The algebra of multiple divisor functions and applications to multiple zeta values

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1 The algebra of multiple divisor functions and applications to multiple zeta values ESF Workshop: New Approaches To Multiple Zeta Values ICMAT Madrid - 30th September 2013 joint work: H.B., Ulf Kühn, arxiv: [math.nt]

8 Multiple divisor functions As a generalization of the classical divisor sums we define for r 1,..., r l 0 the multiple divisor sum by σ r1,...,r l (n) := u 1v 1+ +u l v l =n u 1> >u l >0 v r vr l l. With this we define for s 1,..., s l > 0 the multiple divisor function of weight s s l and length l by [s 1,..., s l ] := 1 (s 1 1)!... (s l 1)! σ s1 1,...,s l 1(n)q n Q[[q]]. n>0

9 Multiple divisor functions As a generalization of the classical divisor sums we define for r 1,..., r l 0 the multiple divisor sum by σ r1,...,r l (n) := u 1v 1+ +u l v l =n u 1> >u l >0 v r vr l l. With this we define for s 1,..., s l > 0 the multiple divisor function of weight s s l and length l by [s 1,..., s l ] := 1 (s 1 1)!... (s l 1)! σ s1 1,...,s l 1(n)q n Q[[q]]. n>0 Example I For l = 1 these are the classical divisor functions [2] = n>0 σ 1 (n)q n = q + 3q 2 + 4q 3 + 7q 4 + 6q q 6 + 8q q

10 Multiple divisor functions As a generalization of the classical divisor sums we define for r 1,..., r l 0 the multiple divisor sum by σ r1,...,r l (n) := u 1v 1+ +u l v l =n u 1> >u l >0 v r vr l l. With this we define for s 1,..., s l > 0 the multiple divisor function of weight s s l and length l by 1 [s 1,..., s l ] := σ s1 1,...,s (s 1 1)!... (s l 1)! l 1(n)q n Q[[q]]. Example II [4, 2] = 1 σ 3,1 (n)q n = n>0 n>0 ( ) q 3 + 3q q 5 }{{} +27q6 + 78q σ 3,1(5) = = 15, because 5 = = = = =

11 Multiple divisor functions As a generalization of the classical divisor sums we define for r 1,..., r l 0 the multiple divisor sum by σ r1,...,r l (n) := u 1v 1+ +u l v l =n u 1> >u l >0 v r vr l l. With this we define for s 1,..., s l > 0 the multiple divisor function of weight s s l and length l by [s 1,..., s l ] := 1 (s 1 1)!... (s l 1)! σ s1 1,...,s l 1(n)q n Q[[q]]. n>0 Example III [4, 4, 4] = 1 ( q 6 + 9q q q q q ), 216 [3, 1, 3, 1] = 1 ( q q q q q q ) 4

12 Multiple divisor functions - Filtration Weight and length filtration We define the vector space MD to be the Q vector space generated by [ ] = 1 Q[[q]] and all multiple divisor sums [s 1,..., s l ]. On MD we have the increasing filtration Fil W given by the weight and the increasing filtration Fil L given by the length, i.e., we have for A MD Fil W k (A) := [s 1,..., s l ] A 0 l k, s1 + + s l k Q Fil L l (A) := [s 1,..., s r ] A r l Q. If we consider the length and weight filtration at the same time we use the short notation Fil W,L k,l := Fil W k Fil L l. As usual gr W k (A) = FilW k (A)/ Fil W k 1(A) will denote the graded part (similar gr L l and gr W,L k,l ).

13 Multiple divisor functions - Algebra structure Theorem The Q-vector space MD has the structure of a bifiltered Q-Algebra (MD,, Fil W, Fil L ), where the multiplication is the natural multiplication of formal power series and the filtrations Fil W and FilL are induced by the weight and length, in particular Fil W,L k 1,l 1 (MD) Fil W,L k 2,l 2 (MD) Fil W,L k 1+k 2,l 1+l 2 (MD). It is a (homomorphic image of a) quasi-shuffle algebra in the sense of Hoffman. The first products of multiple divisor functions are given by [1] [1] = 2[1, 1] + [2] [1], [1] [2] = [1, 2] + [2, 1] + [3] 1 2 [2], [1] [2, 1] = [1, 2, 1] + 2[2, 1, 1] + [2, 2] + [3, 1] 3 [2, 1]. 2

14 Multiple divisor functions - Algebra structure The structure of being a quasi-shuffle algebra is as follows: Let Q A be the noncommutative polynomial algebra over Q generated by words with letters in A = {z 1, z 2,... }. Let be a commutative and associative product on QA. Define on Q A recursively a product by 1 w = w 1 = w and z a w z b v := z a (w z b v) + z b (z a w v) + (z a z b )(w v). The commutative Q-algebra (Q A, ) is called a quasi-shuffle algebra. The harmonic algebra (Hoffman) is an example of a quasi-shuffle algebra with z a z b = z a+b which is closely related to MZV.

15 Multiple divisor functions - Algebra structure In our case we consider the quasi-shuffle algebra with a b z a z b = z a+b + λ j a,b z j + λ j b,a z j, j=1 j=1 where with Bernoulli numbers B n λ j a,b = ( 1)b 1 ( a + b j 1 a j ) Ba+b j (a + b j)!. Proposition The map [. ] : (Q A, ) (MD, ) defined on the generators by z s1... z sl [s 1,..., s l ] is a homomorphism of algebras, i.e. it fulfils [w v] = [w] [v].

16 Multiple divisor functions - Modular forms Proposition The ring of modular forms M Q (SL 2 (Z)) and the ring of quasi-modular forms M Q (SL 2 (Z)) are subalgebras of MD. For example we have G 2 = [2], G 4 = [4], G 6 = [6]. The proposition follows from the fact that M Q (SL 2 (Z)) = Q[G 4, G 6 ] and M Q (SL 2 (Z)) = Q[G 2, G 4, G 6 ]. Remark Due to an old result of Zagier we have M Q (SL 2 (Z)) Fil W,L k,2 (MD).

17 Multiple divisor functions - Derivation Theorem The operator d = q d dq Fil W,L k+2,l+1 (MD). Examples: is a derivation on MD and it maps FilW,L k,l (MD) to d[1] = [3] + 1 [2] [2, 1], 2 d[2] = [4] + 2[3] 1 [2] 4[3, 1], 6 d[2] = 2[4] + [3] + 1 [2] 2[2, 2] 2[3, 1], 6 d[1, 1] = [3, 1] [2, 1] + 1 [1, 2] + [1, 3] 2[2, 1, 1] [1, 2, 1]. 2 The second and third equation lead to the first linear relation between multiple divisor functions in weight 4: [4] = 2[2, 2] 2[3, 1] + [3] 1 3 [2].

18 Multiple divisor functions - Derivation In the lowest length we get several expressions for the derivative: Proposition Let k N, then for any s 1, s 2 1 with k = s 1 + s 2 2 we have the following expression for d[k]: ( ) ( ) k s 1 1 d[k] k k s 1 1 [k + 1] = [s 1] [s 2] a+b=k+2 (( ) ( )) a 1 a 1 + [a, b]. s 1 1 s 2 1 Observe that the right hand side is the formal shuffle product. The possible choices for s 1 and s 2 give k 2 linear relations in FilW,L k,2 (MD), which are conjecturally all relations in length two.

19 Multiple divisor functions - qmz We define the space of all admissible multiple divisor functions qmz as qmz := [s 1,..., s l ] MD s1 > 1 Q. Theorem The vector space qmz is a subalgebra of MD. We have MD = qmz[ [1] ]. The algebra MD is a polynomial ring over qmz with indeterminate [1], i.e. MD is isomorphic to qmz[ T ] by sending [1] to T.

20 Multiple divisor functions - qmz We define the space of all admissible multiple divisor functions qmz as qmz := [s 1,..., s l ] MD s1 > 1 Q. Theorem The vector space qmz is a subalgebra of MD. We have MD = qmz[ [1] ]. The algebra MD is a polynomial ring over qmz with indeterminate [1], i.e. MD is isomorphic to qmz[ T ] by sending [1] to T. Sketch of proof: The first two statements can be proven by induction using the quasi-shuffle product. The algebraic independency of [1] for the third statements follows from the fact that near q = 1 one has [1] log(1 q) 1 1 q but [s 1,..., s l ] (1 q) s 1 + +s l for [s 1,..., s l ] qmz.

21 Multiple divisor functions - Connections to MZV For the natural map Z : qmz MZ of vector spaces given by we have a factorization Z([s 1,..., s l ]) = ζ(s 1,..., s l ) qmz k gr W k (qmz) Z MZ, k Z k Remark If we endow qmz = k gr W k (qmz) with the induced multiplication given on equivalence classes, then the products in qmz satisfy the stuffle relations. Moreover the right hand arrow is a homomorphism of algebras.

22 Multiple divisor functions - Connections to MZV Extending Z by setting Z([s 1,..., s l ][1] r ) = ζ(s 1,..., s l )T r we obtain a homomorphisms of vector spaces MD = qmz[[1]] qmz[[1]] Z k Z k MZ[T ], such that the image of a multiple divisor function [s 1,..., s l ] with s 1 = 1 equals the stuffle regularised ζ(s 1,..., s l ) as given by Ihara, Kaneko and Zagier (2006). Remark Again the right hand arrow is a homomorphism of algebras.

23 Multiple divisor functions - Connections to MZV All relations between stuffle regularised multiple zeta values of weight k correspond to elements in the kernel of Z k. We approach ker(z k ) MD by using an analytical description of Z k. Proposition For [s 1,..., s l ] Fil W k (qmz) we have the alternative description In particular we have Z k ([s 1,..., s l ]) = lim q 1 (1 q) k [s 1,..., s l ]. Z k ([s 1,..., s l ]) = { ζ(s1,..., s l ), s s l = k, 0, s s l < k.

24 Multiple divisor functions - Connections to MZV Sketch of proof: The multiple divisor functions can be written as [s 1,..., s l ] = 1 (s 1 1)!... (s l 1)! l n 1> >n l >0 j=1 q nj P sj 1 (q nj ) (1 q nj ) sj, where P k (t) is the k-th Eulerian polynomial. These polynomials have the property that P k (1) = k! and therefore lim q 1 q n P k 1 (q n ) (k 1)! (1 q) k (1 q n ) k = 1 n k. Now doing some careful argumentation to justify the interchange of limit and summation one obtains Z s1+ +s l ([s 1,..., s l ]) = lim q 1 (1 q) s1+ +s l [s 1,..., s l ] = ζ(s 1 + +s l ).

25 Multiple divisor functions - Connections to other q-analogues Another example of a q-analogue of multiple zeta values are the modified multiple q-zeta values. They are defined for s 1 > 1,s 2,..., s l 1 as ζ q (s 1,..., s l ) = n n 1> >n l >0 j=1 Similar to the multiple divisor functions one derives q nj(sj 1) (1 q nj ) sj. lim q 1 (1 q)s1+ +s l ζ q (s 1,..., s l ) = ζ(s 1,..., s l ). They seem to have a close connection to multiple divisor functions an if all entries s i > 1 they are indeed elements of MD, e.g. ζ q (4) = [4] [3] [2]. But if at least one entry is 1 this connection is conjectural.

26 Multiple divisor functions - Connections to MZV Theorem For the kernel of Z k Fil W k (MD) we have If s s l < k then Z k ([s 1,..., s l ]) = 0. For any f Fil W k 2(MD) we have Z k (d(f)) = 0. If f Fil W k (MD) is a cusp form for SL 2 (Z), then Z k (f) = 0.

27 Multiple divisor functions - Connections to MZV Sketch of proof: First we extend the Z k to a larger space. We define for ρ R { } Q ρ = a n q n R[[q]] a n = O(n ρ 1 ) and n>0 { } Q <ρ = a n q n R[[q]] ε > 0 with a n = O(n ρ 1 ε ). n>0 For ρ > 1 define the map Z ρ for a f = n>0 a nq n R[[q]] by Z ρ (f) = lim (1 q) ρ a n q n. q 1 n>0 Lemma Z ρ is a linear map from Q ρ to R Q <ρ ker Z ρ. d Q <ρ 1 ker(z ρ ), where as before d = q d dq. For any s 1,..., s l 1 we have [s 1,..., s l ] Q <s1+ +s l +1.

28 Multiple divisor functions - Connections to MZV Example I: We have seen earlier that the derivative of [1] is given by d[1] = [3] + 1 [2] [2, 1] 2 and because of the theorem it is d[1], [2] ker Z 3 from which ζ(2, 1) = ζ(3) follows.

29 Multiple divisor functions - Connections to MZV Example II: (Shuffle product) We saw that for s 1 + s 2 = k + 2 ( k ) d[k] s 1 1 k = [s1] [s2] + ( ) ( k ( s 1 1 [k + 1] a 1 ) ( s a 1 s 2 1) ) [a, b] a+b=k+2 Applying Z k+2 on both sides we obtain the shuffle product for single zeta values ζ(s 1 ) ζ(s 2 ) = a+b=k+2 (( ) a 1 + s 1 1 ( )) a 1 ζ(a, b). s 2 1

30 Multiple divisor functions - Connections to MZV Example III: For the cusp form S 12 ker(z 12 ) we derived the representation = 168[5, 7] + 150[7, 5] + 28[9, 3] [2] [4] [6] [8] [12]. Letting Z 12 act on both sides one obtains the relation 5197 ζ(12) = 168ζ(5, 7) + 150ζ(7, 5) + 28ζ(9, 3). 691

31 Multiple divisor functions - Connections to MZV To summarize we have the following objects in the kernel of Z k, i.e. ways of getting relations between multiple zeta values using multiple divisor functions. Remark Elements of lower weights, i.e. elements in Fil W k 1(MD). Derivatives Modular forms, which are cusp forms Since 0 ker Z k, any linear relation between multiple divisor functions in Fil W k (MD) gives an element in the kernel. The number of admissible ) multiple divisor functions [s 1,..., s l ] of weight k and length l equals. Since we have MD = qmz [ [1] ], it follows that knowing ( k 2 l 1 the dimension of gr W,L relations. k,l (qmz) is sufficient to know the number of independent

32 Multiple divisor functions - Dimensions k l ?? ????? ?????? ???????? dim Q gr W,L k,l (qmz): proven, conjectured.

33 Multiple divisor functions - Dimensions We observe that d k := dim Q gr W k (qmz) satisfies: d 0 = 1, d 1 = 0, d 2 = 1 and d k = 2d k 2 + 2d k 3, for 5 k 11. We see no reason why this shouldn t hold for all k > 11 also, i.e. we ask whether k 0 dim Q gr W k (qmz)x k = k 0 d kx k? = 1 X 2 + X 4 1 2X 2 2X 3. (1)

34 Multiple divisor functions - Dimensions We observe that d k := dim Q gr W k (qmz) satisfies: d 0 = 1, d 1 = 0, d 2 = 1 and d k = 2d k 2 + 2d k 3, for 5 k 11. We see no reason why this shouldn t hold for all k > 11 also, i.e. we ask whether k 0 dim Q gr W k (qmz)x k = k 0 d kx k? = 1 X 2 + X 4 1 2X 2 2X 3. (1) Compare this to the Zagier conjecture for the dimension d k of MZ k k 0 dim Q gr W k (MZ)X k = k 0 d k X k? = 1 1 X 2 X 3 k d k d k

35 Multiple divisor functions - Summary Multiple divisor functions are formal power series in q with coefficient in Q coming from the calculation of the fourier expansion of multiple Eisenstein series. The space spanned by all multiple divisor functions form an differential algebra which contains the algebra of (quasi-) modular forms. A connection to multiple zeta values is given by the map Z k whose kernel contains all relations between multiple zeta values of weight k. Questions: What is the kernel of Z k? Is there an analogue of the Broadhurst-Kreimer conjecture? Is there a geometric/motivic interpretation of the multiple divisor functions?

36 In the context of a generalisation of the Broadhurst-Kreimer conjecture in our context one may ask whether the modular forms replace the even zeta values. Both may be the only subalgebras which are nilpotent with respect to the length, this is due to the following reformulation of an old result by Zagier. Proposition As a subalgebra of MD the algebra of modular forms is graded with respect to the weight and filtered with respect to the length. We have k dim Q gr W,L k,l M(SL 2 (Z)) x k y l = 1 + x4 1 x 2 y + x 12 (1 x 4 )(1 x 6 ) y2, in particular dim Q M k (SL 2 (Z)) x k 1 = (1 x 4 )(1 x 6 ). k

Multiple divisor functions, their algebraic structure and the relation to multiple zeta values

Multiple divisor functions, their algebraic structure and the relation to multiple zeta values Kyushu University - 13th November 2013 joint work: H.B., Ulf Kühn, arxiv:1309.3920 [math.nt] Multiple