are only interested in periodic functions. The next proposition guarantees that they have a height.
Proof Since β is periodic, we can rewrite the sum over β( j) j α as It is clear by Theorem 2.1 that for any c the expressions

Proposition 2.5 Let d ≥ 0 be an integer and
Proof Since β is N -periodic we know by Proposition 2.4 that the expressions define polynomials for all integers values 0 ≤ α. We need to show, that these have degree at most α − d. We obtain Since the sum over v starts at d + 1, by and each (non-principal) even character has height at least 1, since then we additionally have Then we obtain for the value P ( ) (1): Our investigations rest on the properties of some explicit polynomials. They are similar, but simpler as the sums in (2.1). For a fixed non-negative integer α we define a sequence by

Lemma 2.8
The sequence ( p T (α; x)) T ∈N converges to some polynomial function on the interval [0, 1) from below for all α ≥ 0. In particular, the terms p T are uniformly bounded in the sense This uniform boundedness is a very important property as we will see later.
Proof It is clear that

Remark 2.9
In fact, one can give an explicit formula for the Q α in terms of Eulerian numbers, but we will not need such a precise description for our applications.
and after further manipulations The right hand side is greater than the left hand side for x = 1, since On the other hand, the left hand side is unbounded and monotonically decreasing in the interval (0, 1]. Hence, there is exactly one solution for the above equation in this area and the claim follows.
Before we can go on to the next lemma of this section we recall: Lemma 2.11 Let a k be a sequence of complex numbers and b k and c k sequences of positive real numbers such that 0 ≤ b k+1 ≤ b k and c k+1 ≥ c k ≥ 0 for all k. Then we have for all n ≥ 1: Proof The first statement is called Abel's inequality, so we will only prove the second one. We set A n = n k=1 a k and obtain by partial summation Hence the lemma is proved.
Our strategy will be to expand ω(z) in (2.1) into a Fourier series. With this we will obtain a double series, which is on the one hand more complicated. On the other hand, this simplifies the occurring summands drastically. Partial summation and Abel's inequalities are then the key tools when estimating sums of this type, as the next boundedness lemma shows.

Lemma 2.12
Let M, L, T > 1 and w ≥ 0 be integers, ζ j M = 1 be a root of unity, 0 ≤ X , Y ≤ 1 be real numbers and c k be a monotonically increasing (or decreasing) sequence (that may depend on X and Y ), which is bounded by 0 ≤ c k ≤ B and B does not depend on X , Y , L and j. Then we have uniformly for L, X , Y , j, where C w is the constant defined in Lemma 2.8.
Proof Without loss of generality, we assume c k to be an increasing sequence. In the case that c k is decreasing the proof works similar. By Lemma 2.11 we first obtain In the case c k is decreasing we could switch 2B by B, but since B ≤ 2B the estimate works in both cases. To estimate the inner sum for any value I with 1 ≤ I ≤ L, we will use the fact, that the p T are monotonically increasing first in some interval [0, ξ w,T ] and then monotonically decreasing in [ξ w,T , 1], as it was shown in Lemma 2.10. For any I choose the Note that in the case Y = 1 the second condition is empty. Then, using the triangle inequality, we see We apply Lemma 2.11 on the first sum to obtain where C w is the constant given in Lemma 2.8. The inner sum can be estimated again with Similarly, we obtain with Lemma 2.11 This proves the lemma.
The next lemma can be seen as an analogous result to the previous lemma.
The right hand side is obviously bounded for 0 ≤ x ≤ ∞ and only depends on j, M, N and p, so we have found a possible D j,M,N , p .
We now have all the tools to prove the main theorem of this section. Proof For y = 0 the inequality holds since in the case α ≤ d the left hand side is always zero (note that ω(0) exists) and otherwise the right hand side is +∞ from the right. Let y > 0. We then have In the first step we will only deal with the inner sums. We obtain with partial summation On the other hand, we have For the right sum we obtain with Lemma 2.11 and (2.4) (note that 1−e −2π ky is monotonous): 1 − e −2π k N y u+1 .
After multiplying and dividing by 1 − e −2π k N y α−d+1 , this equals Put Y := e −2π N y . There is a constant A > 0 not depending on y and k such that For k > 0 the sequence

is decreasing and bounded between 0 and
This gives us when putting X := Y u and using Lemma 2.12. On the other hand, when putting Z := e −2π y , we obtain for the right sum in (2.6) and since β is N -periodic this equals Note that we always have 0 ≤ c k (u) ≤ c k+1 (u) ≤ 1. Since β has height d, by Lemma 2.7, there are coefficients δ α,β,u (w) such that The sequence y α−d 1 − Z k d−α in k is bounded by some V α−d and monotonous. Hence we obtain with Lemma 2.11 that the above estimate is smaller or equal to and by Lemma 2.12 this is smaller or equal to for some F α,β,ω > 0 only depending on α, β and ω. By considering (2.4), (2.5), (2.7) and (2.8) and using the triangle inequality in (2.3) (note that the constants do not depend on L), the theorem is proved.
Since we have assumed β to be N -periodic it might come from a weak function η ∈ W N , i.e., β := β η . The purpose of the next section will be to use the Dominated convergence theorem to improve regions of convergence of L-functions assigned to products of weak modular forms. (3.1) In the case the set S is clear, we simply write | · |. For example, S could be the set of integral ideals of a number field and | · | their norm. Let a(t m ) m∈N a sequence of complex numbers. We define the corresponding formal Dirichlet series by In the case that the series converges for all s ∈ C with Re(s) > 0, one can check using partial summation that such Dirichlet series converge (if they do) on half planes and represent holomorphic functions in these regions. This is for example the case, if the |t n | increase monotonously. Since we have (3.1), one can show that F(s) will converge in some point s 0 if and only if a(t) = O(|t| ν ) for some ν ∈ R. Definition 3.1 Let F(s) = t∈S a(t)|t| −s be a Dirichlet series, Q a totally ordered countable set together with a surjective map w : Q → S with finite fibres. We also assume that F converges to a holomorphic function on some half plane {Re(s) > σ 0 }. The order of Q shall respect the order of S, this means u 1 ≤ Q u 2 implies w(u 1 ) ≤ S w(u 2 ) for all u 1 , u 2 ∈ Q. We define an integer map on Q via |u| Q := |w(u)| S . In other words, all elements in the same fibre of a t ∈ S are associated to the same integer. By a splitting of F we mean a Dirichlet series F(s) = u∈Q b(u)|u| −s Q that has the following properties: (i) F(s) converges to a holomorphic function in some half plane {Re(s) > σ 0 }. (ii) We have for all t ∈ S the summation formula u∈w −1 (t) b(u) = a(t).
We may think of splittings in the following way: we have Q = t∈S σ −1 (t) and therefore t∈S a(t)|t| −s =

Remark 3.2 Consider the number theoretic function
Note that normally one considers tuples in Z 4 but to keep things simple in this example we use N 4 0 . Then the ordinary Dirichlet series As a result, the series x, x −s is a possible splitting of D 4 . Note, that this series also converges (independent from the chosen order) on Re(s) > 2 and represents a holomorphic function in this region, which shows that also condition (i) of Definition 3.1 is satisfied.
Splittings that are obtained by maps N × N → N and (x, y) → x, y will play the key role in the rest of this section. Throughout, we will omit the construction details as they were presented in the last example.
The next definition provides kind of an inverse concept for splittings.

Definition 3.3
Let S = ∞ j=1 S j be a disjoint covering with finite S j . We say that a Dirichlet series F(s) = t∈S a(t)|t| −s respects the rearrangement (S j ) j∈N , if the series is given by the partial sums If there might be danger with confusion we simply write Obviously, F(s) and (F, (S j ) j∈N )(s) coincide in all regions of absolute convergence. In the case of S j = {t ∈ S | |t| = j}, (F, (S j ) j∈N )(s) is an ordinary Dirichlet series b(n)n −s -we call this the standard rearrangement. The next proposition makes clear why rearrangements makes splitting undone in some situations.

Proposition 3.4 Let F be a splitting of F over Q. Define the disjoint union Q j
If we now sum F with respect to (Q j ) j∈N we obtain F. Proof This follows directly from the definitions. Definition 3. 5 We call (T j ) j∈N a sub-rearrangement of (S j ) j∈N , if there is a sequence of integers 0 < k 1 < k 2 < k 3 < · · · such that T 1 = S 1 ∪ · · · ∪ S k 1 , T 2 = S k 1 +1 ∪ · · · ∪ S k 2 and so on.
In the following we define for any rearrangement the abscissa of convergence σ ((F, (S j ) j∈N )) to be the infimum real value σ 0 , such that for all complex values s ∈ C with Re(s) > σ 0 the series converges and represents a holomorphic function in this region.

Remark 3.6
One easily checks σ ((F, (T j ) j∈N )) ≤ σ ((F, (S j ) j∈N )) for (T j ) j∈N a subrearrangement of (S j ) j∈N . Hence Proposition 3.4 shows that splitting does not improve the area of convergence. However, when rearranging a split series the situation might look different.
Let R(F) the set of all rearrangements of F. We define an equivalence relation on R(F) by putting two coverings in the same class if the resultant series have the same abscissa of convergence. We collect this data in R(F)/ ∼. We would like to study R(F)/ ∼, in particular, we are interested in the following question:

Question 3.7 What is the value σ (F) := inf G∈R(F)/∼ σ (G)?
There is no simple answer to this question. It rather strongly depends on the Dirichlet series itself, as the next examples demonstrate.
(i) If a(t) ≥ 0 globally, the region of convergence can not be improved by rearranging the Dirichlet series. Hence |R(F)/ ∼ | = 1 and σ (F) = σ (F). (ii) Although the set of possible rearrangements is large, − σ (F) does not have to be unbounded even in the case that F is entire. If χ is an even real non-principal character modulo M, one can show that σ (L(χ; s)) = −1 if L(χ; −1) / ∈ Z. In this case the "best" rearrangement of L(χ; s) is given by We conclude L(χ, 0) = 0. Since all inner summands in the rearrangements are integers when s = −1, there is indeed no better choice if L(χ, −1) / ∈ Z, as the reader may easily check. A similar argument shows σ (L(χ; s)) = σ 0 = 0 if χ is real, odd and L( n s for Re(s) > 1 is well-known and elementary. Here μ(n) is the Möbius function. Since μ(n) has sign changes, it makes sense to look at possible rearrangements. However, it seems to be extremely difficult to find improvements of σ = 1, since there is no progress in this area until today! We have 1 2 ≤ σ (ζ −1 ) ≤ 1 and σ (ζ −1 ) = 1 2 implies the Riemann hypothesis.

Remark 3.8
In the case of (ii), where the coefficients are well-studied, there are of course even more powerful tools for analytic continuation using series transformations, that can be seen as generalized rearrangements in the sense that we allow the splitting sets S n to have infinite order. For example, when using Euler summation, we find the right hand series will converge globally for non-principal characters χ.
Let k = (k 1 , . . . , k ) and f ∈ U k ( 1 (M, N )) be a weak modular form. In the following we give a natural splitting for L( f ; s) in terms of the overset Q = N × N . After this, when applying the Dominated convergence theorem from the last section we can find good rearrangements of these splittings to give estimates for the size defined in Question 3.7. Let G ( ) Note that each ψ extends multiplicatively to a map ψ : Z → C × . For k ∈ N also define the (multiplicative) map k (n) = n k 1 −1 1 · · · n k −1 . [M, d], where 0 ≤ d. Then there is some constant C h > 0, only depending on h, such that we have uniformly for x ∈ [0, 1]:

Lemma 3.9 Let h be a M-periodic function in
Proof We use partial summation again. We obtain: By Proposition 2.7 there is a constant C h such that uniformly on [0, 1]: On the other hand, we uniformly have This proves the lemma.
In the following, consider the subsets is the vector valued Fourier transform. Here, the order of summation respects the rearrangement (T M, p ) p∈N .

Proof We have
We obtain with Lemma 3.9: uniformly for x ∈ [0, 1], where C h,u > 0 only depends on the functions h 1 , . . . , h and the vector u. As a result, the integral converges absolutely to a holomorphic function for Re(s) > − j=1 c j and we may switch it with summation in this region: where we respect the rearrangement (T M, p ) p∈N in the last sum. This proves the lemma.
In the following, we will look at L-functions corresponding to higher weak modular forms. Let f be a modular form in M k ( 1 (M, N )) with Fourier expansion Then we remember that its corresponding L-function is given by One can show that this series converges on some half plane and we have the relation  ( 1 (M, N )) be a higher weak modular form, such that Here we assume that sgn(h α, j t α, j ) = (−1) k j for all j = 1, . . . , . Then, for all complex numbers s with Re(s) > |k|, we have where the coefficients a(u, v) are given by Proof The series on the right of (3.2) converges absolutely on the half plane {s ∈ C | σ > |k|}, since and on the other hand, for all ε > 0, and hence Since t j (0) = h j (0) = 0 for all 1 ≤ j ≤ , all involved weak functions have a removable singularity in z = 0 and so have their product. We have for all Hence, due to absolute convergence, we obtain for all s with σ > |k|: Together with Lemma 3.10 we obtain, that this equals This proves the proposition.
Proposition 3.11 provides us coefficients a(u, v) that belong to splittings of L( f ; s) over Q = N × N . We may use this to define a linear map from "splitting coefficients" to modular forms. Firstly, consider the vector space Secondly, look at the subspace B M,N ,k ⊂ A k of functions that generate L-functions of higher weak modular forms in U k ( 1 (M, N ) M, N )).

For all values s
where t(u) := t 1 (u 1 ) · · · t (u ) and (F ( ) Here, a(u, v) are the standard coefficients obtained in Proposition 3.11, for the set a+ M,N ,k see also Proposition 3.12.

Proof
The series on the right of (3.2) converges absolutely for all s with Re(s) > |k|. Since t j (0) = h j (0) = 0 for all 1 ≤ j ≤ , all involved weak functions have a removable singularity in z = 0 and so have their product. We have for all The functions t 1 , . . . , t have heights d 1 , . . . , d which means by Theorem 2.14 that there is a constant C > 0 such that for all T ∈ N and 0 ≤ x ≤ 1: In the proof of the Dominated convergence theorem the upper bound was independent of the choice of the partial sums for the series of ω. Hence, together with Lemma 3.10 we obtain for Re(s) > −c: Since the order of summation in the partial sums respects the rearrangement (U M,N ,m ) m∈N , note that the theorem is proved.
From this we obtain a much more general result as (ii) presented in the above examples. Proof Put k = d + 1. Choose h = 0 such that sgn(t · h) = (−1) k . Then we obtain with Theorem 3.13 that the series converges for all s ∈ C with Re(s) > 0 to a holomorphic function. Since the claim follows.
One consequence of this observation is an application to infinite products.
Together with Corollary 3.14 we conclude that ∞ n=0 4 j=1 a 4 (4n + j)(4n + j) −s converges to a holomorphic function for all s ∈ C with Re(s) > −1 and we find .
The next final corollary provides natural generalized Dirichlet series representations for Lfunctions associated to products of Eisenstein series for non-principal primitive Dirichlet characters. Proof Since all characters are primitive, we have Hence we obtain with Theorem 3.13 We can simplify the expression (F ( ) so we obtain The extended domain of convergence follows, because of the rearrangement, with Theorem 3.13 and the fact that the height of ψ j is given by 1 2 (ψ j (−1) + 1).
Note that this representation of the L-function of the considered product is more natural since it is a direct generalization of the formula for L(E k (χ, ψ; τ ), s) in the case = 1, where the series directly splits into a product of two Dirichlet L-functions: The region of convergence may be improved when summing with respect to the rearrangement (U M,N ,m ) m∈N . In this case we end up with By Corollary 3.14 this converges, if k ≥ 2, for Re(s) > k −1 if ψ is odd and for Re(s) > k −2 if ψ is even (and of course, non-principal). In the case k = 1 we have convergence in the region Re(s) > −1 if and only if ψ and χ are both even and for Re(s) > 0 else. Finally, we give an example.
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