Generating series of multiple divisor sums and other interesting q-series

Transcription

1 Generating series of multiple divisor sums and other interesting q-series 1th July 2014

2 Content of this talk We are interested in a family of q-series which arises in a theory which combines multiple zeta values, partitions and modular forms. There are a lot of linear relations between these q-series. For example: n 1>n 2>0 q n1 n 2 q n2 (1 q n1 )(1 q n2 ) = 1 n 2 q n 2 1 q n 1 2 n>0 n>0 nq n 1 q n n>0 First we start by an elementary definition of these q-series using partitions... nq n (1 q n ) 2.

3 Partitions By a partititon of a natural number n with l different parts we denote a representation of n as a sum of l different numbers. For example 15 = is a partition of 15 with 4 different parts.

4 Partitions By a partititon of a natural number n with l different parts we denote a representation of n as a sum of l different numbers. For example 15 = is a partition of 15 with 4 different parts. ( We identify a partition of n with l u ) different parts with a tupel v, with u, v N l. The u j are the l different summands. The v j count their appearence in the sum. The above partition is therefore given by ( u v ) ( = 4,3,2,1 ). 2,1,1,2

5 Partitions We denote the set of all partition of n with l different parts by P l (n), i.e. we set {( ) } u P l (n) := N l N l n = u 1 v 1 u l v l, u 1 > > u l > 0. v An element in P l (n) can be represented by a Young tableau. For example ( ) 4, 3, 2, 1 = P 4 (15) 2, 1, 1, 2 With this it is easy to see that this element gives rise to a different element P 4 (15):

6 Partitions We denote the set of all partition of n with l different parts by P l (n), i.e. we set {( ) } u P l (n) := N l N l n = u 1 v 1 u l v l, u 1 > > u l > 0. v On the set P l (n) we have an involution ρ given by the conjugation of partitionen ( ) 4, 3, 2, 1 = 2, 1, 1, 2 ρ = ( ) 6, 4, 3, 2 1, 1, 1, 1 On P l (n) this ρ is explicitly given by ρ (( )) ( u v = u v ), where u j = v 1 v l j1 and v j = u l j1 u l j2 with u l1 := 0, i.e. ( ) u1,..., u l ρ : v 1,..., v l ( v1 v l,..., v 1 v 2, v 1 u l, u l 1 u l,..., u 1 u 2 ).

7 Partitions & q-series We now want to construct q-series n>0 a nq n, where the coefficients a n are given by sums over P l (n). The map ρ will then give linear relations between these series. Example Consider the following series v 1 v 2 q n n>0 ( u v) P 2(n)

8 Partitions & q-series We now want to construct q-series n>0 a nq n, where the coefficients a n are given by sums over P l (n). The map ρ will then give linear relations between these series. Example Consider the following series v 1 v 2 q n = u2 (u 1 u 2) q n n>0 ( u v) P 2(n) n>0 ρ(( u v))=( u v ) P2(n) = u 2 u 1 q n u 2 2 q n. n>0 ( u v) P 2(n) n>0 ( u v) P 2(n)

9 bi-brackets Definition For r 1,..., r l 0, s 1,..., s l > 0 and c := (r 1!(s 1 1)!... r l!(s l 1)!) 1 we define th following q-series [ ] s1,..., s l := c r 1,..., r l = u 1> >u l >0 v 1,...,v l >0 u r1 1 n>0 r 1!... url l r 1! ( u v) P l (n) u r1 1 vs u r l l vs l 1 l q n v s v s l 1 l (s 1 1)!... (s l 1)! qu1v1 u lv l Q[[q]] which we call bi-brackets of weight r 1 r k s 1 s l, upper weight s 1 s l, lower weight r 1 r l and length l. By BD we denote the Q-vector space spanned by all bi-brackets and 1.

10 bi-brackets The bi-brackets can also be written as [ ] s1,..., s l = c r 1,..., r l n 1> >n l >0 where the P k 1 (t) are the Eulerian polynomials defined by n r1 1 P s 1 1(q n1 )... n r l l P s l 1(q n l ) (1 q n1 ) s1... (1 q n l ) s l, P k 1 (t) (1 t) k = d k 1 t d. d>0 Examples: P 0 (t) = P 1 (t) = t,p 2 (t) = t 2 t, P 3 (t) = t 3 4t 2 t, [ ] 1, 1 = q n1 n 2 q n2 0, 1 (1 q n1 )(1 q n2 ). n 1>n 2>0

11 multiple divisor sums and modular forms For r 1 = = r l = 0 we also write [ ] s1,..., s l 1 = [s 1,..., s l ] =: σ s1 1,...,s 0,..., 0 (s 1 1)!... (s l 1)! l 1(n)q n. n>0 and denote the space spanned by all [s 1,..., s l ] and 1 by MD. We call the coefficients σ s1 1,...,s l 1(n) multiple divisor sums. In the case l = 1 these are the classical divisor sums σ k 1 (n) = d n dk 1 and [k] = 1 (k 1)! σ k 1 (n)q n. n>0

12 multiple divisor sums and modular forms For r 1 = = r l = 0 we also write [ ] s1,..., s l 1 = [s 1,..., s l ] =: σ s1 1,...,s 0,..., 0 (s 1 1)!... (s l 1)! l 1(n)q n. n>0 and denote the space spanned by all [s 1,..., s l ] and 1 by MD. We call the coefficients σ s1 1,...,s l 1(n) multiple divisor sums. In the case l = 1 these are the classical divisor sums σ k 1 (n) = d n dk 1 and [k] = 1 (k 1)! σ k 1 (n)q n. n>0 These function appear in the Fourier expansion of classical Eisenstein series which are modular forms for SL 2 (Z), for example G 2 = 1 24 [2], G 4 = [4], G 6 = [6]. We will see that we have an inclusion of algebras M Q (SL 2 (Z)) M Q (SL 2 (Z)) MD BD, where M Q (SL 2 (Z)) = Q[G 4, G 6 ] and M Q (SL 2 (Z)) = Q[G 2, G 4, G 6 ] are the algebras of modular forms and quasi-modular forms.

13 bi-brackets - examples [2] = n>0 σ 1 (n)q n = q 3q 2 4q 3 7q 4 6q 5 12q 6..., [ ] 2, 2 = 2q 3 7q 4 23q 5 42q 6 89q 7 142q 8 221q 9 342q 10..., 1, 0 [ ] 2, 2 = q 3 3q 4 10q 5 16q 6 35q 7 52q 8 78q 9 120q 10..., 0, 1 [ ] 1, 1, 1 = 1 ( 12q 6 28q 7 96q 8 481q 9 747q q ), 1, 2, 3 12 [4, 4, 4] = 1 ( q 6 9q 7 45q 8 190q 9 642q q ), 216 [3, 1, 3, 1] = 1 ( q 10 2q 11 8q 12 16q 13 43q 14 70q ). 4

14 bi-brackets There are a lot of relations between bi-brackets. For example the discussion from the beginning gives [ ] 2, 2 = 0, 0 [ ] 1, 1 2 1, 1 [ ] 1, 1. 0, 2 To obtain more relations we have to study the algebraic structure of the space BD which is "similar" to the algebraic structure of the space of multiple zeta values. The brackets [s 1,..., s l ] have a direct connection to multiple zeta values and the Fourier expansion of multiple Eisenstein series.

15 Multiple zeta values Definition For natural numbers s 1 2, s 2,..., s l 1 the multiple zeta value (MZV) of weight s 1... s l and length l is defined by ζ(s 1,..., s l ) = n 1>...>n l >0 1 n s ns l l The product of two MZV can be expressed as a linear combination of MZV with the same weight (stuffle product). e.g: ζ(r) ζ(s) = ζ(r, s) ζ(s, r) ζ(r s). MZV can be expressed as iterated integrals. This gives another way (shuffle product) to express the product of two MZV as a linear combination of MZV. These two products give a number of Q-relations (double shuffle relations) between MZV. Conjecturally these are all relations between MZV.

16 Multiple zeta values Example: ζ(2, 3) 3ζ(3, 2) 6ζ(4, 1) shuffle = ζ(2) ζ(3) stuffle = ζ(2, 3) ζ(3, 2) ζ(5). = 2ζ(3, 2) 6ζ(4, 1) double shuffle = ζ(5). But there are more relations between MZV, which can be proven by using (extended) double shuffle relations. e.g.: ζ(4) = ζ(2, 1, 1), ζ(5) = ζ(4, 1) ζ(3, 2) ζ(2, 3), 16ζ(3, 2, 2) = 18ζ(5, 2) 21ζ(4, 3) 2ζ(7), 5197 ζ(12) = 168ζ(5, 7) 150ζ(7, 5) 28ζ(9, 3). 691

17 bi-brackets - filtrations Filtrations On BD we have the increasing filtrations Fil W given by the upper weight,fild given by the lower weight and Fil L given by the length, i.e., we have for A BD Fil W k (A) := [ ] s 1,..., s l A 0 l k, s1 s l k r 1,..., r Q l Fil D k (A) := [ ] s 1,..., s l A 0 l k, r1 r l k r 1,..., r Q l Fil L l (A) := [ ] s 1,..., s l A r l r 1,..., r Q. l 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 and simlar for the other filtrations.

18 bi-brackets - algebra structure Theorem The space BD is a filtered differential Q-algebra with the differential given by d = q d dq and Fil W,D,L k 1,d 1,l 1 (BD) Fil W,D,L k 2,d 2,l 2 (BD) Fil W,D,L k 1k 2,d 1d 2,l 1l 2 (BD). As in the case of multiple zeta values we also have two different ways, also called stuffle and shuffle, of writing the product of two bi-brackets. Examples: [1] [1] = 2[1, 1] [2] [1] [ ] [ ] [ ] [ ] 1 st 1, 1 1, 1 1 [1] = 1 0, 1 1, 0 1 [ ] [ ] 1 sh 1, 1 [1] = 1 [ ] [ ] , [ ] [ ] [ ] 1, 2 2, 2 1, 3 d = , 4 4, 4 3, 5 [ ] 2 1

19 bi-brackets - algebra structure That the space BD is closed under d = q d dq is easy to see, since d n>0 a nq n = n>0 na nq n one obtains: Proposition The operater d on [ ] s1,..., s l d = r 1,..., r l [ s1,...,s l ] r is given by 1,...,r l l ( [ ]) s1,..., s j 1, s j 1, s j1,..., s l s j (r j 1). r 1,..., r j 1, r j 1, r j1,..., r l j=1 Example: [ ] [ ] k 1 s1 1, s 2 d[k] = k, d[s 1, s 2 ] = s 1 1 1, 0 [ ] s1, s 2 1 s 2. 0, 1

20 bi-brackets - generating series To prove that BD is closed under multiplication we will consider the generating series of bi-brackets Definition For the generating function of the bi-brackets we write X 1,..., X l Y 1,..., Y l := [ ] s 1,..., s l r 1 1,..., r l 1 s 1,...,s l >0 r 1,...,r l >0 X s X s l 1 l Y r Y r l 1 l

21 bi-brackets - generating series For n N set L n (X) := Theorem ex q n Q[[q, 1 e X q n X]]. For all l 1 we have the following two expressions for the generating series of bi-brackets X 1,..., X l Y 1,..., Y l = = (where X l1 := 0) u 1> >u l >0 j=1 u 1> >u l >0 j=1 l e ujyj L uj (X j ) l e uj(x l1 j X l2 j ) L uj (Y 1 Y l j1 ) Corollary (partition relation) For all l 1 we have X 1,..., X l Y 1,..., Y l = Y 1 Y l,..., Y 1 Y 2, Y 1 X l, X l 1 X l,..., X 1 X 2

22 bi-brackets - partition relation The proof of the theorem uses the conjugation ρ which was given by ( ) ( ) u1,..., u l v1 v l,..., v 1 v 2, v 1 ρ : v 1,..., v l u l, u l 1 u l,..., u 1 u 2 on the set of partitions P l (n). It is therefore not surprising that we have the similar looking equality X 1,..., X l Y 1,..., Y l = Y 1 Y l,..., Y 1 Y 2, Y 1 X l, X l 1 X l,..., X 1 X 2.

23 bi-brackets - partition relation Corollary (partition relation in length 1 and 2) For r, r 1, r 2 0 and s, s 1, s 2 > 0 we obtain for the coefficients in the corollary before [ ] [ ] s r 1 =, r s 1 [ ] s1, s 2 = ( )( )[ ] ( 1) k s1 1 k r2 j r2 j 1, r 1 j 1. r 1, r 2 k j s 2 1 k, s 1 1 k 0 j r 1 0 k s 2 1 Examples: [ ] [ ] [ ] 1, 1 2, 2 3, 1 = 2, 1, 1 0, 0 0, 0 [ ] [ ] [ ] [ ] 3, 3 1, 1 1, 1 1, 1 = 6 3, 0, 0 0, 4 1, 3 2, 2 [ ] [ ] [ ] [ ] [ ] 2, 2 2, 2 2, 2 3, 1 3, 1 = , 1 0, 2 1, 1 0, 2 1, 1

24 bi-brackets - algebra structure Lemma For L n (X) = ex q n 1 e X q n we have ( X Y L n(x) L n(y ) = coth 2 = k>0 ) Ln(X) Ln(Y ) 2 B k k! (X Y )k 1 ( L n(x) ( 1) k 1 L n(y ) ) Ln(X) Ln(Y ) 2 Ln(X) Ln(Y ) X Y Proof: By definition it is and by direct calculation L n (X) L n (Y ) = coth(x) = ex e X e X e X = 1 2 e 2X 1 1 e X Y 1 L 1 n(x) e Y X 1 L n(y ). This gives the first equation and the second one follows by the generating series of the X Bernoulli numbers e X 1 = B n n 0 n! X n.

25 bi-brackets - algebra structure Proposition The product of the generating series in length one can be written as: ("stuffle product of the generating series in length one") ( ) X 1 Y 1 X 2 Y 2 = X 1, X 2 Y 1, Y 2 X 2, X 1 Y 2, Y 1 1 X 1 X 1 X 2 Y 1 Y 2 X 2 Y 1 Y 2 ( ) B k k! (X 1 X 2 ) k 1 X 1 Y 1 Y 2 X 2 ( 1)k 1 Y 1 Y 2. k=1 ("shuffle product of the generating series in length one") X 1 Y 1 X 2 Y 2 = X 1 X 2, X 1 Y 2, Y 1 Y 2 X 1 X 2, X 2 Y 1, Y 2 Y 1 ( ) 1 X 1 X 2 Y 1 Y 2 Y 1 X 1 X 2 Y 2 ( ) B k k! (Y 1 Y 2 ) k 1 X 1 X 2 Y 1 X 1 X 2 ( 1)k 1 Y 2. k=1

26 bi-brackets - algebra structure Sketch of the proof: For the stuffle product consider X 1 X 2 Y 1 Y 2 = e n1y1 L n (X 1 ) e n2y2 L n (X 2 ) n 1>0 n 2>0 =... n 1>n 2>0 n 2>n 1>0 n 1=n 2>0 = X 1, X 2 Y 1, Y 2 X 2, X 1 Y 2, Y 1 e n(y1y2) L n (X 1 )L n (X 2 ) n>0 and then use the lemma for the term L n (X 1 )L n (X 2 ). For the shuffle product first use the partition relation on the left hand side, i.e. use X1 Y 1 = Y 1 X and then use the partition relation again on the right hand side. 1

27 bi-brackets - algebra structure Idea of proof for the algebra structure: In order to proof that an arbitrary product of bi-brackets is again an element in BD one considers the product of the generating series in general X 1,..., X m X m1,..., X l Y 1,..., Y m Y m1,..., Y l = n 1> >n m>0 n m1> >n l >0 where we again consider all possible sums over shuffles of n 1,..., n m and n m1,..., n l plus the sums with equalities n j = n i with 1 j m and m 1 i l. = n 1>n 2>0 n 3>0 n 1>n 2>n 3>0 n 1=n 3>n 2>0 n 1>n 3>n 2>0 n 1>n 3=n 2>0. n 3>n 1>n 2>0 For each equality n j = n i one uses the lemma to rewrite L nj (X j ) L ni (X i ).

28 bi-brackets - stuffle product Corollary (stuffle product) For s 1, s 2 > 0 and r 1, r 2 0 we have [ ] [ ] [ ] [ ] ( )[ ] s1 s2 st s1, s 2 s2, s 1 r1 r 2 s1 s 2 = r 1 r 2 r 1, r 2 r 2, r 1 r 1 r 1 r 2 ( ) r1 r 2 s1 ( 1) s2 1 ( )[ ] B s1s 2 j s1 s 2 j 1 j r 1 (s j=1 1 s 2 j)! s 1 j r 1 r 2 ( ) r1 r 2 s2 ( 1) s1 1 ( )[ ] B s1s 2 j s1 s 2 j 1 j r 1 (s 1 s 2 j)! s 2 j r 1 r 2 j=1 Notice: If r 1 = r 2 = 0, i.e. when the two brackets are elements in MD, all elements on the right hand side are also elements in MD.

29 bi-brackets - shuffle product Corollary (shuffle product) For s 1, s 2 > 0 and r 1, r 2 0 we have [ ] [ ] s 1 s 2 sh = ( )( ) s 1 s 2 j 1 r 1 r 2 k r 1 r 2 s 1 j r 1 1 j s 1 0 k r 2 1 j s 2 0 k r 1 ( s 1 s 2 j 1 s 1 1 )( ( )[ ] s 1 s 2 2 s 1 s 2 1 s 1 1 r 1 r 2 1 ( ) s 1 s 2 2 r1 s 1 1 j=0 ( ) s 1 s 2 2 r2 s 1 1 j=0 r 1 r 2 k r 1 k ( 1) r 2 B r1 r 2 j1 (r 1 r 2 j 1)! ( 1) r 1 B r1 r 2 j1 (r 1 r 2 j 1)! ) ( 1) r 2 k ( 1) r 1 k [ ] s 1 s 2 j, j k, r 1 r 2 k [ ] s 1 s 2 j, j k, r 1 r 2 k ( )[ ] r 1 r 2 j r 1 j s 1 s 2 1 j ( )[ ] r 1 r 2 j s 1 s 2 1 r 2 j j

30 bi-brackets - stuffle & shuffle product Using the shuffle and stuffle product we obtain linear relations in BD. Example: [ ] [ ] [ ] [ ] 1 2 st 1, 2 2, 1 = 35 [ ] [ ] , 3 4 3, 4 4, [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 2 sh 1, 2 1, 2 1, 2 1, 2 2, 1 = , 7 1, 6 2, 5 3, 4 1, 6 [ ] [ ] [ ] 2, 1 2, 1 2, [ ] 2 1 [ ] 2 2, 5 3, 4 4, [ ] 2 8

31 bi-brackets This procedure works in general. For example the stuffle product for the generating series of length one and length two is given by X 1 Y 1 X 2, X 3 Y 2, Y 3 = X 1, X 2, X 3 Y 1, Y 2, Y 3 X 2, X 1, X 3 Y 2, Y 1, Y 3 X 2, X 3, X 1 Y 2, Y 3, Y 1 ( ) 1 X 1, X 3 X 1 X 2 Y 1 Y 2, Y 3 X 2, X 3 Y 1 Y 2, Y 3 ( ) 1 X 2, X 1 X 1 X 3 Y 2, Y 1 Y 3 X 2, X 3 Y 2, Y 1 Y 3 k=1 k=1 B k (X1 X2)k 1 k! B k (X1 X3)k 1 k! ( ) X 1, X 3 Y 1 Y 2, Y 3 X 2, X 3 ( 1)k 1 Y 1 Y 2, Y 3 ( ) X 2, X 1 Y 2, Y 1 Y 3 X 2, X 3 ( 1)k 1 Y 2, Y 1 Y 3

32 bi-brackets - relations The partition relation and the two ways of writing the product give a large family of linear relations in BD and conjecturally these are all relations. Indeed there are so many relations that numerical experiments suggest the following conjecture: Conjecture The algebra BD of bi-brackets is a subalgebra of MD and in particular it is Fil W,D,L k,d,l (BD) Fil W,L kd,ld (MD). This conjecture is interesting, because the elements in MD have a connection to multiple zeta values.

33 bi-brackets - connections to mzv Denote the space of all admissible brackets by qmz := [s 1,..., s l ] MD s1 > 1 Q. Proposition For [s 1,..., s l ] Fil W k (qmz) define the map Z k by then it is 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 1 s l = k, 0, s 1 s l < k. The map Z k is linear on Fil W k (qmz), i.e. relations in Fil W k (qmz) give rise to relations between MZV. Example: [4] = 2[2, 2] 2[3, 1] [3] 1 3 [2] Z = 4 ζ(4) = 2ζ(2, 2) 2ζ(3, 1).

34 bi-brackets - connections to mzv All relations between MZV are in the kernel of Z k and therefore we are interested in the elements of it. Theorem For the kernel of Z k we have For s 1 s l < k it is Z k ([s 1,..., s l ]) = 0. If f Fil W k 2(MD) then Z k (d(f)) = 0. Every cusp form f Fil W k (MD) is in the kernel of Z k. But these are not all elements in the kernel of Z k. There are elements in the kernel of Z k which can t be "described" by just using elements of MD in the list above.

35 bi-brackets - connections to mzv In weight 4 one has the following relation of MZV ζ(4) = ζ(2, 1, 1), i.e. it is [4] [2, 1, 1] ker Z 4. But this element can t be written as a linear combination of cusp forms, lower weight brackets or derivatives. But one can show that [4] [2, 1, 1] = 1 2 (d[1] d[2]) 1 [ ] 2, 1 3 [2] [3] 1, 0 [ 2,1 and 1,0] ker Z4. [ s1,...,s In general one can show that most of the bi-brackets l ] r where at least one 1,...,r l r j 0 is in the kernel of Z k. Conjecture (rough version) Every element in the kernel of Z k can be desribed by using bi-brackets.

36 Summary bi-brackets are q-series whose coefficients are rational numbers given by sums over partitions. The space BD spanned by all bi-brackets form a differential Q-algebra and there are two different ways to express a product of bi-brackets. This give rise to a lot of linear relations between bi-brackets and conjecturally every element in BD can be written as a linear combination of elements in MD. The elements in MD have a connection to multiple zeta values and elements in the kernel of Z k give rise to relations between them. Conjecturally the elements in the kernel of Z k can be described by using bi-brackets. In a recent joint work with K. Tasaka it turns out that bi-brackets are also necessery to give a definition of "shuffle regularized multiple Eisenstein series".