aa r X i v : . [ m a t h . N T ] M a r Multiple Dedekind Zeta Functions
Ivan Horozov ∗ Department of Mathematics,Washington University in St. Louis,One Brookings Drive, Campus box 1146Saint Louis, MO 63130, USA
Abstract
In this paper we define multiple Dedekind zeta values (MDZV), using a new typeof iterated integrals, called iterated integrals on a membrane. One should considerMDZV as a number theoretic generalization of Euler’s multiple zeta values. Overimaginary quadratic fields MDZV capture, in particular, multiple Eisenstein series[GKZ]. We give an analogue of multiple Eisenstein series over real quadratic field andan alternative definition of values of multiple Eisenstein-Kronecker series [G2]. Eachof them is a special case of multiple Dedekind zeta values. MDZV are interpolatedinto functions that we call multiple Dedekind zeta functions (MDZF). We show thatMDZF have integral representation, can be written as infinite sum, and have analyticcontinuation. We compute explicitly the value of a multiple residue of certain MDZFover a quadratic number field at the point (1 , , , Contents ∗ E-mail: [email protected] Analytic properties and special values 30
Multiple Dedekind zeta functions generalize Dedekind zeta functions in the same waythe multiple zeta functions generalize the Riemann zeta function. Let us recall knowndefinitions of the above functions. The Riemann zeta function is defined as ζ ( s ) = X n> n s , where n is an integer. Multiple zeta functions are defined as ζ ( s , . . . , s m ) = X In Subsection 1.1, we recall definitions of multiple zeta values and of polylogarithmsby giving many explicit formulas. In Subsection 1.2, we generalize the previous formulasto multiple Dedekind zeta values over the Gaussian integers via many examples.In Section 2.1, we give two Definitions of iterated integrals on a membrane. Thefirst definition is more intuitive. It can be used to generalize the first few formulas forMDZV over the Gaussian integers from Subsection 1.2. The second Definition is theone needed for the definition of multiple Dedekind zeta values. It is needed in order toexpress special values of the multiple Eisenstein series via MDZV, when the modularparameter has a value in an imaginary quadratic field.In Section 2.2, we use some basic algebraic number theory (see [IR]), in order toconstruct the functions that we integrate. We use an idea of Shintani (see [Sh], [C])for defining a cone. We associate a product of geometric series to every unimodularsimple cone. This is the type of functions that we integrate. Lemma 2.15 shows thata fundamental domain for the non-zero integer O K − { } modulo the units U K can bewritten as a finite union of unimodular simple cones.In Section 3, we define Dedekind polylogarithms associated to a positive unimod-ular simple cone. Theorem 3.2 expresses Dedekind zeta values in terms of Dedekindpolylogarithms. The heart of the section is Definition 3.4 of multiple Dedekind zetavalues (MDZV) as an iterated integral over a membrane and Definition 3.6 of multipleDedekind zeta functions (MDZF) in terms of an integral representation. Theorems 3.5and 3.7 express MDZV and MDZF as an infinite sum. At the end of the Section 3,we give many examples. Examples 1 and 2 are the simplest multiple Dedekind zetavalues. Example 3 expresses partial Eisenstein-Kronecker series associated to an imag-inary quadratic ring as multiple Dedekind zeta values (see [G2], section 8.1). Example4 considers multiple Eisenstein-Kronecker series (for an alternative definition see [G2],Section 8.2). Examples 5 give the simplest multiple Dedekind zeta functions. Example6 is a double Dedekind zeta function.In Section 4, we prove an analytic continuation of multiple Dedekind zeta functions,which allows us to consider special values of multiple Eisenstein series, examined byGangl, Kaneko and Zagier (see [GKZ]), as values of multiple Dedekind zeta functions,(see Examples 7, 8, 9 in Subsection 4.1). Examples 10 and 11 are particular cases3f analytic continuation and of a multiple residue at (1 , . . . , We are going to present several examples of Riemann zeta values and multiple zeta valuesin order to introduce key examples of multiple Dedekind zeta value as iterated integrals.Instead of considering the ring of integers in a general number field, which we will do inthe later sections, we will examine only the ring of Gaussian integers. Also, here we willignore questions about convergence. Such questions will be addressed in Subsection 2.2. Let us recall the m -th polylogarithm and its relation to Riemann zeta values.If the first polylogarithm is defined as Li ( x ) = Z x dx − x = Z x (1 + x + x + . . . ) dx = x + x x . . . and the second polylogarithm is Li ( x ) = Z x Li ( x ) dx x = x + x + x + . . . (Note that ζ (2) = Li (1)), then the m -th polylogarithm is defined by iteration Li m ( x m ) = Z x m Li m − ( x m − ) dx m − x m − . (1.3)This is a presentation of the m -th polylogarithm as an iterated integral. By a directcomputation it follows that Li m ( x ) = x + x m + x m + . . . and the relation ζ ( m ) = Li m (1)is straightforward. Using Equation 1.3, we can express the m -th polylogarithm as Li m ( x m ) = Z 2) = ∞ X n ,n =1 n ( n + n ) = Z Li , ( x ) dx x . Thus, an integral representation of ζ (1 , 2) is ζ (1 , 2) = Z t >t >t > dt ( e t − ∧ dt ( e t − ∧ dt . (1.7)Similarly, ζ (2 , 2) = Z t >t >t >t > dt ( e t − ∧ dt ∧ dt ( e t − ∧ dt . (1.8)5 .2 Dedekind polylogarithms over the Gaussian integers In this Subsection, we are going to construct analogues of polylogarithms (and of somemultiple polylogarithms), which we call Dedekind (multiple) polyologarithms over theGaussian integers. We will denote by f m the m -th Dedekind polylogarithm, which willbe an analogue the m -th polylogarithm Li m ( e − t ) with an exponential variable. Each ofthe analogues will have an integral representation, resembling an iterated integral andan infinite sum representation, resembling the classical Dedekind zeta values over theGaussian integers. We also draw diagrams that represent integrals in order to give ageometric view of the iterated integrals on membranes in dimension 2. We will giveexamples of multiple Dedekind zeta values (MDZV) over the Gaussian integers, usingthe Dedekind (multiple) polylogarithms.We are going to generalize Equations (1.5) and (1.6) for (multiple) polylogarithms totheir analogue over the Gaussian integers. We will recall some properties and definitionsrelated to Gaussian integers. For more information one may consider [IR].By Gaussian integers we mean all numbers of the form a + ib , where a and b areintegers and i = √− 1. The ring of Gaussian integers is denoted by Z [ i ]. We call thefollowing set C a cone C = N { i, − i } = { α ∈ Z [ i ] | α = a (1 + i ) + b (1 − i ); a, b ∈ N } , where N denotes the positive integers. Note that 0 does not belong to the cone C , sincethe coefficients a and b are positive integers. We are going to use two sequences ofinequalities t > u > v > w and t > u > v > w , when we deal with a small number of iterations. The reason for introducing them isto make the examples easier to follow. However, for generalizations to higher order ofiteration we will use the following notation for the two sequences t , > t , > t , > t , and t , > t , > t , > t , . We are going to define a function f , which will be an analogue of Li ( e − t ). Let f ( C ; t , t ) = X α ∈ C exp( − αt − αt ) . (1.9) f ( C, u , u ) = Z u ∞ Z u ∞ f ( C ; t , t ) dt ∧ dt . We can draw the following diagram for the integral representing f .+ ∞ t u + ∞ t u f dt ∧ dt f ( C ; t , t ) dt ∧ dt , depending on thevariables t and t , subject to the restrictions + ∞ > t > u and + ∞ > t > u .We need the following: Lemma 1.1 (a) Z u ∞ e − kt dt = e − ku k ; (b) Let N ( α ) = αα. Then Z u ∞ Z u ∞ exp( − αt − αt ) dt ∧ dt = exp( − αu − αu ) N ( α ) . The proof is straight forward.Using the above Lemma, we obtain f ( C ; u , u ) = X α ∈ C exp( − αu − αu ) N ( α )We define a Dedekind dilogarithm f by f ( C ; v , v ) = Z v ∞ Z v ∞ f ( C ; u , u ) du ∧ du == Z t >u >v ; t >u >v f ( C ; t , t ) dt ∧ dt ∧ du ∧ du (1.10)We can associate a diagram to the integral representation of the Dedekind dilogarithm f (see Equation (1.10)). + ∞ t u v + ∞ t u v f dt ∧ dt du ∧ du The diagram represents that the variables under the integral are t , t , u , u , subjectto the conditions + ∞ > t > u > v and + ∞ > t > u > v . Also, the function f inthe diagram depends on the variables t and t .Similarly to Equation (1.3), we define inductively the m -th Dedekind polylogarithmover the Gaussian integers f m ( C ; t ,m , t ,m ) = Z t ,m ∞ Z t ,m ∞ f m − ( C ; t ,m − , t ,m − ) dt ,m − ∧ dt ,m − , (1.11)7here t , > t , > · · · > t ,m − > t ,m and t , > t , > · · · > t ,m − > t ,m . The aboveintegral is the key example of an iterated integral over a membrane, which is the topicof Subsection 2.1.From Equation (1.11), we can derive an analogue of the infinite sum representationof a polylogarithm (see Equation (1.5)). f m ( C ; t ,m , t ,m ) = X α ∈ C exp( − αt ,m − αt ,m ) N ( α ) m . (1.12)The above Equation gives an infinite sum representation of the m -th Dedekind polylog-arithm over the Gaussian integers.We derive one relation between the Dedekind m -polylogarithm f m , a Dedekind zetavalue over the Gaussian integers and a Riemann zeta value. For arithmetic over theGaussian integers one can consider [IR]. Lemma 1.2 For the Dedekind polylogarithm f m , associated to the above cone C , wehave f m ( C ; 0 , 0) = 2 − m ( ζ Q ( i ) ( m ) − ζ (2 m )) , where ζ Q ( i ) ( m ) is a Dedekind zeta value and ζ (2 m ) is a Riemann zeta value.Proof. We are going to prove the following equalities, which give the lemma. f m ( C ; 0 , 0) = X α ∈ C N ( α ) m = 2 − m X ( α ) =(0) ⊂ Z [ i ] − X α ∈ N N (( α )) m == 2 − m ( ζ Q ( i ) ( m ) − ζ (2 m )) , (1.13)The first equality follows from (1.12). The second and the third equalities relate ourintegral to classical zeta values. The second equality uses two facts: (1) for the Gaussianintegers the norm of an element α , N ( α ), is equal to the norm of the principal idealgenerated by α , denoted by N (( α )), namely N ( α ) = N (( α )). Recall that for the Gaussianintegers the norm of an element α , is N ( α ) = αα , and the norm of a principal ideal N (( α )) is equal to the number of elements in the quotient module N (( α )) = | Z [ i ] / ( α ) | , where ( α ) = α Z [ i ] = { µ ∈ Z [ i ] | µ = αβ for some β ∈ Z [ i ] } is view as a Z [ i ]-submodule of Z [ i ]. (2) the set of non-zero principal ideals can beparametrized by the non-zero integers modulo the units. Since the units are ± , ± i ,we have that ( α ) ⊂ Z [ i ], ( α ) = (0) can be parametrized by elements of the Gaussianintegers with positive real part and non-negative imaginary part, which we will denoteby C . Multiplying each element of C by 1 − i , we obtain the union of the cone C andthe set { a + ai | a ∈ N } . Summing over C gives the Dedekind zeta value. Summingover (1 − i ) C gives 2 − mζ Q ( i ) ( m ). Such a sum can be separated to a sum over C , whichcontributes f m and a sum over the set { a + ai | a ∈ N } , which gives 2 − m ζ (2 m ). ✷ Li , (1 , e − t ) over the Gaussianintegers, using the following integral representation f , ( C ; v , v ) = Z v ∞ Z v ∞ f ( C ; u , u ) f ( C ; u , u ) du ∧ du , called a Dedekind double logarithm. Such an integral will be considered as an exampleof an iterated integral over a membrane in Subsection 2.1. As an analog for Equation(1.10), we can express f , only in terms of f by f , ( C ; v , v ) = Z t >u >v ; t >u >v ( f ( C ; t , t ) dt ∧ dt ) ∧ ( f ( u , u ) du ∧ du ) . It allows us to associate a diagram to the Dedekind double logarithm f , :+ ∞ t u v + ∞ t u v f dt ∧ dt f du ∧ du The variables t , t , u , u in the diagram are variables in the integrant. They are subjectto the conditions t > u > v and t > u > v . Also, the lower left function f in thediagram depends on the variables t and t and the upper right function f depends on u and u .The similarity between f , ( C ; v , v ) and Li , (1 , e − t ) can be noticed by the infinitesum representation in the following: Lemma 1.3 f , ( C ; v , v ) = X α,β ∈ C exp( − ( α + β ) v − ( α + β ) v ) N ( α ) N ( α + β ) . Proof. f , ( C ; v , v ) = Z v ∞ Z v ∞ f ( C ; u , u ) f ( C ; u , u ) du ∧ du == Z v ∞ Z v ∞ X α ∈ C exp( − αu − αu ) N ( α ) X β ∈ C exp( − βu − βu ) du ∧ du == Z v ∞ Z v ∞ X α,β ∈ C exp( − ( α + β ) u − ( α + β ) u ) N ( α ) du ∧ du == X α,β ∈ C exp( − ( α + β ) v − ( α + β ) v ) N ( α ) N ( α + β ) . f , , we define a multiple Dedekind polylog-arithm f , ( C, w , w ) = Z w ∞ Z w ∞ f , ( C ; v , v ) dv ∧ dv . We can associate the following diagram to the multiple Dedekind polylogarithm f , + ∞ t u v w + ∞ t u v w f dt ∧ dt f du ∧ du dv ∧ dv The diagram represents the following: The variables of the integrant are t , t , u , u , v , v . The variables are subject to the conditions t > u > v > w and t > u > v > w .The lower left function f depends on the variables t and t . And the middle function f depends on u and u . Thus, the diagram represents the following integral: f , ( C ; w , w ) = (1.14)= Z D w ,w ( f ( C ; t , t ) dt ∧ dt ) ∧ ( f ( C ; u , u ) du ∧ du ) ∧ ( dv ∧ dv ) , where the domain of integration is D w ,w = { ( t , t , u , u , v , v ) ∈ R | t > u > v > w and t > u > v > w } A direct computation leads to f , ( C ; w , w ) = X α,β ∈ C exp( − ( α + β ) w − ( α + β ) w ) N ( α ) N ( α + β ) . We define a multiple Dedekind zeta value as ζ Q ( i ); C (1 , 1; 2 , 2) = f , ( C ; 0 , 0) = X α,β ∈ C N ( α ) N ( α + β ) . Now let us give a relation between multiple Dedekind zeta values and iterated inte-grals. We use the following pair of inequalities in the following Sections t , > t , > , and t , > t , > t , , instead of t > u > v > w and t > u > v > w , sinceusing such notation it is easier to write higher order iterated integrals. In this notation,from Equation (1.11) , we obtain f ( C ; t , , t , ) = (1.15)= Z t , >t , >t , ; t , >t , >t , ( f ( C ; t , , t , ) dt , ∧ dt , ) ∧ ( dt , ∧ dt , ) . and f , ( C ; t , , t , ) = (1.16)= Z t , >t , >t , ; t , >t , >t , ( f ( C ; t , , t , ) dt , ∧ dt , ) ∧ ( f ( C ; t , , t , ) dt , ∧ dt , ) . In the next Section, we generalize the (iterated) integrals appearing in Equations(1.11) (1.14), (1.15), and (1.16), called iterated integrals over a membrane, (see Definition2.1).The next two Examples are needed in order to relate multiple Dedekind zeta valuesto values of Eisenstein series and values of multiple Eisenstein series (see [GKZ], see alsoExamples 7, 8, 9 at the end of Section 3). The integrals below will present another typeof iterated integral on a membrane, (see Definition 2.3) leading to a multiple Dedekindzeta values.We define the following iterated integral to be a multiple Dedekind zeta value ζ Q ( i ); C (3; 2) = Z t >u >v > t >u > f ( C ; t , t ) dt ∧ du ∧ dv ∧ dt ∧ du . (1.17)The reason for such a definition is its infinite sum representation ζ Q ( i ); C (3; 2) = X α ∈ C α α , (1.18)which can be achieved essentially in the same way as for the other multiple Dedekindzeta values. We can associate the following diagram to the integral representation of ζ Q ( i ); C (3; 2) in Equation (1.17).+ ∞ t u v ∞ t u f dt ∧ dt du ↑ → du → dv f ( C ; t , t ) dt followedby du and dv , The integration with respect to the variables t , u , v leads to 1 /α in the summation of Equation (1.18). In vertical direction, we have a double iteration.First we have f ( C ; t , t ) dt followed by u . That leads to 1 /α in the summation inEquation (1.18).Diagrams associated to the integral representation of ζ Q ( i ); C (3; 2) (Equations (1.17))are not unique. Alternatively, we could have used the diagrams+ ∞ t u v ∞ t u f dt ∧ dt du ↑→ du → dv or + ∞ t u v ∞ t u f dt ∧ dt → du ∧ du ↑ dv → Consider the following diagram, associated to a more complicated MDZV, for thepurpose of establishing notation. 12 ∞ t u v w ∞ t u v f dt ∧ dt dv ↑ du → f dv ∧ du dw → The diagram encodes that + ∞ > t > u > v > w > ∞ > t > u > v > f dt ,followed by du , f dv and dw . That gives an analogue of ζ ( a, c ) = ζ (2 , 2) in horizontaldirection for ( a, c ) = (2 , f dt , followed by f du and dv . That gives an analogue of ζ ( b, d ) = ζ (1 , 2) in vertical direction, for ( b, d ) = (1 , ζ Q ( i ); C,C ( a, b ; c, d ) = ζ Q ( i ); C,C (2 , 1; 2 , ζ Q ( i ); C,C (2 , 1; 2 , 2) = X α,β ∈ C α α ( α + β ) ( α + β ) . In Section 3, we use iterated integrals over a membrane to define multiple Dedekindzeta values associated to any number field. Let D be a domain defined in terms of the real variables t i,j for i = 1 , . . . , n and j =1 , . . . , m , by D = { ( t , , . . . , t n,m ) ∈ R nm | t i, > t i, > · · · > t i,m > i = 1 , . . . , n } . For each j = 1 , . . . , m , let ω j be a differential n -form on C n . Let g : (0 , + ∞ ) n → C n 13e a smooth map, whose pull-back sends the coordinate-wise foliation on C n to acoordinate-wise foliation on (0 , + ∞ ) n . We will call such a map a membrane . Oneshould think of the n -forms g ∗ ω j as an analogue of f ( C ; t , t ) dt ∧ dt from Equation(1.9). Definition 2.1 An iterated integral on a membrane g , in terms of n -forms ω j , j =1 , . . . , m , is defined as Z g ω . . . ω m = Z D m ^ j =1 g ∗ ω j ( t ,j , . . . , t n,j ) . (2.19) Definition 2.2 A shuffle between two ordered sets S = { , . . . , p } and S = { p + 1 , . . . , p + q } is a permutation τ of the union S ∪ S , such that1. for a, b ∈ S , we have τ ( a ) < τ ( b ) if a < b ;2. for a, b ∈ S , we have τ ( a ) < τ ( b ) if a < b ;We denote the set of shuffles between two ordered sets of orders p and q , respectively, by Sh ( p, q ) . The definition of an iterated integral on a membrane is associated with the followingobjects:1. g : (0 , + ∞ ) n → C n , a membrane (that is a smooth map, whose pull-back sends thecoordinate-wise foliation on C n to a coordinate-wise foliation on (0 , + ∞ ) n ).2. ω , . . . , ω m differential n -forms on C n ;3. m i copies of differential 1-forms d z i on C n , for i = 1 , . . . , n ;4. a shuffle τ i ∈ Sh ( m, m i ) for each i = 1 , . . . , n ;5. τ = ( τ , . . . , τ n ), the set of n shuffles τ , . . . , τ n . Definition 2.3 Given the above data, we define an iterated integral on a membrane g ,involving n -forms and -forms, as Z ( g,τ ) ω . . . ω m (d z ) m . . . (d z n ) m n == Z D m Y j =1 g ∗ ω j ( t ,τ ( j ) , . . . , t n,τ n ( j ) ) n ^ i =1 m + m i ^ j =1 g ∗ dz i,j , (2.20) where t i,j = g ∗ z i,j and also t i,j belong to the domain D = { ( t , , . . . , t n,m ) ∈ R mn | t i, > t i, > · · · > t i,m + m i > } . emark: Comparing the Definitions 2.1 and 2.3, one can notice that there is nosign occurring. The reason for that is the following:1. In Definition 2.1 we use a domain D , whose coordinates are ordered by t , , . . . , t n, , t , , . . . , t n, , . . . , t ,m , . . . , t n,m . It is the same as the order of the differential 1-formsunder the integral in Equation (2.19).2. In Definition 2.3 we use a domain D , whose coordinates are ordered by t , , . . . , t ,m + m ,t , , . . . , t ,m + m , . . . , t n, , . . . , t n,m + m n . It is the same as the order of the differential1-forms under the integral in Equation (2.20).Thus, if m = · · · = m n = 0, both definitions lead to the same value, since the per-mutation of the differential forms coincides with the permutation of the coordinates ofthe domain of integration. Thus, the change of orientation of the domain of integrationcoincides with the sign of permutation acting on the differential forms. Theorem 2.4 (homotopy invariance) The iterated integrals on membranes from Def-inition 2.3 are homotopy invariant, when the homotopy preserves the boundary of themembrane.Proof. Let g be a homotopy between the two membranes g and g . LetΩ = m Y j =1 ω j ( z ,τ ( j ) , . . . , z n,τ n ( j ) ) n ^ i =1 m + m i ^ j =1 dz i,j Note that Ω is a closed form, since ω i is a form of top dimension and since dz i,j isclosed. By Stokes Theorem, we have0 = Z s =1 s =0 Z D g ∗ d Ω == Z ( g ,τ ) Ω − Z ( g ,τ ) Ω ± (2.21) ± Z s =1 s =0 n X i =1 m + m i X j =1 Z D | ( z i,j = z i,j +1 ) g ∗ Ω ± (2.22) ± Z s =1 s =0 n X i =1 Z D | ( z i,m + mi =0) g ∗ Ω (2.23)We want to show that the difference in the terms in (2.21) is zero. It is enough to showthat each of the terms (2.22) and (2.23) are zero. If z i,j = z i,j +1 , then the wedge ofthe corresponding differential forms will vanish. Thus the terms in (2.22) are zero. If z i,m + m i = 0 then dt i,m + m i = 0, defined via the pull-back g ∗ . Then the terms (2.23) areequal to zero. ✷ .2 Cones and geometric series Let n = [ K : Q ] be the degree of the number field K over Q . Let O K be the ring ofintegers in K . And let U k be the group of units in K . We are going to use an idea ofShintani [C] by examining Dedekind zeta functions in terms of a cone inside the ring ofintegers.We define cone for any number ring. The meaning of cones is roughly the following:summation over the elements of finitely many cones would give multiple Dedekind zetavalues or multiple Dedekind zeta functions. Definition 2.5 We define a cone C to be C = N { e , . . . , e k } = { α ∈ O K | α = a e + · · · a k e k for e i ∈ O K and a i ∈ N } with generators e , . . . , e k . For the next definition, we are going to use that a number field K can be viewed asan n -dimensional vector space over the rational numbers Q . Definition 2.6 An unimodular cone is a cone with generators e , . . . , e k such that e , . . . , e k as elements of K are linearly independent over Q , when we view the field K as a vectorspace over Q . Note that if C is an unimodular cone then 0 / ∈ C , since e , . . . , e k are linearly inde-pendent over Q and the coefficients a , . . . , a n are positive integers. Definition 2.7 We call C an unimodular simple cone if for any embedding σ i of K intothe complex numbers, σ i : K → C and a suitable branch of the functions arg( z ) , we havethat the closure of the set arg( σ ( C )) is an interval [ θ , θ ] , such that its lengths is lessthan π , namely, θ − θ ∈ [0 , π ) . In particular, the cone C = { α ∈ Z [ i ] | α = a (1 + i ) + b (1 + i ) , a, b ∈ N } , considered in Subsection 1.2, is an unimodular simple cone, since arg( σ ( α )) ∈ ( − π/ , π/ σ ( α )) ∈ ( − π/ , π/ α ∈ C . The maps σ and σ are complex conju-gates of each other. Definition 2.8 (Dual cone) For an unimodular simple cone C with generators e , . . . , e k ,we define a dual cone of C to be C ∗ = { ( z , . . . , z n ) ∈ C n | Re ( z i σ i ( e j )) > for i = 1 , . . . , n, and j = 1 . . . , k } Clearly, if C is an unimodular simple cone then the dual cone C ∗ is a non-empty set.One can prove that by considering each coordinate of C ∗ , separately.16 efinition 2.9 For an unimodular simple cone C , we define a function f ( C ; z , . . . , z n ) = X α ∈ C exp( − n X i =1 σ i ( α ) z i ) , (2.24) where σ , . . . , σ n are all embeddings of the number field K into the complex numbers C and the domain of the function f is the dual cone C ∗ . Lemma 2.10 The function f is uniformly convergent for ( z , . . . , z n ) in any compactsubset B of the dual cone C ∗ of an unimodular simple cone C .Proof. From the Definition 2.8, we have Re( σ i ( e j ) z i ) > 0. Let y j = n Y i =1 exp( − σ i ( e j ) z i ) . (2.25)Then | y j | < B . Moreover, | y i | achieves a maximum on the compactsubset B . Let | y j | ≤ c j < B for some constant c j . Then the rate ofconvergence of the geometric sequence in y j is uniformly bounded by c j on the compactset B . Therefore, we have a uniform convergence of the geometric series in y j . Thefunction f ( C ; z , . . . , z n ) is a product of k geometric series in the variables y , . . . , y k each of which is uniformly bounded in absolute value by the constants c , . . . , c k on thedomain B , respectively. Then, we obtain that f ( C ; z , . . . , z n ) = k Y j =1 y j − y j . (2.26) ✷ Corollary 2.11 The function f ( C ; z , . . . , z n ) has analytic continuation to all valuesof z , . . . , z n , except at n X i =1 σ i ( e j ) z i ∈ πi Z , for j = 1 , . . . , k. Proof. Using the product formula 2.26 in terms of geometric series in y j , we see that theright hand side of 2.26 makes sense for all y j = 1. This gives analytic continuation fromthe domain C ∗ to the domain consisting of points ( y , . . . , y n ) with y i = 1. ✷ Definition 2.12 (positive cone) We call C a positive unimodular simple cone if C is anunimodular simple cone and the product of the positive real coordinates is in C ∗ , namely ( R > ) n ⊂ C ∗ as subsets of C n . emma 2.13 If C is an unimodular simple cone then for some α ∈ O K we have that αC = { αβ | β ∈ C } is a positive unimodular simple cone.Proof. In order to find such an elements α , we need to recall properties of real or complexembeddings of a number field K .The degree of a number field n = [ K : Q ] is the dimension of K as vector space over Q . Then there are exactly n distinct embeddings K → C . Let the first r embeddings, σ , . . . , σ r , be the ones whose image is inside the real numbers. They are called realembeddings. Let the next r embeddings be complex embeddings, which are not pair-wisecomplex conjugates of each other. Let us denote them by σ r +1 , . . . , σ r + r . Let the last r embeddings be the complex conjugates of previously counted complex embeddings,namely, σ r + r + i ( β ) = σ r + i ( β ) , for i = 1 , . . . , r . We also have that n = r + 2 r .Let V R be a n dimensional real vector subspace of C n defined in the following way: V R = { ( z , . . . , z n ) ∈ C n | ( z , . . . , z r ) ∈ R r , ( z r +1 , . . . , z r + r ) ∈ C r , and( z r + r +1 , . . . , z r +2 r ) = ( z r +1 , . . . , z r + r ) } Now, we proceed with the proof of the Lemma in six Steps.Step 1. K is dense in V R .Step 2. V R ∩ C ∗ in non-empty.Step 3. V R ∩ C ∗ is an open subset of V R .Step 4. K ∩ C ∗ is non-empty.Step 5. O K ∩ C ∗ is non-empty.Step 6. αC is a positive unimodular simple cone for any α ∈ O K ∩ C ∗ .(Step 1) Recall also the product of the n embeddings of K to the space V R , n Y i =1 σ i : K → V R , mapping β ∈ K to ( σ ( β ) , . . . , σ n ( β )) ∈ V R has a dense image.(Step 2) Indeed, let z i be the i -th coordinate of C ∗ . The first r coordinates z , . . . , z r of C ∗ can be real numbers (positive or negative), since σ i ( K ) ⊂ R for i = 1 , . . . , r . Thus,the first r i coordinates can be both in V R and in C ∗ . For the coordinates z r +1 , . . . , z r + r of C ∗ there are no restrictions when we intersect C ∗ with V R . For the last r coordinatesof C ∗ we must have that z r + r + i = z r + i in order for the coordinates to be in theintersection C ∗ ∩ V R . Since, σ r + r + i ( β ) = σ r + i ( β ), we have the conditions on the( r + i )-coordinate and on the ( r + r + i )-coordinate of a point in C ∗ to be in V R are Re ( z r + i σ r + i ( β )) > , β ∈ C and z r + r + i = z r + i . The last condition implies that Re ( z r + r + i σ r + r i ( β )) = Re ( z r + i σ r + i ( β )) > . Thus, such a point ( z , . . . , z n ) is in C ∗ ∩ V R .(Step 3) It is true, since C ∗ is an open subsets of C n (Step 4) Since K is dense in V R (Step 1) and V R ∩ C ∗ is open in V R (Steps 2 and 3),we have that K ∩ C ∗ is non-empty.(Step 5) If α ∈ K ∩ C ∗ then for some positive integer L , we have that Lα ∈ O K , andalso, Lα ∈ C ∗ , since C ∗ is invariant under rescaling by a positive (real) number L .(Step 6) Let ( t , . . . , t n ) ∈ R n> and let β ∈ C . Put z i = t i σ i ( α ). Then ( z , . . . , z n ) ∈ C ∗ . Re ( t i σ i ( αβ )) = Re ( t i σ i ( α ) σ i ( β )) = Re ( z i σ i ( β )) > . ✷ In this Subsection, we are going to examine union of cones that give a fundamentaldomain of the ring of integers O K modulo the group of units U K . We also examine afundamental domain of an ideal a modulo the group of units U K . Definition 2.14 We define M as a fundamental domain of O K − { } mod U k . For an ideal a , let M ( a ) = M ∩ a . Lemma 2.15 For any ideal a the set M ( a ) can be written as a finite disjoint union ofunimodular simple cones.Proof. It is a simple observation that M ( a ) can be written as a finite union of unimodularcones. We have to show that we can subdivide each of the unimodular cones into finiteunion of unimodular simple cones.Let σ , . . . , σ r be the real embeddings of the number field K and let σ r +1 , . . . , σ r + r be the non-conjugate complex embeddings of K . We define T = {− , } r × ( S ) r . Let C be an unimodular cone. We define a map J by J : C → Tα (cid:18) σ ( α )) | σ ( α )) | , · · · , σ r + r ( α )) | σ r + r ( α )) | (cid:19) Denote it by ¯ C the closure of the image of J in T . Then one can cut the cone C intofinitely many cones C i such that for C i and any embedding σ of K into C , we have that19rg( σ ( C )) ∈ [ θ , θ ], for θ − θ ∈ [0 , π ). Then C i is an unimodular simple cone. Thus,the cones C i ’s are finitely many unimodular simple cones, whose (disjoint) union givesthe set M ( a ) . ✷ For an element α in a ring of integers O K , denote by ( α ) the principal ideal generatedby α . Then N K/ Q (( α )) denoted the norm of the principal ideal generated by α . We havethat N K/ Q (( α )) is a positive integer equal to the number of elements in the quotient O K / ( α ). Also N K/ Q ( α ) is the norm of the algebraic number α . This is equal to theproduct of all of its Galois conjugates, which is an integer, possibly a negative integer.We always have that N K/ Q (( α )) = | N K/ Q ( α ) | .However, for elements of an unimodular simple cone, we can say more. Lemma 2.16 Let C be an unimodular simple cone. Then for every α ∈ C , we have N K/ Q (( α )) = ǫ ( C ) N K/ Q ( α ) , where ǫ ( C ) = ± depends only on the cone C , not on α .Proof. Note that on the left we have a norm of an ideal and on the right we have a normof a number. Since C is a simple cone, we have that for all real embeddings σ : K → R ,the signs of σ ( α ) and σ ( β ) are the same for all α and β in C . Let ǫ σ be the sign of σ ( α )for each real embedding σ . Then the product over all real embeddings of ǫ σ is equal to ǫ ( C ). ✷ Let us recall the Dedekind zeta values ζ K ( m ) = X a =(0) N K/ Q ( a ) m , where a is an ideal in O K .We are going to express the summation over elements, which belong to a finite unionof positive unimodular simple cones. We will define a Dedekind polylogarithm associatesto a positive unimodular simple cone. The key result in this subsection will be that aDedekind zeta value can be expressed as a Q linear combination of values of the Dedekindpolylogarithms.We also define a partial Dedekind zeta function by summing over ideals in a givenideal class [ a ] ζ K, [ a ] ( m ) = X b ∈ [ a ] N K/ Q ( b ) − m , Let us consider a partial Dedekind zeta functions ζ K, [ a ] − ( m ), corresponding to anideal class [ a ] − , where a is an integral ideal. For every integral ideal b in the class [ a ] − ,we have that ab = ( α ) , α ∈ a . Then N K/ Q ( b ) = N K/ Q ( a ) − N K/ Q (( α )) . Let M ( a ) = n ( a ) [ i =1 C i ( a ) , where n ( a ) is a positive integer and C i ( a )’s are unimodular simple cones. Let α i be anelement of the intersection of O K with the dual cone C i ( a ) ∗ , then α i C i ( a ) is a positiveunimodular simple cone (see Lemma 2.13).Then, ζ K, [ a ] − ( m ) = X b ∈ [ a ] − N K/ Q ( b ) − m == N K/ Q ( a ) m n ( a ) X i =1 ǫ ( C i ( a )) m N ( α i ) m X α ∈ α i C i ( a ) N K/ Q ( α ) − m , (3.27)where ǫ ( C i ( a )) = ± 1, depending on the cone, N K/ Q ( a ) is a norm of the ideal a and N ( α i )is the norm of the algebraic integer α i .We are going to give an example of higher dimensional iteration in order to illustratethe usefulness of this procedure. For a positive unimodular simple cone C , we define f m ( C ; u , . . . , u n ) = Z u ∞ . . . Z u n ∞ f m − ( C ; t , . . . , t n )d t ∧ · · · ∧ d t n , where t i ∈ ( u i , + ∞ ). This is an iteration, giving the simplest type of iterated integralson a membrane. We start the induction on m from m = 0 . Recall that f was introducedin Definition 2.9.Note that a norm of an algebraic number α can be expresses as a product of itsembeddings in the complex numbers σ ( α ) , . . . , σ n ( α ) .N K/ Q ( α ) = σ ( α ) . . . σ n ( α ) . Integrating term by term, we can express f m as an infinite sum f m ( C ; t , . . . , t n ) = X α ∈ C exp( − P ni =1 σ i ( α ) t i ) N K/ Q ( α ) m . Note that a cone C is a linear combination of its generators so that the coefficients ofthe generators are positive integers. In particular, 0 is not an element of an unimodularsimple cone C , since then the generators are linearly independent over Q . Thus, thereis no division by 0. Definition 3.1 We define an m -th Dedekind polylogarithm, associated to a number field K and a positive unimodular simple cone C , to be Li Km ( C ; X , . . . , X n ) = f m ( C ; − log( X ) , . . . , − log( X n )) . heorem 3.2 Dedekind zeta value at s = m > can be written as a finite Q -linearcombination of Dedekind polylogarithms evaluated at ( X , . . . , X n ) = (1 , . . . , .Proof. If a , . . . , a h are integral ideals in O K , representing all the ideal classes, then usingEquation (3.27), we obtain ζ K ( m ) = h X j =1 N K/ Q ( a j ) m n ( a ) j X i =1 ǫ ( C i ( a j )) m N ( α i,j ) m f m ( α i,j C i ( a j ) , , . . . , , where C i ( a j ) are unimodular simple cones such that n ( a ) j [ i =1 C i ( a j ) = M ( a j )and ǫ ( C i ( a j )) = ± 1, depending on the cone (see Definition 2.14 and Lemma 2.15). Let α i,j ∈ C i ( a j ) ∗ ∩ O K be an algebraic integer in the dual cone of C i ( a j ). Then by Lemma2.13 we have that α i,j C i ( a j ) is a positive unimodular simple cone. The iterated integralsare hidden in the functions f m . Consider Definition 2.1 with differential forms ω = f ( α i,j C i ( a j ); z , . . . , z n )d z ∧ · · · ∧ d z n ,ω = ω = · · · = ω m = d z ∧ · · · ∧ d z n . And let g be inclusion of (0 , ∞ ) n in C n . Then the corresponding iterated integral on amembrane gives f m ( α i,j C i ( a j ); t , . . . , t n ) . ✷ We recall an integral representation of a multiple zeta value ζ ( k , k , . . . , k m ) = X 1) The relation between the shuffle τ and the set of integers k , . . . , k m is the following: 1 = τ (1) k + 1 = τ (2) k + k + 1 = τ (3) · · · k + · · · + k m − + 1 = τ ( m ) k + · · · + k m − + k m = number of differential 1-formsThe integers 1 , k + 1 , k + k + 1 , . . . , k + · · · + k m − + 1, are the values of the index i , where the analogue of the form dt i / ( e t i − 1) appears under the integral, not the form dt i . (see Equation (3.29))In order to define multiple Dedekind zeta values, we will use n shuffles of pairs ofordered sets, where n = [ K : Q ] is the degree on the number field.Let m , . . . , m n be positive integers. (The positive integer m i will denote the numberof times the differential form d z i occurs.) We define the following ordered sets: S = { , , . . . , m } ,S i = { m + 1 , . . . , m + m i } , Definition 3.3 Denote by Sh ( p, q ) the subset of all shuffles τ ∈ Sh ( p, q ) of the twosets { , . . . , p } and { p + 1 , . . . , p + q } such that τ (1) = 123or the definition of multiple Dedekind zeta values at the positive integers, we useDefinition 2.3, where we take the n -forms to be ω j = f ( C j , z , . . . , z n )d z ∧ · · · ∧ d z n , for j = 1 , . . . , m , where C , . . . , C m are positive unimodular simple cones, and the 1-formsto be d z i on C n occurring m i times for i = 1 , . . . , n . Definition 3.4 (Multiple Dedekind zeta values) For each i = 1 , . . . , n , let τ i ∈ Sh ( m, m i ) .We define the integers k i,j and m i in terms of the shuffle τ i via the following relations τ i (1) (3.30) k i, + 1 = τ i (2) (3.31) k i, + k i, + 1 = τ i (3) (3.32) · · · k i, + · · · + k i,m − + 1 = τ i ( m ) (3.33) k i, + · · · + k i,m − + k i,m = m + m i (3.34) We define multiple Dedekind zeta values at the positive integers by ζ K ; C ,...C m ( k , , . . . , k ,m ; . . . ; k n, , . . . , k n,m ) = Z ( g,τ ) ω . . . ω m (d z ) m . . . (d z n ) m n Theorem 3.5 For the general form of a multiple Dedekind zeta value, we need: a num-ber field K ; positive unimodular simple cones C , . . . , C m in O K ; elements α j ∈ C j for j = 1 , . . . , m ; complex embeddings of the elements α i,j = σ i ( α j ) . Then a multipleDedekind zeta value has the following representation as an infinite sum ζ K ; C ,...C m ( k , , . . . , k ,m ; . . . ; k n, , . . . , k n,m ) == X α ∈ C · · · X α m ∈ C m n Y i =1 m Y j =1 ( α i, + · · · + α i,j ) − k i,j . (3.35) Proof. There are n different embedding σ , . . . , σ n of K into C . Given k i, , . . . , k i,m wefind m i using Equation (3.34). Then we find τ i by the values at 1 , , . . . , m obtainedfrom Equations (3.31), (3.32), (3.33).Now we use Definition 3.4 of a multiple Dedekind zeta value in terms of iteratedintegrals on a membrane from Definition 2.3. We are going to follow closely Equation(3.29). The variable t , enters as a variable of the function f ( C ; · · · ), the variables t , , . . . , t ,k , appear as differential 1-forms dt , , . . . , dt ,k , since τ (2) = k , + 1 (seeEquation (3.31)). Recall that σ : K → C is an embedding of K into the complexnumbers and α ,j = σ ( α j ). Thus integrating with respect to t , , . . . , t ,k , gives usdenominators α k , , , associated to each α ∈ C Then t ,k , +1 enters as a variable in thefunction f ( C ; · · · ), since τ (2) = k , + 1. Then the variables t ,k , +2 , . . . , t ,k , + k , appear as differential 1-forms dt ,k , +2 , . . . , dt ,k , + k , , since τ (3) = k , + k , + 1 (seeEquation (3.32)). Thus integrating with respect to t , , . . . , t ,k , gives us a denominators24 k , , ( α , + α , ) k , , associated to each α ∈ C and each α ∈ C . Continuing thisprocess to the variable t ,m + m , we obtain a denominator m Y j =1 ( α , + · · · + α ,j ) k ,j , associated to each m -tuple ( α , . . . , α m ), where α j ∈ C j . There are n different embed-dings σ , . . . , σ n of K into C , where n = [ K : Q ] is the degree of the number field. Sofar we have considered the contribution of the first embedding. The contribution of thefirst and the second embedding is obtained in essentially the same way as for the firstembedding. It gives a denominator m Y j =1 ( α , + · · · + α ,j ) k ,j ( α , + · · · + α ,j ) k ,j , associated to each m -tuple ( α , . . . , α m ), where α j ∈ C j . Similarly, after integrating withrespect to all the variables t i,j we obtain a denominator n Y i =1 m Y j =1 ( α i, + · · · + α i,j ) k i,j , associated to each m -tuple ( α , . . . , α m ), where α j ∈ C j . Then the numerators are allequal to 1 since the lower bound for the variables under the exponents in f ( C j ; · · · ) is0. Thus, the exponents become equal to 1.The following examples of MDZV give analogues of Dedekind zeta function and of(multiple) Eisenstein-Kronecker series. Examples: 1. Let C be a positive unimodular simple cone in the ring of integers O K of a number filed K . In m = 1 and if all values k i, are equal to k , then ζ K ; C ( k, . . . , k ) = X α ∈ C N K/ Q ( α ) k . Note that the number 0 does not belong to any unimodular simple cone C .2. Let m = 2, and let k = k , = · · · = k n, and l = k , = · · · = k n, be positive integers greater that 1. Finally, let C and C be positive unimodular simplecones in the ring of integers O K of a number field K . Then the corresponding multipleDedekind zeta value can be written both as a sum and as an integral: ζ K ; C ,C ( k, . . . , k ; l, . . . , l ) = X α ∈ C ,β ∈ C N K/ Q ( α ) k N K/ Q ( α + β ) l (3.36)25. Let K be an imaginary quadratic field. Let C be a positive unimodular simplecone in O K . We can represent the cone C as an N -module: C = N { µ, ν } , for µ, ν ∈ O K .Put z = µ/ν . Consider then ζ K ; C ( k, k ) = X α ∈ C N ( α ) k = | ν | − k X a,b ∈ N | az + b | k , where the last sum is a portion of the k -th Eisenstein-Kronecker series. Such series couldbe found in [W] E k ( z ) = X a,b ∈ Z ;( a,b ) =(0 , | az + b | k . 4. With the notation of Example 3, we obtain an analogue of values of multipleEisenstein-Kronecker series ζ K ; C,C ( k, k ; l, l ) = | ν | − k − l X a,b,c,d ∈ N | az + b | k | ( a + c ) z + ( b + d ) | l . An alternative generalization was considered in [G2], Section 8.2. We will try to give some intuition behind the integral representation of the multiplezeta functions (see in [G1]). After that we will generalize the construction to define thenumber field analogues - multiple Dedekind zeta functions. In order to do that, we givetwo examples - one for ζ (3) and another for ζ (1 , ζ (3) = Z 3) = Z 1) == Z t >t > d t Γ(1)( e t − ∧ t − d t Γ(3)( e t − 1) == Z (0 , ∞ ) u − u − d u ∧ d u Γ(1)Γ(3)( e u + u − e u − . The first two equalities are of the same type as in the previous example. For the thirdequality we use Equation (3.37). For the last equality we use the change of variable t = u ,t = u + u , where u > u > 0. Following [G1], we can interpolate the multiple zeta values by ζ ( s , . . . , s d ) = Γ( s ) − . . . Γ( s d ) − Z (0 , ∞ ) d u s − . . . u s d − d d u ∧ · · · ∧ d u d ( e u + ··· + u d − e u + ··· + u d − . . . ( e u d − . If we denote by f ( N ; t ) = X a ∈ N e − at , then f ( N , t ) = 1 e t − ζ ( s , . . . , s d ) = Γ( s ) − . . . Γ( s d ) − Z (0 , ∞ ) d d ^ j =1 f ( N ; u i + · · · + u d ) u s j − j d u j Let n = [ K : Q ] be the degree of the number field. We recall Definition 2.9 of f , f ( C ; t , t , . . . , t n ) = X α ∈ C e − P ni =1 σ i ( α ) t i , where σ i : K → C run through all embeddings of the field K into the complex numbers.Let C and C be two unimodular simple cones. We want to raise an algebraic integer α to a complex power s as a portion of the multiple Dedekind zeta function. We define α s = e s log( α ) for one element α ∈ C and α = σ ( α ). Choose a branch of the logarithmic function bymaking a cut of the complex plane at the negative real numbers. Since C is a positive27nimodular simple cone we have that R n> ⊂ C ∗ , the function σ i composed with log iswell defined on a positive unimodular simple cone C .Then, we define a double Dedekind zeta function as ζ K ; C ,C ( s , , . . . , s n, ; s , , . . . , s n, ) = (3.38)= Γ( s , ) − . . . Γ( s n, ) − Z (0 , + ∞ ) n f ( C ; ( u , + u , ) , . . . , ( u n, + u n, )) ×× f ( C ; ( u , , . . . , u n, ) n ^ i =1 u s i, − i, d u i, ∧ n ^ i =1 u s i, i, d u i, . This definition combines both double zeta function and multiple Dedekind zeta valueswith double iteration. More generally, we can interpolate all multiple Dedekind zetavalues into multiple Dedekind zeta functions so that multiple zeta functions are particularcases. Again, we define u s i,j i,j by u s i,j i,j = e s i,j log( u i,j ) , along the branch of logarithm described above. Definition 3.6 (Multiple Dedekind zeta functions) Let n = [ K ; Q ] be the degree of thenumber field. Let C , . . . , C m be m positive unimodular simple cones in O K . Let u i,j ∈ (0 , ∞ ) for i = 1 , . . . , n and j = 1 , . . . , m . We define multiple Dedekind zeta functions bythe integral ζ K ; C ,...,C m ( s , , . . . , s n, ; . . . ; s ,m , . . . , s n,m ) == ( n,m ) Y ( i,j )=(1 , Γ( s i,j ) − × (3.39) × Z (0 , + ∞ ) mn m ^ j =1 f ( C j ; ( u ,j + · · · + u ,m ) , . . . , ( u n,j + · · · + u n,m )) n ^ i =1 u s i,j − i,j d u i,j , when Re ( s i,j ) > . Theorem 3.7 (Infinite Sum Representation) For the general form of a multiple Dedekindzeta function, we need: a number field K ; positive unimodular simple cones C j in O K ,for j = 1 , . . . , m ; elements α j ∈ C j for j = 1 , . . . , m ; complex embeddings of the ele-ments α i,j = σ i ( α j ); Then, a multiple Dedekind zeta function has the following infinitesum representation ζ K ; C ,...,C m ( s , , . . . , s n, ; . . . ; s ,m , . . . , s n,m ) == X α ∈ C · · · X α m ∈ C m n Y i =1 m Y j =1 ( α i, + · · · + α i,j ) − s i,j , when Re ( s i,j ) > . roof. We have ζ K ; C ,...,C m ( s , , . . . , s n, ; . . . ; s ,m , . . . , s n,m ) == ( n,m ) Y ( i,j )=(1 , Γ( s i,j ) − ×× Z (0 , + ∞ ) mn m ^ j =1 f ( C j ; ( u ,j + · · · + u ,m ) , . . . , ( u n,j + · · · + u n,m )) n ^ i =1 u s i,j − i,j d u i,j == ( n,m ) Y ( i,j )=(1 , Γ( s i,j ) − X α ∈ C · · · X α m ∈ C m Z (0 , + ∞ ) mn m ^ j =1 n ^ i =1 e − ( α i, + ··· + α i,j ) u i,j u s i,j − i,j du i,j = X α ∈ C · · · X α m ∈ C m n Y i =1 m Y j =1 ( α i, + · · · + α i,j ) − s i,j . The following examples give a bridge between Dedekind zeta function and values ofEisenstein series (Example 5), and between multiple Dedekind zeta function and valuesof multiple Eisenstein series. More about values of multiple Eisenstein series will appearin Examples 7, 8, 9 on pages 29 and 30. Examples: 5. Let K be any number field, let m = 1 and let C be a positive unimodular simplecone in O K . Then ζ K ; C ( s , , . . . , s n, ) = X α ∈ C Q ni =1 α s i, i , (3.40)where α i = σ i ( α ) is the i -th embedding in the complex numbers. In particular, if allvariables s i, , for i = 1 , . . . , n have the same value s , then ζ K ; C ( s, . . . , s ) = X α ∈ C N K/ Q ( α ) s . (3.41)6. Now, let m = 2. Then we have a double iteration. Let K be any number field.Let C and C be two positive unimodular simple cones. Then ζ K ; C ,C ( s , , . . . , s n, ; s , , . . . , s n, ) = X α ∈ C ,β ∈ C Q ni =1 α s i, i ( α i + β i ) s i, . (3.42)In particular, if s j = s ,j = · · · = s n,j for j = 1 , 2, then ζ K ; C ,C ( s , . . . , s ; s , . . . , s ) = X α ∈ C ,β ∈ C N K/ Q ( α ) s N K/ Q ( α + β ) s . (3.43)29 Analytic properties and special values Assuming the analytic continuation (Theorem 4.2), we can consider values of the multipleDedekind zeta functions, when one or more of the arguments are zero, which allows usto express special values of multiple Eisenstein series (see [GKZ]) as multiple Dedekindzeta values. This is presented in the following three examples. Examples: 7. Let K be an imaginary quadratic field. Let C be a positive unimod-ular simple cone in O K . We can represent the cone C as an N -module: C = N { µ, ν } , for µ, ν ∈ O K . Put z = µ/ν . Consider ζ K ; C ( k , , k , ) at k , = k and k , = 0.Then ζ K ; C ( k, 0) = X α ∈ C α k = ν − k X a,b ∈ N az + b ) k , where the last sum is a portion of the k -th Eisenstein series. E k ( τ ) = X a,b ∈ Z ;( a,b ) =(0 , az + b ) k . is an analogue of Eisenstein series.8. Let K be an imaginary quadratic field. Let C be a positive unimodular simplecone in O K . We can represent C as C = N { µ, ν } = { α ∈ O K | α = aµ + bν, a, b ∈ N } . Put z = µ/ν . Then, we obtain a value of multiple Eisenstein series ζ K ; C,C ( k, l, 0) = ν − k − l X a,b,c,d ∈ N az + b ) k (( a + c ) z + ( b + d )) l . 9. Similarly, one can define analogue of values of the above Eisenstein series over realquadratic field K , by setting E k ( z ) = ν k ζ K ; C ( k, 0) = X α ∈ C α k , where C = N { µ, ν } is a positive unimodular simple cone in a real quadratic ring ofintegers O K . The following examples of analytic continuations are based on a Theorem of Gelfand-Shilov. The constructions in example 11 is central for this Section. Using Example 11,we change the variables in a way that we can apply Gelfand-Shilov’s Theorem (Theorem4.1) that gives analytic continuation of MDZF. Moreover, in Example 11 we computea multiple residue at (1 , , , heorem 4.1 ([GSh]) Let φ ( x ) be a test function on R , which decreases rapidly (expo-nentially) when x → ∞ and let x + = (cid:26) x if x > if x ≤ Then the value of the distribution x s − dx Γ( s ) on the test function φ , namely, Z R φ ( x ) x s − dx Γ( s ) is an analytic function in the variable s . In Examples 10 and 11, we express a multiple Dedekind zeta function (MDZF) as a testfunction times a distribution when s i,j > s i,j aftermultiplying by a suitable Γ-factors. Using this method, we compute the multiple residueat (1 , . . . , Example 10. Let K be a quadratic field and let C = N { α, β } be a positive uni-modular simple cone. Put α , α and β , β be the images under the two embeddingsinto C of α and β , respectively. We will compute the residue of ζ K ; C ( s , s ) = X µ ∈ C µ s µ s at the hyperplane s + s = 2 and evaluated at s = s = 1.We have ζ K ; C ( s , s ) = Γ( s ) − Γ( s ) − Z ∞ Z ∞ t s − t s − dt ∧ dt ( e α t + α t − e β t + β t − . Set t = x (1 − x ) and t = x x . ThenΓ( s + s − − ζ K ; C ( s , s )is the value of the distribution Dx = x s + s − x s (1 − x ) s − dx dx Γ( s + s − s )Γ( s )at the test function φ = x ( e x ( α +( α − α ) x ) − e x ( β +( β − α ) x ) − . Using Theorem 4.1 we obtain that Γ( s + s − − ζ K ; C ( s , s ) is an analytic function.The residue of ζ K ; C ( s , s ) at s + s = 2 is Z α + ( α − α ) x )( α + ( β − β ) x ) · x s (1 − x ) s − dx dx Γ( s )Γ( s ) . s = 1, the integral becomes Z dx ( α + ( α − α ) x )( α + ( β − β ) x )Thus, the residue of ζ K ; C ( s , s ) at s + s = 2, evaluated at ( s , s ) = (1 , 1) is given bythe above integral. After evaluating it, we obtain( Res s + s =2 ζ K ; C ( s , s )) | ( s ,s )=(1 , = log (cid:16) α α (cid:17) − log (cid:16) β β (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) α β α β (cid:12)(cid:12)(cid:12)(cid:12) . In particular, if β = 1 we obtain( Res s + s =2 ζ K ; C ( s , s )) | ( s ,s )=(1 , = log( α ) − log( α ) α − α . (4.44)Note that if K is a real quadratic field and α is a generator of the group of units, then | log( α ) − log( α ) | = 2 | log( α ) | is two times the regulator of the number field K and α − α is an integer multiple ofthe discriminant of K . For a definition of a discriminant and a regulator of a numberfield, one may consult [IR]. Equation (4.44) is true for any quadratic field, not only forreal quadratic fields.The following Example gives key constructions needed for the proof of the analyticcontinuation of MDZF (Theorem 4.2). It is also a case study of Conjecture 4.4 aboutthe multiple residue of a multiple Dedekind zeta function at (1 , . . . , Example 11. Let K be a quadratic extension of Q . Let C = N { , α } and C = N { , γ } be two positive unimodular simple cones. Let ζ K ; C ,C ( s , s ; s ′ , s ′ ) = X µ ∈ C ; ν ∈ C µ s µ s ( µ + ν ) s ′ ( µ + ν ) s ′ . An integral representation can be written as ζ K ; C ,C ( s , s ; s ′ , s ′ )Γ( s )Γ( s )Γ( s ′ )Γ( s ′ ) == Z t >t ′ > t >t ′ > ( t − t ′ ) s − ( t − t ′ ) s − ( t ′ ) s ′ − ( t ′ ) s ′ − dt ∧ dt ∧ dt ′ ∧ dt ′ ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − . (4.45)We compute the residue of a double Dedekind zeta function by taking the multipleresidues of six functions ζ ( a ) , . . . , ζ ( f ) and considering their sum.We shall write the last differential form in Equation (4.46) as Dt . Then, we have ζ K ; C ,C ( s , s ; s ′ , s ′ ) = (Γ( s )Γ( s )Γ( s ′ )Γ( s ′ )) − Z t >t ′ > t >t ′ > Dt (4.46)32e are going to express the above integral as a sum of six integrals, which are enumeratedby all possible shuffles of t > t ′ > t > t ′ > 0. In other words, each of the sixnew integrals will be associated to each linear order among the variables t , t , t ′ , t ′ thatrespect the above two inequalities among them. Thus, all possible cases are(a) t > t ′ > t > t ′ > t > t > t ′ > t ′ > t > t > t ′ > t ′ > t > t > t ′ > t ′ > t > t > t ′ > t ′ > t > t ′ > t > t ′ > t > t ′ > t > t ′ > ζ K ; C ,C ( s , s ; s ′ , s ′ ) = (Γ( s )Γ( s )Γ( s ′ )Γ( s ′ )) − Z t >t ′ > t >t ′ > Dt = (Γ( s )Γ( s )Γ( s ′ )Γ( s ′ )) − ×× Z t >t ′ >t >t ′ > + Z t >t >t ′ >t ′ > ++ Z t >t >t ′ >t ′ > + Z t >t >t ′ >t ′ > ++ Z t >t >t ′ >t ′ > + Z t >t ′ >t >t ′ > ! Dt == ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ ) + ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ )++ ζ ( c ) K ; C ,C ( s , s ; s ′ , s ′ ) + ζ ( d ) K ; C ,C ( s , s ; s ′ , s ′ )++ ζ ( e ) K ; C ,C ( s , s ; s ′ , s ′ ) + ζ ( f ) K ; C ,C ( s , s ; s ′ , s ′ ) . We define ζ ( a ) , . . . , ζ ( f ) to be the above six integrals, corresponding to the domains ofintegration given by (a),...,(f). The reason for defining them is to take multiple residues ofthe multiple Dedekind zeta function. It is easier to work with the functions ζ ( a ) , . . . , ζ ( f ) for the purpose of proving analytic continuation and taking residues.Thus, we compute the residue of a double Dedekind zeta function by taking themultiple residues of six functions ζ ( a ) , . . . , ζ ( f ) and considering their sum.Consider the domain of integration( a ) t > t ′ > t > t ′ > . We will compute the residues of ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ )Γ( s )Γ( s )Γ( s ′ )Γ( s ′ ) == Z t >t ′ > >t >t ′ > ( t − t ′ ) s − ( t − t ′ ) s − ( t ′ ) s ′ − ( t ′ ) s ′ − dt ∧ dt ∧ dt ′ ∧ dt ′ ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − . u = t − t ′ , u = t ′ − t , u = t − t ′ , u = t ′ . Theiradmissive values are in the interval (0 , + ∞ ). Let us make the following substitution u = x (1 − x ) ,u = x x (1 − x ) ,u = x x x (1 − x ) ,u = x x x x . We are going to express the above integral in terms of the variables x , x , x and x .We have α t + α t = α ( u + u + u + u ) + α ( u + u ) = x ( α + α x x , ) t + t = x (1 + x x ) ,γ t ′ + γ t ′ = γ ( u + u + u ) + γ u = x x (1 + x x ) ,t ′ + t ′ = x x (1 + x x ) ,t − t ′ = u = x (1 − x ) ,t − t ′ = u = x x x (1 − x ) ,t ′ = u + u + u = x x ,t ′ = u = x x x x . For the change of variables in the differential forms, we have dt ∧ dt ∧ dt ′ ∧ dt ′ = du ∧ du ∧ du ∧ du and du u ∧ du u ∧ du u ∧ du u = dx ∧ dx ∧ dx ∧ dx x x x x (1 − x )(1 − x )(1 − x ) . Then ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ )Γ( s )Γ( s )Γ( s ′ )Γ( s ′ ) == Z t >t ′ > >t >t ′ > ( t − t ′ ) s − ( t − t ′ ) s − ( t ′ ) s ′ − ( t ′ ) s ′ − dt ∧ dt ∧ dt ′ ∧ dt ′ ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ( u ,...,u ) ∈ (0 , ∞ ) u s − u s − ( u + u + u ) s ′ − u s ′ − du ∧ du ∧ du ∧ du ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ( u ,...,u ) ∈ (0 , ∞ ) u s u s ( u + u + u ) s ′ − u s ′ u du u ∧ du u ∧ du u ∧ du u ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ∞ Z (0 , [ x (1 − x )] s [ x x x (1 − x )] s [ x x ] s ′ [ x x x x ] s ′ [ x x (1 − x )]Ω( e x (1+ x x ) − )( e x ( α + α x x ) − )( e x x (1+ x x ) − e x x ( γ + γ x x ) − , where Ω = du u ∧ du u ∧ du u ∧ du u = dx ∧ dx ∧ dx ∧ dx x x x x (1 − x )(1 − x )(1 − x ) . 34e can express the last integral as a distribution evaluated at a test function. Put φ ( x , x , x , x ) == x x ( e x (1+ x x ) − )( e x ( α + α x x ) − )( e x x (1+ x x ) − e x x ( γ + γ x x ) − Dx be a distribution defined by Dx = x s + s + s ′ + s ′ − Γ( s + s + s ′ + s ′ − x s + s ′ + s ′ − Γ( s + s ′ + s ′ − × (4.47) × x s + s ′ − Γ( s + s ′ ) x s ′ − Γ( s ′ ) (1 − x ) s − (1 − x ) s − . (4.48)Then we have ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ ) = Γ( s + s + s ′ + s ′ − s + s ′ + s ′ − s + s ′ )Γ( s )Γ( s )Γ( s ′ ) ×× Z φDx Using Theorem 4.1 we prove analytic continuation of ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ ) everywhereexcept at the poles of Γ( s + s + s ′ + s ′ − s + s ′ + s ′ − s + s ′ ).We will take the residue at s + s + s ′ + s ′ = 4 and then evaluate at ( s , s , s ′ , s ′ ) =(1 , , , . Note that φ (0 , x , x , x ) = 1(1 + x x )( α + α x x )(1 + x x )( γ + γ x x )Then we can compute( Res s + s + s ′ + s ′ =4 ζ ( a ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == Z (0 , x dx ∧ dx ∧ dx (1 + x x )( α + α x x )(1 + x x )( γ + γ x x ) . Note that we cannot take a double residue at the point (1 , , , t > t > t ′ > t ′ > ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ )Γ( s )Γ( s )Γ( s ′ )Γ( s ′ ) == Z t >t >t ′ >t ′ > ( t − t ′ ) s − ( t − t ′ ) s − ( t ′ ) s ′ − ( t ′ ) s ′ − dt ∧ dt ∧ dt ′ ∧ dt ′ ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − . u = t − t ,u = t − t ′ ,u = t ′ − t ′ ,u = t ′ . Their admissive values are (0 , + ∞ ). Let u = x (1 − x ) ,u = x x (1 − x ) ,u = x x x (1 − x ) ,u = x x x x . We will express the above integral in terms of x , . . . , x . We have α t + α t = α ( u + u + u + u + 4) + α ( u + u + u ) = x ( α + α x ) ,t + t = x (1 + x ) ,γ t ′ + γ t ′ = γ ( u + u ) + γ u = x x x ( γ + γ x ) ,t ′ + t ′ = x x x (1 + x ) ,t − t ′ = u + u = x (1 − x x ) ,t − t ′ = u + u = x x (1 − x x ) ,t ′ = u + u = x x u ,t ′ = u = x x x x . For the differential forms, we have dt ∧ dt ∧ dt ′ ∧ dt ′ = du ∧ du ∧ du ∧ du du u ∧ du u ∧ du u ∧ du u = dx ∧ dx ∧ dx ∧ dx x x x x (1 − x )(1 − x )(1 − x ) . Then ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ )Γ( s )Γ( s )Γ( s ′ )Γ( s ′ ) == Z t >t >t ′ >t ′ > ( t − t ′ ) s − ( t − t ′ ) s − ( t ′ ) s ′ − ( t ′ ) s ′ − dt ∧ dt ∧ dt ′ ∧ dt ′ ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ( u ,...,u ) ∈ (0 , ∞ ) ( u + u ) s − ( u + u ) s − ( u + u ) s ′ − u s ′ − du ∧ du ∧ du ∧ du ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ( u ,...,u ) ∈ (0 , ∞ ) ( u + u ) s − ( u + u ) s − ( u + u ) s ′ − u s ′ u u u du ∧ du ∧ du ∧ du u u u u ( e t + t − e α t + α t − e t ′ + t ′ − e γ t ′ + γ t ′ − Z ∞ Z (0 , [ x (1 − x x )] s − [ x x (1 − x x )] s − [ x x x ] s ′ − [ x x x x ] s ′ X Ω( e x (1+ x x ) − e x ( α + α x x ) − e x x (1+ x x ) − e x x ( γ + γ x x ) − , X = x x x (1 − x )(1 − x )(1 − x )and Ω = du ∧ du ∧ du ∧ du u u u u = dx ∧ dx ∧ dx ∧ dx x x x x (1 − x )(1 − x )(1 − x )Now we can express the last integral as a distribution evaluated at a test function. Put φ ( x , x , x , x ) = x x x ( e x (1+ x ) − e x ( α + α x ) − e x x x (1+ x ) − e x x x ( γ + γ x ) − , to be a test function. Let Dx be a distribution defined by Dx = x s + s + s ′ + s ′ − Γ( s + s + s ′ + s ′ − x s + s ′ + s ′ − Γ( s + s ′ + s ′ − × (4.49) × x s + s ′ − Γ( s + s ′ − x s ′ − Γ( s ′ ) (1 − x x ) s − (1 − x x ) s − (4.50)Then we have ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ ) = Γ( s + s + s ′ + s ′ − s + s ′ + s ′ − s + s ′ − s )Γ( s )Γ( s ′ ) ×× Z φDx Using Theorem 4.1 we prove analytic continuation of ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ ) everywhereexcept at the poles of Γ( s + s + s ′ + s ′ − s + s ′ + s ′ − s + s ′ − s + s + s ′ + s ′ = 4 and at s ′ + s ′ = 2. And then, wewill evaluate at ( s , s , s ′ , s ′ ) = (1 , , , φ (0 , x , , x ) = 1(1 + x )( α + α x )(1 + x )( γ + γ x )Then we can compute( Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == Z (0 , dx ∧ dx (1 + x )( α + α x )(1 + x )( γ + γ x ) == log (cid:16) α α + α (cid:17) α − α · log (cid:16) γ γ + γ (cid:17) γ − γ . For the cases (c), (d) and (e), we obtain( Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ζ ( c ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == Z (0 , dx ∧ dx (1 + x )( α + α x )( x + 1)( γ x + γ ) == − log (cid:16) α α + α (cid:17) α − α · log (cid:16) γ γ + γ (cid:17) γ − γ . Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ζ ( d ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == Z (0 , dx ∧ dx ( x + 1)( α x + α )(1 + x )( γ + γ x ) == − log (cid:16) α α + α (cid:17) α − α · log (cid:16) γ γ + γ (cid:17) γ − γ . ( Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ζ ( e ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == log (cid:16) α α + α (cid:17) α − α · log (cid:16) γ γ + γ (cid:17) γ − γ . Case (f) is similar to case (a), namely, there is no double residue at the point(1 , , , Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ζ K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == Res s ′ + s ′ =2 Res s + s + s ′ + s ′ =4 ( ζ ( b ) K ; C ,C ( s , s ; s ′ , s ′ ) + ζ ( c ) K ; C ,C ( s , s ; s ′ , s ′ )++ ζ ( d ) K ; C ,C ( s , s ; s ′ , s ′ ) + ζ ( e ) K ; C ,C ( s , s ; s ′ , s ′ )) | (1 , , , == log( α ) − log( α ) α − α · log( γ ) − log( γ ) γ − γ . Note that if K is a real quadratic field and α is a generator of the group of units, then | log( α ) − log( α ) | = 2 log | α | is two times the regulator of the number field K and α − α is an integer multiple of the discriminant of the field K . For a definition of a discriminantand a regulator of a number field one may consult with [IR]. The above formula is truefor any quadratic field, not necessarily for a real quadratic field. Theorem 4.2 Multiple Dedekind zeta functions ζ K ; c ,...,C m ( s , , . . . , s n, ; . . . ; s ,m , . . . , s n,m ) have an analytic continuation from the region Re ( s i,j ) > for all i and j to s i,j ∈ C withexception of hyperplanes. The hyperplanes are defined by sum several of the variables s i,j without repetitions being set equal to an integer. roof. Recall that f ( C j ; t ,j , . . . , t n,j ) is used to define the multiple Dedekind zetafunctions, where the domain of integration is D = { ( t i,j ) ∈ R mn | t i, > t i, > · · · > t n,j > } .J i = { k i, , . . . , k i,m } . Note that there are n sets J , . . . , J n , and each of them has m elements, | J i | = m . Let τ run through all the shuffles of the ordered sets J , . . . , J n , τ ∈ Sh ( J , . . . , J n ) . Let t , t . . . , t mn be the variables t , , . . . , t n,m , written in decreasing order. There arefinitely many ways of arranging the variables in decreasing order. More precisely, thenumber of such arrangements is equal to the number of shuffles in Sh ( J , . . . , J n ). Weneed to consider all such shuffles in order to express the multiple Dedekind zeta functionas a sum of partial multiple Dedekind zeta functions, corresponding to each shuffle τ (see Example 11). Let u k = t k − t k +1 , for k = 1 , . . . , mn − u mn = t mn . Let u = x (1 − x ) u = x x (1 − x ) · · · u mn − = x . . . x mn − (1 − x mn ) u mn = x . . . x mn . Note that ( x , . . . , x mn ) ∈ R × [0 , mn − Then each of the linear factors in the denominator of f ( C j ; · · · ) can be written as x . . . x k g j,l,τ . for some positive integer k , k ≤ mn and a polynomial g j,l,τ , in the variables x , . . . , x mn ,vanishing at the origin. Also, the indices i and j of the polynomial g j,l,τ are associatedto the l -th generator of the cone C j , and τ is a shuffle of ordered sets Sh ( J , . . . , J n ).Example of polynomials g j,l,τ , can be found in Equations (4.48) and (4.50). If thepolynomial g j,l,τ , in terms of x , . . . , x mn , has a pole on the domain of integration R × [0 , mn − then we move the membrane of integration to a membrane D ′ ∈ C mn so thatthe real coordinates of D ′ give the domain R × [0 , mn − . (This is analogue of contourintegration.) The integral representation in terms of x , . . . , x mn (similar to the ones inExample 11, giving ζ ( a ) ), are a type of zeta function that we call partial MDZF timesΓ-factors. Using Theorem 4.1, we find that the partial MDZF together with the Γ-factors is an analytic function. The Γ-factors give hyperplanes where the poles of thepartial MDZF occur. Expressing a MDZF as a finite sum of partial MDZF we obtainthe analytic continuation from the domain Re ( s i,j ) > s i,j ∈ C with poles alonghyperplanes coming from Γ-factors. ✷ .4 Final remarks In this final Subsection, we proof that certain multiple residue of a multiple Dedekindzeta functions is a period in the sense of algebraic geometry. Based on Theorem 4.3, westate two conjectures. One of the conjectures is about the exact values of the multipleresidue and the other conjecture is about values of the multiple Dedekind zeta functionsat other integers. Theorem 4.3 The multiple residue of a multiple Dedekind zeta function at the point ( s , , . . . , s n, ; . . . ; s ,m , . . . , s n,m ) = (1 , . . . , is a period over Q .Proof. We use the notation introduced in the proof of Theorem 4.2 and of Example 11in Subsection 4.2.The m -fold residue at (1 , . . . , 1) can be computed via an integral of a rational function,which is a product of the functions representing the hyperplanes, where f vanishes,expressed in terms of the variables x i . The value is a period. In general, after we takethe multiple residues at (1 , . . . , g j,l,τ over j and l , for j = 1 , . . . , m , where l signifies the l -generator of thecone C j ). Note that τ is a shuffle. So that different shuffles τ correspond to differentpartial MDZF. The boundaries of the integral (after taking the multiple residues) forma unit cube. Therefore, the value of the multiple residue at (1 , . . . , 1) of a multipleDedekind zeta function is a period. ✷ For a more precise interpretation see Conjecture 4.4 and Examples 10 and 11.From Examples 10 and 11, we know that a multiple residue of multiple Dedekind zetafunction is a product of residues of partial Dedekind zeta functions, for quadratic fieldsand double iteration. For unimodular simple cones C , . . . , C m , we consider a multipleDedekind zeta function ζ K ; C ,...,C m ( s , . . . , s d ) = X α ∈ C ,...α d ∈ C m N ( α ) s N ( α + α ) s · · · N ( α + · · · + α m ) s m . We expect that Conjecture 4.4 The multiple residue of ζ K ; C ,...,C m ( s , . . . , s m ) at the point (1 , . . . , ,namely Res s m =1 . . . Res s + ...s m = m ζ K ; C ,...,C m ( s , . . . , s m ) = m Y j =1 Res s =1 ζ K ; C j ( s ) . The conjecture is proven for a quadratic fields K and double iteration in Examples 10and 11.We do expect that multiple Dedekind zeta values should be periods over Q .40 onjecture 4.5 Let K be a number field. For any choice of unimodular simple cones C , . . . , C m , in the ring of integers of a number field K , we have that the multipleDedekind zeta values (see Definition 3.4) ζ K ; C ,...,C m ( k , , . . . , k n, ; . . . ; k ,m , . . . , k n,m ) are periods over Q , when the k , , . . . , k m,n are natural numbers greater than . The reasons for this conjecture are the following:1. We have that multiple zeta values are periods;2. Dedekind zeta values are periods;3. From Theorem 4.3, we have that the multiple residue of a multiple Dedekind zetafunction at (1 , . . . , 1) is a period.4. The main reason is the representation of multiple Dedekind zeta values as iteratedintegrals on membranes. We will give a semi-algebraic relations among the variablesin such integrals.We use Equations (2.25) and (2.26). Recall that y j = n Y i =1 exp( − σ i ( e j ) z i ) . If we set x i = exp( − z i ) , then a multiple Dedekind zeta value is an iterated integral on a membrane of the n -form Y j dy j − y j and the 1-forms dx i x i , which mostly resembles polylogarithms. However, the relations between x i ’s and y j ’s are(semi)-algebraic, namely, dy j y j = n X i =1 σ i ( e j ) dx i x i , which are not algebraic. Explicitly, they are given bylog( y j ) = n X i =1 σ i ( e j ) log( x i ) , which involves the logarithmic function. Note that the logarithmic function is a homo-topy invariant function on a path space. In this setting, the above logarithmic functions41an be considered as a function on the path space of an affine n -space without somedivisors. One may take a simplicial scheme as a model of the path space so that itrestricts well onto the loop space of a scheme as a simplicial scheme. Hopefully, thatwould interpret the above (semi)-algebraic relations in terms of logarithms in an algebraiccontext. Acknowledgements: This paper owes a lot to many people. Ronald Brown gaveinspiring talk on higher cubical categories. I few days after that talk, I had a gooddefinition of iterated integrals on a membrane. With Alexander Goncharov I had a lot offruitful discussions. His great interest in my different approaches to multiple Dedekindzeta functions encouraged me to continue my search of the right ones. From MladenDimitrov I learned about the cones constructed by Shintani. Anton Deitmar asked manyquestions on the subject that helped me to clear up (at least to myself) the structure ofthe paper. Matthew Kerr clarified many questions I had about Hodge theory. Dev Sinhaexplained simplicial de Rham structures on mapping spaces of a manifold, in particular,a path space or a loop space of a variety. Finally, I would like to thank the referee whoseeffort greatly improved the presentation.I acknowledge the great hospitality of Durham University and the generous financialsupport from the European Network Marie Currie that allowed me to invite RonaldBrown for several days. I would like to thank Tubingen University and in particular,Anton Deitmar and his grant “Higher modular forms and higher invariants” for thefinancial support and great working conditions.