Rational functions, cotangent sums and Eichler integrals

With the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms.


Introduction
In this paper we continue the study of so called weak functions that were introduced in [8] in the context of modular forms. Weak functions are essentially rational functions R(X) that are holomorphic outside roots of unity with zeros in 0 and ∞ and only simple poles. After a substitution X = e 2π iz we hence find for the corresponding ω(z) := R(e 2π iz ) that: (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| → ∞.
On the other hand, when starting with such an ω(z), by Liouville's theorem one quickly sees that it is essentially just a rational function R(X) with the above properties. This induces a 1:1 correspondence. 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 residue function β ω (x) := −2πires z=x ω(z). Note that such a residue function will be 1-periodic and have discrete support ⊂ R. Vice versa, we say that ω β is the weak function induced by the residue function β. In [8] we have proved the following construction theorem for modular forms for the congruence subgroup , In the case that k = 1 and 2 the map stays well-defined under the restriction that the function z → z k−1 η(z)ω(zτ ) has a removable singularity in z = 0.
The question which modular forms are precisely caught in the image of this construction map is partially addressed by the author in [6]. For example, we conclude for non-principal characters modulo M and N : The discrete Fourier transform F N is defined in Definition 4.2 below. In particular, if χ and ψ are primitive and hence conjugate up to a constant under the Fourier transform, this simplifies to The Fourier coefficients of E k (χ, ψ; τ ) are well-known and given by This allows an extension to the weights k = 1, 2. Using this it can be verified quite easily that all modular forms constructed in Theorem 0.1 are part of the Eisenstein space. But still, considering Theorem 0.1, several questions naturally appear.
(1) What happens, if we allow weak functions to have poles at arbitrary real numbers? (2) What happens, if we allow weak functions to have poles of arbitrary order? (3) Are there any generalizations for non-positive integers k?
Of course one can also combine the first questions, by asking for poles of arbitrary order at arbitrary real numbers, but this seems to be rather difficult for now. The goal is now to consider a more general class of weak functions, the pre-weak functions, that are allowed to have poles at real numbers and only have to be bounded as the imaginary part of the argument tends to ±∞. We collect these functions in the vector space W pre . What we get is the following transformation law.
Hereω(z) = ω(−z) and W ⊗ (k) denotes a sufficiently good subspace of W pre ⊗ W pre and is explained below.
This transformation law is just a straight generalization of the key transformation theorem in [8]. However, due to a lack of rapid decay in the situation of non-positive k we are faced with some technical issues and must be careful with the analysis. A first example of application of this generalized transformation law is the definition and investigation of generalized Eisenstein series. These functions can be written as a holomorphic limit of modular forms and have a Fourier series t∈R >0 a(t)q t representation with real exponents. This, provided k 1, gives a first answer to question (1). The objects which come across naturally when answering question (1) are a certain type of generalized L-functions. They are defined by Here, β ω is the 1-periodic coefficient function of the pre-weak function ω. In order to isolate information on its values at integer arguments, we analyze the case ω ≡ 1, hence 1 ⊗ η, in more detail in Section 2. In this situation it makes sense to choose k 0 in the above setting. As an application we achieve detailed descriptions of cotangent sums in terms of the L-functions defined above. A (general) cotangent sum is a finite sum of the form A famous example for a cotangent sum is given in [10, p. 262]: Note that the sum is always rational independent of the choice of N . Generally, it turns out that the arithmetic nature of such cotangent sums is strongly tied with the arithmetic nature of corresponding L-functions. The key idea is to construct (for fixed ω) a lower diagonal isomorphism A m ∈ Q m×m between the spaces {a 1 C(β ω ; 1)+· · ·+a m C(β ω ; m)|a j ∈ Q} and {b 1 L * (β ω ; 1)/π + · · · + b m L * (β ω ; m)/π m |b j ∈ Q}, where the L * are essentially Lfunctions. A consequence of this construction is the following theorem.
For example, with ζ (2k) ∈ Qπ 2k an easy consequence of Theorem 0.3 is This is well-known and was verified by Berndt and Yeap (see [1, p. 6]). We can use Theorem 0.3 to show some more interesting relations for cotangent sums. Furthermore, with our method it is possible to derive explicit formulas for the cotangent sums C(χ; m) where χ is an arbitrary primitive character. These will be stated in Corollary 2.19. Similarly, we can give (rather complicated) formulas for Dirichlet series with trigonometric coefficients at integer arguments, see Corollary 2.22 and Remark 2.23. Explicit formulas in this spirit are worked out in [7]. Finally, using Fourier analysis and the generalized Clausen functions one can derive closed formulas for cotangent sums presented by Berndt and Yeap [1] involving sine and cosine functions. Here we use explicit terms (described in Theorem 2.15) of the rational isomorphisms briefly described above.
Although it can be done, we do not address question (2) in full generality and concentrate on the case of poles only at rational arguments. As a result we obtain a quite simple criterion for a certain type of q-series to satisfy a transformation law under τ → − 1 τ . This is achieved done in Section 3. A quite full answer to question (2) is that, beginning with higher degree weak functions, one is able to build q-series that satisfy generalized modular identities in the spirit of (3.10). Note that these functions are linked to derivatives of modular forms due to the technical Lemma 3.8. The whole idea behind this is that one splits the spaces of pre-weak functions with higher degree poles into differentiated spaces of ordinary pre-weak functions, see Corollary 3.3.
It turns out that the negative weight cases are also closely tied to modular forms, at least if the weak functions only have poles at rational points. The connection is given by Eichler integrals. To any modular form of weight k 2, that vanishes in the cusps in τ = 0 and i∞, we can associate an Eichler integral. It has the form where c k is some normalizing constant. This integral represents a holomorphic and periodic function on the upper half plane and is tied to the so called period polynomial p(f ; τ ) of f by the functional equation Explicitly, we have a connection to the critical values of the L-function associated to f by These period polynomials are very important objects in number theory. For example, they appear in the context of a conjecture by Delinge-Beilinson-Scholl which makes an assertion about the nature of values of derivatives of L-functions of Hecke cuspforms f , see also [12]. Also, an immediate implication of the Eichler-Shimura isomorphism, see [11], applied to the period polynomial is Manin's Periods Theorem [14], which provides important information about the arithmetic nature of critical L-values. For a detailed investigation of the values of Eichler integrals at algebraic points, also in the context of Ramanujan identities for L-values at integer arguments, see [9]. Finally, a fairly good introduction to the so called Riemann hypothesis for period polynomials attached to derivatives of L-functions is given in [5].
In the case the pre-weak functions ω ⊗ η only live on points of finite order, the rational function at the end of the transformation refers to a period polynomial if k < 0. It implies that critical values of L-functions attached to Eisenstein series are just residues of elementary functions. Moreover, in the last section, we prove a duality result that concerns pre-weak functions. We prove that the k − 1-fold integrals of ϑ k can be expressed as linear combinations of expressions ϑ j (with some negative j) and apply this to Eisenstein series. This is realized by an injective linear map that is given in terms of several Fourier transforms and will be specified below. For the different denotations of the several spaces of pre-weak functions the reader is advised to consult the notation list below. In particular, we prove the following theorem, which gives a satisfactory answer for question (3) for most values of k.
is the space of Fourier series of the form ∞ n=1 a(n)q n/M that converge to holomorphic functions on the upper half plane.
Here, k−1 denotes the k − 1-fold integral as it is defined in Definition 4.1. The proof of this theorem essentially relies on a weak form of Bol's identity which is presented in Lemma 4.9. The idea is that applying k − 1-fold integration on a function that comes from rational functions and essentially equals ϑ k (ω ⊗ η; τ ) can be expressed by other functions of this type. As a consequence, for example, we can find the following interpretation of critical values of L-functions associated to Eisenstein series. Theorem 0.6 (cf. Theorem 4.15) Let k 3 be an integer, χ and ψ be two primitive Dirichlet characters with χ(−1)ψ(−1) = (−1) k and f (τ ) = E k (χ, ψ; τ ). We then have the following identity between rational functions: It is unlikely that Theorem 4.15 can be generalized to arbitrary cusp forms in this simple form.

Summary
The paper is organized as follows. First we define the term pre-weak function and consider generalized Eisenstein series, that have q-series expansions t 0 a(t)q t with real exponents. We prove the functional equations of the associated generalized L-functions.
In the second section we apply the theory of pre-weak functions to cotangent sums and generalize some results by Berndt, Yeap and Zaharescu. In the third section we expand our main ideas to pre-weak functions of higher degree, which means that we allow poles of higher order. Finally, in the last section, we generalize the ideas of the second section to period polynomials and prove a duality result.
Notation General As usual we denote N, Z, Q, R and C as the sets of natural numbers, integer numbers, rational numbers, real numbers and complex numbers, respectively. Also we write N 0 = N ∪ {0}. We set Re(s) and Im(s) as the real and imaginary part of s. We denote H as the upper half plane, i.e. τ ∈ H if and only if Im(τ ) > 0. We abbreviate F d := Z/dZ for any number d ∈ N. For positive integers N we sometimes write F N −1 := Z[N −1 ]/Z. Several complex variables with different meanings will appear. We put e(z) := e 2π iz . However, in the case of variable τ we use the common notation q := e 2π iτ . Throughout the paper k, N 1 and M 1 will denote integers.
Spaces For each set L (for example the real or the complex numbers) we define L C 0 as the space of all functions f : L → C, that are zero everywhere except finitely many x ∈ L. The subspace L C 0 0 ⊂ L C 0 is given by all f satisfying x∈L f (x) = 0. In the case 0 ∈ L we write L C 0,0 for the subspace of functions with f (0) = 0. As usual, we define M k ( ) to be the space of holomorphic modular forms of weight k for the congruence subgroup . By C + [[X]] we define the space of all power series ∞ n=0 a(n)X n with radius of convergence at least 1. In our applications, we will put X := q or more generally X := q we denote the kernel of f → a(0). In the context of weak and pre-weak functions there will be lots of notation for slightly different C-vector spaces, so it is convenient to provide an overview: • For any positive integer N , W N denotes the space of weak functions with level N , see p. 1. • The spaces W weak and W pre collect all weak and pre-weak functions of degree 1, respectively, see p. 9. • The space W ⊗ (k) is generated by all tuples ω ⊗ η of pre-weak functions, that allow a well-defined formulation of the transformation law for ϑ k (ω ⊗ η; τ ). For a precise definition see Definition 1.1.
• The spaces W weak,a and W pre,a collect all weak and pre-weak functions of degree a, respectively. In the case a = ∞, these spaces collect functions of arbitrary (but finite) degree. See also Definition 3.1. • By V ± we always mean the subspaces of a space V of complex functions consisting of all even and odd functions, respectively. We write ω = ω + + ω − to indicate the even and odd part of a function, respectively. We use the notation sgn(f ) = ±1 to indicate that f is an even or odd function, respectively. • By W ±i∞ we denote the subspace of functions f ∈ W with the property f (±i∞) = 0.
• By W 0 we denote all functions f ∈ W that have a removable or no singularity in z = 0.

Functions
Several times we shall use differential operators. To avoid any confusing situation with possible inner functions we stress at this point that we always mean and f (1) (g(x)) = f (g(x)) (the same for nth derivatives). We write ∂ τ := 1 2π i ∂ ∂τ and ∂ z : If the variable is clear we only write ∂. We will write S(f ) ⊂ U for the set of poles of a meromorphic function f : U → C.
There are several functions in this paper, so it is also convenient to present an overview: • As usual, we denote by ζ (s) :=    • The term F N , where N is an integer, defines the Fourier transform of a periodic function or pre-weak function. The term F −1 N denotes the inverse of F N . For a definition, see Definition 4.2. The more general convention F is given in Definition 1.6. • The expression [β ⊗ γ ] k,j (τ ) is an abbreviation for a Fourier series in the variable τ , constructed using periodic functions β and γ , see Definition 4.4. • To each pre-weak function we can associate a cotangent sum C(ω; m) for some integer m, see (2.1). For a character χ we similarly have C(χ; m), see (0.5). • The coefficients δ ν (u) and δ * ν (u) frequently appear in the context of cotangent sums, and are defined explicitly in (2.7) and (2.17), respectively.

Pre-weak functions, Eisenstein series and generalized periodic L-functions
We denote the vector space of all generalized weak functions of degree 1 (this means, that only poles os degree 1 are allowed) by W weak . I.e., each function ω ∈ W weak has period 1, is meromorphic in C and of rapid decay as |Im(z)| → ∞ and only has poles of degree at most 1 at real values.
We now call a 1-periodic pre-weak, if it has all properties of a weak function except that it is just bounded as y → ±∞ in the strip {0 x < 1}. In other words, we have the exact sequence The subspaces W ±i∞ pre ⊂ W pre contain all pre-weak functions that additionally vanish in z = ±i∞. All introduced notations for weak functions will also apply to pre-weak functions, if appropriate. Note that each ω ∈ W pre also has a representation , where the sum is of course finite. Now consider the homomorphism The holomorphic functions on the right will be called periodic L-functions (since the input function lives on the 1-torus). We have the decomposition is the Hurwitz zeta function. By analytic continuation we may consider the subspace The aim of this section is to assign periodic L-functions to generalized Eisenstein series that satisfy certain transformation properties. These Eisenstein series E k (ω ⊗ η; τ ) arise from tuples of weak functions ω ⊗ η with real (but not necessarily rational) poles. In this situation we are not able to assign ω and η a level. Hence, the functions E k (ω ⊗ η; τ ) will not be modular forms (except of course they identically vanish).
Definition 1.1 We will use the notation W ± to indicate the sub-spaces spanned by odd and even functions. What we need is the following: for k ∈ Z we define The reader should note that this tensor product notation, which will be used frequently, does not commute in general, so in general there is a difference between ω ⊗ η and η ⊗ ω even if ω and η are elements of the same space.
Consider the following linear map between pairs of pre-weak functions and holomorphic functions A proof that this is well-defined is given in Proposition 1.4.
Note that W ⊗ (0) is also spanned by the spaces W + pre ⊗ W − pre and W − pre ⊗ W + pre that entirely map to the constant zero function by Remark 1.2. But we will still use this notation for formal reasons.

Proposition 1.4
The map ϑ k is well-defined.
Proof Let x ∈ R × and K ⊂ H + ∪ H − be a compact subset. Then we have the estimate We distinguish three cases.
1. In the case k > 0 the claim now follows easily since then ω ∈ W weak and hence there is a δ > 0 (depending on K and ω), such that On the other hand, the term |res z=x η(z)| is bounded since η is periodic. 2. If k < 0 it follows that where the constant C > 0 may be chosen as 1] |res z=λ η(z)|.
Since the sum x∈S(η)\{0} |x| −1−|k| converges the claim follows. 3. In the case k = 0 we note that the map is defined on the subspace W weak ⊗ W pre by the arguments of 1. It is clearly defined for W + pre ⊗ W − pre and W − pre ⊗ W + pre since then all summands cancel each other. So we are left to show that we can define it on W pre ⊗W weak . Without loss of generality we assume that ω ⊗η ∈ W ± pre ⊗W ± weak . First let both functions be even. Then ω = c + ω w with some constant c and ω w ∈ W weak . In conclusion, we only have to show that the sequence Since η is weak, the term N j=1 β η (x j ) is bounded and hence the right hand side converges as N tends to infinity. The odd case works similarly, since then we have ω = c cot(πz)+ω w and hence but since (±x) −1 = ±x −1 we are reduced to the even case. Finally, since in both cases we obtain homomorphisms that coincide on the common subspace W weak ⊗ W weak we may extend it to the resultant space We now obtain the following very general transformation law.
Proof Let y > 0 and τ = iy ∈ H. Define Then g y is a meromorphic function in the plane with simple poles at S(g y ) = S(η)∪S(ω)iy\ {0} (all lying on the real and imaginary axes). Consider the closed contour integrals where R n (y) is a sequence of rectangles that cross the axes half between the respective poles x n and x n+1 . We are left to show I n (y) n→∞ −→ 0 since then the claim follows with the identity and residue theorem. Using periodicity of η, ω and the decay of g y we find that this will certainly be the case for k = 0. So we are left to show it for k = 0. We first consider the case ω ⊗ η ∈ W ± pre ⊗ W ∓ pre . Then the functions ϑ 0 (ω ⊗ η) and ϑ 0 (η ⊗ − ω) are constant zero. Since the product ω(z/iy)η(z)/z is an even function in this case, its residue at z = 0 will be 0. Hence the transformation law is trivially satisfied in this case. Now let ω ∈ W weak . Then the integrals on the right and the left in the rectangle will go to zero because of the exponential decay of ω and the periodicity of η. So we can express I n (y) in the form where 0 < σ n → ∞ and 0 < t n → ∞ are chosen in the sense of R n (y). Now we divide the integrals into three parts: Here, c > 0 is some fixed constant (note that √ n = o(σ n )). There is a constant C > 0 such that we have |η(z)| C for all |Im(z)| 1. Also on the segments [−σ n ± it n , σ n ± it n ] the function ω(z/yi) is uniformly bounded (with respect to n = 1, 2, 3, . . .) by some D > 0 since it is periodic along the imaginary axes. Hence for sufficiently large n we obtain On the other hand, since will certainly converge absolutely and also The first integral in (1.6) tends to zero by the same argumentation. The case ω ⊗ η ∈ W pre ⊗ W weak works analogously. This proves the transformation formula.
Note that the (non-trivial) exponents in the above Fourier series can be irrational numbers too.

Theorem 1.8 Let all assumptions hold as above. The generalized Eisenstein series satisfies the modular identity
Proof We find The claim now follows by Theorem 1.5. Note that in the case k = 2 at least one and in the case k = 1 both of the functions ω β and η γ have a removable singularity in z = 0, such that in every case the rational part in (1.5) vanishes.
Analogous to ordinary Eisenstein series we can assign a generalized L-function to Like in the classical case one can show (for example by Mellin transform, using the transformation law of the Eisenstein series) that these L-functions have a meromorphic continuation to the entire plane and satisfy a functional equation of the standard type.

Proposition 1.9
The generalized L-function associated to E k (β, γ ; τ ) is given by where Li s (z) denotes the polylogarithm. It converges on {s ∈ C|Re(s) > k} and has a meromorphic continuation to the entire plane.
Note that L(β; s) represents a holomorphic function on {s ∈ C|Re(s) > 1} by (1.3) (β is 1-periodic and zero at all but finitely many points) and has a holomorphic continuation to C \ {1} with a possible simple pole in s = 1.
Proof Starting with Definition 1.7 we obtain The function γ is zero almost everywhere. Since by its Fourier transform F(γ )(n) is bounded and hence the corresponding Dirichlet series converges absolutely on {s ∈ C|Re(s) > 1}. We now have The claim follows with the analytic properties of s → Li s (e −2π ix ) and L(β; s).
In the next theorem we prove a functional equation for the completed L-function associated to a generalized Eisenstein series.

Theorem 1.10 The completed L-function
extends to an entire function and satisfies the functional equation Proof By Mellin transformation we obtain By splitting the integral in the intervals [0, 1] and [1, ∞) and making the substitution y → y −1 in the first integral we obtain From this one sees that (β, γ ; s) is entire. The symmetry on the right hand side leads to the desired functional equation.

Cotangent sums
Besides periodic L-functions we may associate other objects to a pre-weak function. For integers m = 1, 2, 3, . . . we define the corresponding cotangent sum The primary goal of this section is to develop a principle which helps to write cotangent sums as rational combinations of L-functions, and vice versa. With this we may conclude several results about cotangent sums using well known results about L-functions, and of course vice versa again. A famous example for a cotangent sum is given in [10, p. 262] : Note that the sum is always rational independent of the choice of N . This was generalized by Chu and Marini in [4] and Berndt and Yeap [1, p. 6].

Theorem 2.1 Let N and n be positive integers. Then
In particular, we have Note that the B n denote the Bernoulli numbers defined by generating series The interesting identity in Theorem 2.1 can be proved by looking at f (z) = cot 2n (πz) cot(πkz) and using contour integration. Another more general result is presented in [1, p. 17] (we have corrected a sign error in the original paper) and looks as follows.
In particular, both s m and c m define sequences of elements in Q[k, a].
In other words, the theories of generalized periodic L-functions and cotangent sums are in some way equivalent. To understand this, we modify the definition (1.2) of a periodic L-function in the following way. In the entire section we denote W 0 pre as the subspace of pre-weak functions that have no singularity in z = 0, which is equivalent to β ω (0) = 0. Consider now the homomorphism between the space of pre-weak functions and an infinite tuple of complete L-values at positive integers In the case k = 1, we interpret the sum as (an even pre-weak function is weak up to a constant and an odd up to a cotangent function). If k = 1 this relation still holds if we restrict to weak functions or odd ω.
Before we move on, we define a sequence of numbers which is of great importance in combinatorics.

Definition 2.4
Let n ∈ N 0 and k ∈ Z. We define the Stirling numbers of the second kind by To find the connection between (generalized) L-functions and cotangent sums we need the following lemma.

Lemma 2.5
Define a sequence δ : and for integers ν, u 0 with ν + u 2: Let a ∈ C \ Z. Then we have in an arbitrary small neighborhood of z = 0 Remark 2.6 (i) The first polynomials P ν are given by (ii) We have for all ν 1 the formulas and for all ν 2 Proof It is clear that the function f (z) = cot(π(z − a)) is holomorphic in a neighborhood of z = 0 in the case a ∈ C \ Z. For the constant term we find cot(π(−a)) = − cot(πa) = π 0 (δ 1 (0) + δ 1 (1) cot(πa)) , and indeed this coefficient is Using the formula in [15, p. 2], (note that in the paper, the sum starts at v = 1 but we have n 1, hence n 0 = 0) and the binomial theorem, for ν 1, we end up with and note that this and conclude Together with (2.9) this proves the formula for δ ν (u), after the index shift ν → ν − 1.
We can use Lemma 2.5 to determine the local Taylor expansion of ω(z) at z = 0. This will later help to explain the relationship between periodic L-functions and cotangent sums.

Lemma 2.7
Let ω ∈ W 0 pre . Then we have Proof With the behavior of the function cot(πz) at z = i∞ we obtain the following canonical representation of ω: and with Lemma 2.5 we obtain The claim now follows with some simple rearrangements.
At this point we stress the simple but important fact, that the coefficients δ ν (u) are independent of the choice of ω. Lemma 2.8 (Generalized Abel's theorem) Let f n : E ∪ {1} → C be a sequence of continuous functions that are holomorphic in the unit disc E, such that f n (z) → f (z) as n → ∞ for all z ∈ E. We assume that f is bounded on [0, 1] and put D : Assume that the f n satisfy the Abelian condition: there is a constant C > 0 such that uniformly for all n > 0 and all 0 t 1: Then we have Note that the important case f n (z) = z n is Abel's theorem.
Proof We show that for each ε > 0 there is an N 0 such that for all N N 0 : This is possible since the series ∞ n=1 a(n) converges. By partial summation we obtain with f n (z) → f (z) and 0 t < 1: On the other hand we have From this follows (2.10) and we conclude the lemma.
We consider the following special case.

Lemma 2.9
Let g be holomorphic on E and a neighborhood U of z = 1. Then f n (z) := g(z n ) satisfies the assertions of Lemma 2.8.
To see the lemma one uses the Cauchy integral formula where the closed and smooth integration path γ ⊂ E ∪ U with length L(γ ) surrounds the compact line [0, 1] once in positive direction. We find a minimum distance ε > 0 between γ and [0, 1]. Hence where C > 0 is independent from a and b. Put a = t n and b = t n+1 for 0 < t < 1. Since g(t n ) converges to g(0) if 0 t < 1 and to g(1) if t = 1, one has D := max{|g(0)|, |g(1)|}.
We are now in the position to prove a result that ties values of L-functions with Taylor coefficients of pre-weak functions.

Proposition 2.10 Let k 1 be an integer and
pre else, such that ω has a removable singularity in z = 0. We then have (2.11) (2.12) Proof First we note that in the case k = 1 (2.11) is trivial for elements ω⊗η in W ± pre ⊗W ∓ pre , since then both the left hand side and the right hand side are zero (note that either ω(0) = 0 or L(η; 1) = 0). Also if ω ∈ η ∈ W + pre ⊗ W − pre both sides vanish according to Remark 2.3 and since η is odd. So we can assume η to be weak in this case. We have ω(z) = R(e(z)) with a rational function R, which fulfills the conditions of Lemma 2.9 (note that ω has a removable singularity in z = 0). We obtain: Since η is weak for k = 1 both series will converge for y = 0 separately. Hence with Lemma 2.8 we conclude Note that we have a homeomorphism between the segments [0, i∞] and [0, 1] given by z → e 2π iz . On the other hand, with Theorem 1.5 we obtain lim τ →0 In the case of k = 1, the first term on the right side vanishes because η is weak. The choice ω = 1 finally proves (2.12).
Throughout our analysis of cotangent sums we assume the first component of the W pre ⊗ W pre to be the function which is constant 1. It is trivial but crucial that this function is even. Since we want to consider all values of completed L-functions simultaneously, we only look at elements 1 ⊗ ω ∈ W + pre ⊗ W 0 weak , W + pre ⊗ W 0,− pre . In other words, throughout, ω it is an odd pre-weak function or weak function-both have a removable singularity in z = 0. Together with Lemma 2.7 we can now suggest closed formulas for cotangent sums in terms of corresponding L-functions at integer arguments.

Proposition 2.11
Let k 1 and ω ∈ W 0 weak , W 0,− pre . We have the formula which is equivalent to Proof First note that δ 1 (0) = 0. In the case ω is odd it is trivial that L(ω; 1) = 0 = C(ω; 1), which proves the formula in this case. So let ω be weak if k = 1. With Lemma 2.7 we see that the residue of z −k ω(z) in z = 0 is given by Multiplying by 2πi proves the claim when using (2.12).

Definition 2.12
For Dirichlet characters χ modulo N we put Remark 2.13 Let k > 0 be an integer. In [2] a relation between the class number h K of the field K = Q( √ −k) and cotangent sums is proved. If χ is odd and the Legendre-Kronecker-character for K , we have The present method now gives a further viewpoint to this equation since by (2.13) we have and by the class number formula L(χ; 1) is directly tied to h K . Here we have put where χ is a character modulo N .
Let ∞ be the linear operator a 2 , a 3 , We can write this formally as an infinite lower triangular matrix: (2.14) Proposition 2.11 provides us a linear system with countable many unknowns. In other words, we can find values for the cotangent sums recursively. We obtain: Note that in the case that ω is weak we have L * (ω; k) = π −k L(ω; k). With δ ν (ν) = 1 [see (2.8)] we see that the system (2.15) is invertible, since we have a lower diagonal operator. In other words, for all positive integers m we have where L m (ω) and C m (ω) denote the first m rows vectors of (2.15) and m the regular major m × m block of the operator. Note that since m ∈ Q m×m we have −1 m ∈ Q m×m . Formally we define (2.17) Therefore we obtain the following theorem.
We see that it turns out that there is an arithmetic connection between cotangent sums and generalized L-functions. Together with Theorems 2.1 and 2.2 we are able to find explicit formulas. Here, the key ingredient is the fact that expressions like (i) There are rational numbers δ k ( ) (defined in Lemma 2.5) and δ * k ( ), independent from the choice of ω, such that

23)
Proof (i) The formula (2.19) follows from Proposition 2.11. Let k m be arbitrarily chosen. Formula (2.20) follows with (2.16) and the fact that −1 m ∈ Q m×m is again a lower triangular matrix, when denoting its coefficients by δ * ν (u) [analogously as it was done in (2.14)]. It is clear that all values δ * ν (u) are independent of m and ω. (ii) We first show by induction that for ν, u 1 the δ * ν (u) vanish if ν + u ≡ 1 (mod 2). This is clear for ν < u, so we assume that u ν. Obviously, with the vanishing of the above triangle in mind, the statement is equivalent to the vanishing of all "odd" lower diagonals We formally write −1 ∞ ∞ = I ∞ . First we show the vanishing of D 1 . Let ν 2. Then we obtain, multiplying the νth row of the operator −1 ∞ with the ν − 1th column of ∞ : Hence δ * ν (ν − 1) = 0, since δ ν−1 (ν − 1) = 1 [note that δ ν (ν − 1) = 0-remember that δ ν (u) = 0 if ν + u ≡ 1 (mod 2) by Remark 2.6 (iii)]. Note that the sum could be reduced to two summands in the first step since we have multiplied two lower diagonal operators. For the induction step, we assume that we have proved vanishing for D 1 , D 3 , . . . , D 2k−1 . We show that under these circumstances we obtain the vanishing of D 2k+1 . Let ν 2k + 2, and multiply the νth Since δ ν−2k−1 (ν − 2k − 1) = 1, we obtain δ * ν (ν − 2k − 1) = 0. To obtain the coefficients δ * explicitly, we could of course simply use invert the operator ∞ , which would not be too bad, since all of its finite "blocks" are lower diagonal with determinant 1. However, there is even a quicker trick that uses a small subset of cotangent sums that are polynomials in the "period" variable N . To prove the formula (2.21) for δ * 2k (2 ) with 1 k choose where N > 1 is a positive integer. A brief calculation shows ω N ∈ W 0,− pre . We have for integers k > 0 and for k = 1 the right sum is understood as in (2.6). Since ω N is not weak, we have to include the terms C(β ω N ; 0) = N − 1. From (2.15) and 2.1 we conclude for all even positive integers 2k (2.25) Both sides are a polynomial in N and since this identity is valid for all N > 1, we obtain Using the classical formula we obtain by comparing coefficients: The identity is an even function. The formula now follows after a simple rearrangement. (iii) Looking again at (2.25) we obtain by comparing the coefficients belonging to N : This proves supplementary law (1). On the other hand, making this comparison for the constant terms we find and using supplementary law (1) we immediately see (2). This completes the proof.
It is clear by the vanishing of δ * and δ for arguments ν + u ≡ 1 (mod 2) that and with (2.20) we obtain (2.23).
We want to apply these theorems to make statements about cotangent sums using L-functions. What we need is the following classical result due to Leopoldt.

Theorem 2.17 Let χ be a primitive character modulo N and k be a positive integer. Put
Here We can use this to determine a closed formula for the character cotangent sums (2.28)

Corollary 2.19
Let χ + be an even and χ − be an odd primitive character modulo N > 1 and m 1 be an integer. We have the explicit formulas In particular, independently of m, one has Then ω χ ± (z) is weak and hence C(ω χ ± ; 0) = 0. By Theorem 2.17 one obtains and similarly Note that we obtain an additionally factor 2 (by symmetry) and N 2 and N 2 −1 (by the residues), respectively, in this calculation. Here, C(χ; m) was defined in (2.28).
Proof For the Legendre symbol χ we have the identity Since χ is real the claim follows with Corollary 2.19.
There has been lots of effort finding closed values for Gauss sums. The reader may wish to consult for example [13] for an elementary overview. Also we can use the results about cotangent sums to derive properties about L-functions having trigonometric coefficients.

Corollary 2.22
Let cot = cot except cot(πn) := 0 for all n ∈ Z. Let N > a 1 and n 1 , n 2 , n 3 0 be integers such that n 1 n 2 = 0. We then have for k 1 with n 1 + n 3 ≡ k (mod 2): Proof The condition n 1 + n 3 ≡ r (mod 2) implies that the coefficients (when extended to Z) define an even/odd function if and only if r is even/odd. The result now follows with the well-known expressions for sin n and cos n in terms of linear combinations of multiple arguments sin and cos functions and Theorems 2.2 and 2.14.
Remark 2.23 Again, using Theorems 2.2 and 2.15, one can find rather complicated explicit formulas for the above Dirichlet series in terms of the values δ ν (u).
We can use this formalism to give a purely Fourier analytic proof for Theorem 2.2. Using standard Fourier analysis one obtains for 0 θ < 1:

Remember the modified Clausen function
We can now use Theorem 2.15 to find the closed formulas provided in Theorem 2.2. To see this, put θ = a k for 0 < a < k. Consider the function which lies in W 0,− pre . Then, we have for even values 2 > 0 and by (2.34) this equals to For odd values 2 − 1 > 0 we find L(ω; 2 − 1) = 0. Note also that the sum C(ω; 0) = −1 for obvious reasons. Hence, with Theorem 2.15 we find On the other hand, a straightforward calculation shows The cosine formula of Theorem 2.2 follows now by summing this over = 1, . . . , m, making the substitution 2 = ν + μ and adding everything together. Note that the (−1) m in (2.35) will cancel with that of (2.36) and that the formula (2.36) due to supplementary law (2) is just the case 2 = μ + ν = 0, completing the sum in (2.4). Similarly, we can show the sine formula (2.3) in full generality.

Remark 2.24
Note that we only have used the polynomials P m and Q m defined by in the proof of Theorem 2.15.

The space W pre,∞ and applications
The proof of Theorem 1.5 did not use the order of the poles that occurred, only their locations. This motivates us to generalize the concept of pre-weak functions in the sense, that we allow them to have poles of arbitrary order. In this section we investigate analogous transformation laws for this kind of situation and will apply this to specific types of q-series, see also Theorem 3.13.

Definition 3.1
We call a meromorphic function ω pre-weak of degree d, if all conditions for pre-weak functions are satisfied except that ω has a pole of order d (and all other poles have order at most d). We denote the vector space of pre-weak functions with degree at most a with W pre,a . We collect all pre-weak functions of arbitrary degree in the space Even in the higher degree situation, we will still use the notation Like in the special case a = 1 it is quite easy to classify all pre-weak functions of degree at most a using elementary complex analytic ideas. For this purpose we abbreviate .
We now find that there are uniquely determined functions β j : R/Z → C, 1 j a, that are zero except finitely many arguments, such that In other words, there is an isomorphism As we will see later, it is natural to study transformations of rational functions when applying the differential ∂ = 1 2π i ∂ ∂z . Note that h x, (z) satisfies the differential equation We define the projection π 1 : W i∞ pre,∞ → W i∞ pre,1 (remember that the i∞ in the exponent means that ω(i∞) = 0) by

Proposition 3.2 Let a 1 be an integer. We have the exact sequences
Here, the W i∞ denotes the subspace of all functions f ∈ W with f (i∞) = 0.
Proof We only give a proof for the case of the integer a, since the exactness of the second sequence is immediate with the proof.
It is clear that π 1 is onto and that the extended homomorphism W pre,a ∂ −→ W pre,a+1 has kernel C. Since W i∞ pre,a ∩ C = 0, it follows that ∂ is injective. To see im(∂) ⊂ ker(π 1 ) we observe by (3.2) that has no non-vanishing term h x,1 (z). Hence, π 1 (∂ω) = 0 for all ω ∈ W i∞ pre,∞ . On the other hand, if ω ∈ ker(π 1 ), it is of the form We will show by induction over the maximal degree 2 r a + 1 that all expressions of the form indeed have an integral in W i∞ pre,a . If r = 2 we are reduced to By (3.2) we find that is an integral of (3.3) and we may choose C = 0 to achieve that this is part of W i∞ pre,a . Next, assume that we have proven that has an integral in W i∞ pre,a for a fixed number 2 r a (if we had r = a + 1, we would be done). We then have (3.4) and the left sum on the right of the equation has an integral I 1 in W i∞ pre,a by assumption. By (3.2) we obtain and the integral on the right can be chosen to be in W i∞ pre,a by assumption again. Hence, the right sum on the right side of (3.4) has an integral I 2 in W i∞ pre,a . Hence 2πi(I 1 + I 2 ) ∈ W i∞ pre,a and ∂(2πi(I 1 + I 2 )) = ω r+1 . The claim now follows by induction.
With this we obtain the following. ∂ n W i∞ pre,1 .
Since we have W pre,a ∼ = C ⊕ W i∞ pre,a and the proof is analogous in the infinite case, the corollary follows.
We obtain a similar result for weak functions.

Corollary 3.4 Let a be an integer. Then we have the decompositions
Proof First note that we can write each ω ∈ W weak,a in the form Putting everything together shows the corollary.
At some stage it will be crucial to change from W weak,∞ to W pre,∞ in the sense of decompositions into derivatives. This is done in the obvious way.

Proposition 3.5
Let ω ∈ W weak,a . Then we have the following identity between decompositions provided by Corollary 3.4:

5)
where λ 0 , ω j ∈ W weak,1 , λ j ∈ W i∞ pre,1 for 1 j a − 1. As a result, we get β ω 0 (y) = β λ 0 (y) and for all 1 j a − 1 the corresponding coefficients The result now follows by comparing coefficients, since the ∂ j h y,1 (z) define a basis of ∂W i∞ pre,a−1 .
The next two lemmas will turn out to be very useful when going to Fourier series of ϑ k , where symmetry in the sense of sgn(ω)sgn(η) = (−1) k is required.
Proof Since we have W weak,1 = W + weak,1 ⊕ W − weak,1 and W i∞ pre,1 = W i∞,+ pre,1 ⊕ W i∞,− pre,1 , we can always write λ j (z) = λ + j (z) + λ − j (z), where of course sgn(λ ± ) = ±1. We put λ ±1 := λ ± . With this we obtain The left side of (3.6) has signum (−1) n and the right side signum (−1) n+1 . Since it is an identity, both sides vanish identically. The claim now follows inductively. The term of highest degree ∂ a−1 λ (z) has to vanish, since otherwise there would be a pole of degree a on the right side that does not cancel. Hence λ Continue with the second highest term and so on and conclude The second lemma is stated and proved similarly. Proof Since we have W weak,1 = W + weak,1 ⊕W − weak,1 , we can write each ω j (z) as a unique sum ω + j (z) + ω − j (z), where of course sgn(ω ± ) = ±1. For purpose of notation write ω ±1 (z) := ω ± (z). We now collect all terms with signum (−1) n on the left hand side.
Note that we have sgn(h 0,2 ) = 1, and hence sgn(∂ j−1 h 0,2 ) = (−1) j−1 . If n + j ≡ 1 mod 2, it follows sgn(∂ j−1 h 0,2 ) = (−1) n+j−1−n = (−1) n . Similarly, we find sgn ∂ j ω (−1) n+j j = (−1) n . Since both sides of (3.7) have different signums, both have to vanish. It is now easy to conclude the claim. Indeed, assume without loss of generality that is part of the right side and its "highest" term. When assuming that ω (−1) n+a a−1 (z) does not vanish, it has to have at least two different poles mod Z, since it is weak. It follows, that its a − 1th derivative has at least two poles of order a. But no other term on the right of (3.7) has poles of order a except c a−1 ∂ a−2 h 0,2 (z), but this only has one (mod Z) single pole of degree a in z = 0. So both summands in (3.8) have to vanish separately, since otherwise the poles of degree a could never cancel each other. The lemma now follows when going over all pairs of summands on the right side in (3.7), from above, inductively.
The next lemma provides some useful differential identities. Lemma 3.8 Let be k ∈ Z and ω ⊗ η ∈ W ⊗ (k) , see Definition 1.1.
As an application of the more general formalism we want to give a description of a special case of the main transformation law in the language of series of rational functions. To make things more explicit, we are going to use differentials of the form and apply the results of Lemma 3.8. Since the lemma tells us it seems reasonable to look at differentials Here j denote the Stirling numbers of the second kind (see Definition 2.4) and for integers b a − 1 the numbers κ a,b (j) are defined by We abbreviate s(n, ) := (2πi) −n−1 n j=0 (−1) n+1 j κ 1−k,n−k (j).
It is remarkable that we still obtain a simple modular relationship between ϑ k (ω ⊗ η; τ ) and ϑ k (η ⊗ ω; τ ), as it was the case in Theorem 1.5.
We then have the identity Proof The proof is essentially the same as the one of Theorem 1.5. We may choose τ = iy with y > 0 and use the rapid decay of the functions ω and η for increasing imaginary parts to show where ε > 0 is fixed and sufficiently small (note that ω and η are periodic and only have real poles). Put g τ (z) := z k−1 η(z) ω(zτ ) and h τ (z) := z k−1 ω(z)η(zτ ). For each τ ∈ H we obtain the functional equation Hence by the linearity of the residue. For the residue in z = 0 we obtain and by (3.9) this implies Multiplying this by (−1) k−1 τ k proves the claim.
This framework can be used to derive transformation laws of "higher" functions ϑ k (ω ⊗ η; τ ), where ω(z) and η(z) are allowed to have poles of higher degree, very explicitly. The outcomes are functions of the form f (τ ) = g 0 (τ ) + τ g 1 (τ ) + · · · + τ n g n (τ ), where the g j (τ ) are Fourier series on the upper half plane, such that the f (τ ) possess non-trivial transformation properties. We will omit the details of this extremely technical setup but will give examples in order to convince the reader of its usefulness. We will not use Theorem 3.9 in full generality and show examples with rational poles and lower degrees. Then we have and hence obtain This equals to One could now use the transformation properties of ϑ k−2 (ω ⊗ ω; τ ) given in Theorem 1.5 to make final conclusions. But we will use Theorem 3.9 to investigate ϑ k (η ⊗ η; τ ). Let n be a non-zero integer. We obtain with the series expansions cot(nπτ ) csc(nπτ ), we obtain by symmetry and Theorem 3.9 [note that sgn(η) = 1], that f k (τ ) with series representation this time. The main difference is that ω(z) has no integral function that is weak, since the integral is given by − 1 2π cot(2πz) + C, compare also the result of Corollary 3.4. Let k 6 be even. Very similar to Example 3.10 we find for the transformation law where each s j (τ ) is a finite sum of q-series of rational type (N, M) and a is the degree of R, which is its maximal pole order.
Proof We are able to present a constructive proof, but we will only sketch the ideas of construction. Without loss of generality, we assume P(n) = n k−1 with an arbitrary integer k > 0. Hence ord X=0 (P) = k − 1. For each (M, N )-rational type series with the additional assumption N j=1 ψ(j) = 0 we find weak functions and ω(z) := R(e(z)) such that With Theorem 3.9 we find polynomials Q −1 (X) and Q 1 (X) such that But since V 1 and V 0 if and only if δ(ψ) = 0, we see that Q 1 has degree at most 1 and 1 only if δ(ψ) = 1. On the other hand, also by Theorem 3.9, Q −1 (X) has degree at most U − k = −ord X=1 (R) − ord X=0 (P) − 1 [note that ord X=0 (P) is the correct measure at this point, since if P had more higher degree terms then the degree of Q −1 would be smaller for these terms]. For any fixed x = Note that we will use exactly the same notation for corresponding functions β : F N −1 → C, i.e., we put We obtain the following. Proof For (i) we note The calculation F N (F −1 N (β))(n) = β(n) works the same. To prove (ii), let β be a function in F C 0,0 N , i.e., β(0) = 0. Then we have Since both spaces F C 0,0 N and (F N ) C 0 0 have co-dimension 1 in F C 0 N , (ii) follows with (i). To see (iii), let β fulfill β(−n) = ±β(n) for all n ∈ F N . Then we obtain for all x ∈ F N : which concludes the proof of the proposition.
Before we start, we shortly introduce the Fourier transform of a pre-weak function with rational poles vanishing in i∞. Let N be an integer. Then we define A simple calculation verifies that the inverse of this isomorphism is given by It is easy to see from Proposition 4.3 that Fourier transforms induce isomorphisms between pre-weak functions vanishing in i∞, that have a removable singularity in z = 0, and weak functions. In particular, they preserve the spaces W 0 weak,1 [T N ] of weak functions with removable singularities in z = 0. To prove Eichler duality we will introduce the following bracket notation which will simplify the Fourier series, that will occur frequently.

Definition 4.4 Let β and γ be functions in
Note that [β ⊗ γ ] k, always represents a holomorphic function on the upper half plane with a zero in τ = i∞.
In the following, we want to find the Fourier expansion of ϑ k (ω ⊗ η; τ ) in the case that η has degree 1. This case is the most important one for most of our applications such as Eichler integrals. One of our main tool is a certain differential equation satisfied by the above introduced Fourier series.

Remark 4.5
From now on, if not defined differently, we assume that if some ω ⊗ η ∈ W pre,∞ ⊗ W pre,∞ is used together with some integer k we have sgn(ω)sgn(η) = (−1) k . Remember that we say that a function ω has signum ±1, if it is even or odd, respectively.
In the applications of this section, we will use the term [β ⊗ γ ] k, (τ ) for functions only referring to F C 0 M and F C 0 N . This means, that if β and γ come from a pre-weak functions of the form we will find it useful to identify β with a N -periodic and γ with a M-periodic function on Z (which is the easier convention for this section), respectively, and put For convergence of the L-term in the case k = 0 see Proposition 1.4. In the case ω(z) = h 0,2 (z) we find for sgn(h 0,2 η) = sgn(η) = (−1) k by symmetry This proves the theorem.
Note that the inverse Fourier transform of 1(x) is given by where δ 0 (x) = 1 if x = 0 (mod Z) and δ 0 (x) = 0 for all other values x ∈ 1 N Z/Z. So we can also write The work we have done so far now provides  Proposition 3.5). Then, the following identity is valid on the upper half plane: Proof Starting with an expression ω = ω 0 + a−1 j=1 ∂ j−1 (∂ω j + c j h 0,2 ) with ω j ∈ W N , we obtain by Lemma 3.8 (i). By Lemma 3.7 we know that sgn(ω j ) = (−1) j sgn(ω) and c j = 0 if (−1) j = sgn(ω). From this it follows sgn(ω j η) = (−1) j sgn(ωη) = (−1) j+k = (−1) k−j and if c j = 0, we necessarily have (−1) −j+1 = sgn(ω) = (−1) k sgn(η). We conclude for this case It follows that in both (relevant) cases we are allowed to apply Proposition 4.7 (note that ω 0 (i∞) = 0), so this simplifies to Note that we have added formally the vanishing c j as well. With Lemma 4.6 we conclude that this equals to where δ 0 (x) = 1 if x ∈ Z and 0 else, and finally with Proposition 3.5, this equals to Hence the theorem is proved.
The next lemma imitates a classical result by Bol, see [3].
Note that the choice of k − 1 is crucial for this kind of formula.
Proof This can be followed by direct calculation and for the convenience of the reader we provide the details.
This proves the claim.
We apply the results to obtain a formula for multi-fold integrals of functions ϑ k in terms of functions ϑ j with j ∈ Z.
Proof First of all, Theorem 4.8 gives us Now with Lemma 4.9 we conclude By Proposition 4.3 (iii) we know that Fourier transforms preserve odd and even functional relations, i.e., the signum of the function. Together with Lemma 3.6 we conclude, that This proves the claim.

Remark 4.11
Note that we can weaken the condition η ∈ W i∞,0 pre,1 [T M ] to η ∈ W i∞ pre,1 [T M ] in Theorem 4.10, if ω ∈ W weak,a [T N ] such that 1 a k − 2. The reason for this is that if 1 a k − 2 it follows that 2 − k + j −1 for all 0 j a − 1, which allows F −1 M η to be a pre-weak function in the last step according to Proposition 4.7.

Definition 4.12 Let k
3 be an integer.
With these tools we are able to prove the main result on Eichler duality.
The other way round we see for a ⊗ b ∈ W i∞ pre, 1 (ii) We prove that the compositions give the same output. Let ω ⊗ η ∈ W weak,a [T N ] ⊗ W i∞ pre,1 [T M ] and ω be given by Since 1 a k − 2, according to Theorem 4.10 and Remark 4.11 we have that On the other hand, we find This proves the theorem.
Before we apply this to several situations we first recall basic facts about Eichler integrals.
Proposition 4.14 Let f : H → C be a holomorphic function with the following properties: (i) f is periodic and has a Fourier expansion of the form f (τ ) = ∞ n=1 a f (n)q n/λ with some λ > 0. (ii) There is an integer k 2 and a dual function f * with a Fourier expansion f * (τ ) = ∞ n=1 a f * (n)q n/λ * with λ * > 0, such that f (−1/τ ) = τ k f * (τ ). We can easily apply this formalism to the simplest case of Eisenstein series. It is wellknown that if χ and ψ are primitive Dirichlet characters modulo N χ > 1 and N ψ > 1 and f (τ ) = E k (χ, ψ; τ ) the corresponding L-function is We assume that both χ and ψ are non-principal. From identity (0.2) and Theorem 4.13 we obtain where F (τ ) = i∞ τ E k (χ, ψ; z)(z − τ ) k−2 dz, this combines to In the sense of Proposition 4.14 we have E * k (χ, ψ; τ ) = χ(−1)E k (ψ, χ; τ ) and this provides since χ(−1) 2 = 1. Now according to Proposition 4.14 we have the functional equation On the other hand, from Theorem 1.5, we obtain And this concludes the following theorem.

Remark 4.16
Note that Theorem 4.15 will initially work for k = 2 as well, since the considered characters all satisfy χ(0) = ψ(0) = 0, so that we may assume that η is weak and has a removable singularity in z = 0 in the proof of Eichler duality. So we are allowed to fulfill step (4.1) even if k − 2 < a.
Finally, we give a numerical example.

Authors' contributions
The author is very grateful to Winfried Kohnen and the anonymous referees for many helpful comments that strongly improved the paper.
Funding Open Access funding enabled and organized by Projekt DEAL.