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Pacific Journal of

Mathematics

MULTIPLE HARMONIC SERIES

MIKE HOFFMAN

Vol. 152, No. 2 February 1992

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PA CIFIC JOURNAL OF MATHEMATICSVol. 152, No. 2, 1992

M U L T I P L E H ARMON IC SERIESMICHAEL E. HOFFMAN

We consider several identities involving the multiple harmonic se-ries

v^ 1" ii1'1 #t' 2 nik '

n>n2>->nk>l n\ n2 nkwhich converge when the exponents / , are at least 1 and i\ > 1.There is a simple relation of these series with products of Riemannzeta functions (the case k = 1) when all the i} exceed 1. There arealso two plausible identities concerning these series for integer expo-nents, which we call the sum and duality conjectures. Both generalizeidentities first proved by Euler. We give a partial proof of the dualityconjecture, which coincides with the sum conjecture in one family ofcases. We also prove all cases of the sum and duality conjectureswhen the sum of the exponents is at most 6.

1. Introduction. The problem of computing the doubly infinite se-ries(1) L

nx>n2>\ 1 2which converges when a > 1 and b > 1, was discussed by Euleran d Goldbach in their correspondence of 1742- 3 [3]. Euler evaluatedseveral special cases of (1) in terms of the Riemann zeta function

1n>\

Later, in a paper of 1775 [2],Euler found a general formula for (1) interms of thezeta function when a and b are positive integers whosesum is odd. Thesimplest such result is h r =which hasbeen rediscovered many times since (see [1, p. 252] and thereferences cited there).

We shall consider multiple series of the formn\ n{ ' "nk

275

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27 6 MICHAEL HOFFMANan d

l ' l , 2 , . , / * ) =

(so (1) is S ( , f t) ) . With th is no ta t ion , S(i) = ( ) = ( / ) . Th erelat ion between the S ' s and A 's sho uld be clear: for exam ple,S(i \ , i2) =

an dS(i\ , 12 , *3) = - 4(l > *2 > k ) + (l'i + h , h )

+ A(i 9 i2 + i3) + A(i + i2 + i3) .N o t e th a t (2) imp l ies ,4(2, 1) = C (3) .I t i s immedia te f rom the def in i t ions tha t

5(11, h)+s ( i2, ii) = c ( oc (/ 2 ) + c (/ i + and

A(h , I 2) + ^ ( / 2 , I 'l) = C (/ l) ( l2) - C(/ l + 12)whenever i \ , i^> 1. M ore general ly, if i , i, . . . , i^ > 1 the sums ^ ( l ) J (ik)) a n d k

is the symmetric group of degree fc) can be expressed in termsof the zeta function. We state and prove such formulas in 2.

T h e r e are also two interest ing general conjectures about the quan-t i t ies A(i\ , . . . , fc) for po sitive int eger expo n en ts i\ , . . . , ik, whichwe cal l the sum and duali ty conjectures. Both general ize the identi tyA(2, 1) = - 4(3) . We sta te them in 3 , and give a part ia l proof of theduali ty conjecture in 4. F u r th e r evidence for the two conjectures isdiscussed in 5.

2. Symmetric sums in terms of the zeta function. T o s ta te ou r resul tswe sha ll requ i re som e n o t a t i on . F o r a pa r t it ion = {P\ , P 2, . . . , P / }of th e set {1, 2, . . . , k }, le t

c( ) = ( c ar d - l ) !( c a r d P 2 - 1 ) ! (cardP/ - 1)!.Also, given su ch a a n d a > tu le i = {i\ , . . . , ik} of exponents ,define

5 = 1

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M U L T I P L E H A R M O N I C S E R IE S 277T H E O R E M 2.1. For any real i\, . . . , /* > 1,

(1) J > ( / ( i ) , . . . , / W )= c( )C(i, ).part i t ions of {1 , . . . , k}

Proof. Assume t ha t the ij are all dis t inct . (There is no loss ofgenerality, since we can take limits.) The left - hand side of (1) can bewrit ten

y y - n>:.>nk>\ n ( \ ) n (2) n{k)

N o w t h i n k of t h e s ym m e t r i c g r o u p ^ as a c t i n g on / c - tup les (n\ , . . . ,nk ) of positive in tegers . A given k - t u p le n = ( i , . . . , J i f c ) has ani s o t ro p y g ro u p ^(n) and an a s s o c i a t e d p a r t i t i o n o f {1 , 2 , . . . , / : }: is the set of eq u i v a l en ce classes of the r e l a t i o n given by / ~ j iffm = j, an d fc(n ) = { e k | (i) ~ i V/ }. N o w the t e r m

(2 ) !\ n2

occurs on theleft- han d side of (1) exa ct ly ca rd ^ ( n ) t im es. It occurson the r ight - hand side in those terms corresponding to par t i t ions t h a t are refinements of : letting >: den ote refinem ent , (2) occurs

t imes . Thus , theconclusion will follow ifcard ^ (n ) = ] c{U)

for any k - t up le n and associated part i t ion . To see th i s, n ote tha tc( ) coun t s thepermutations having cycle-type specified by : sinceany element of ^ (n ) has a u n iqu e cycle- type specified by a part i t iont h a t refines , the result follows.

F or k 3, the theorem says

= ( h ) ( i i ) ( h ) + C(+ (h +h ) ( h ) +2(ix + h + 13)

for i\, i2, h > 1. This is themain result of [7].

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27 8 MICHAEL HOFFMANTo state theanalog of 2.1 for the A % we require onemore bit of

n o ta t i on . For a partition = {P\, ... , P[} of {1, 2, . . . , k}, let

T H E O R E M 2.2. For any real i\ , . . . , i^ > 1,

G fe partitions of { 1 , . . . , k]

Proof. We follow the same line of argument as in the precedingproof. The left- hand side is now

nx>n2>- '>nk>\ n (l)n (2) n (k)and a term (2) occurs on the left- hand side once if all the rti aredistinct, and no t at all otherwise. Thus, it suffices to show(3) c ( ) (w ;- ! I 0, otherwise.To prove this, note first that the sign of c( ) is positive if the permu-tations of cycle- type are even, an d negative if they are odd: thus,th e left- hand side of (3) is thesigned sum of the number of even an dodd permutations in the isotropy group k( ). But such an isotropygroup has equal numbers of even and od d permutations unless it istrivial, i.e.unless theassociated partition is {{1}, {2}, . . . , {k}}.

When all the exponents are 2 we have the following result, whichproves a conjecture of C. Moen.

COROLLARY 2.3. For k > 1, 2k

Proof. Applying Theorem 2.2, we obtaini k

1= 1 partitions {P ,...,/>,} of {!,...,*:}

x JJ(cardP 5 - l)!C(2cardP5).5 = 1

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M U L T I P L E HARMONIC SERIES 279Using the well-known formula for values of the zeta function at evenintegers in terms of Bernoulli numbers (see, e.g., [4]), and writing psfor card Ps, we can express the right-hand side as

/ = ! partitions {j* , . . . , />} of {!,...,}

22k-i2k fc 11/ = 1 partitions {/>,,...,/>,}of {!,...,*} i= l

since the sum of the ps is k. Now for a given (unordered) sequence/?i, ... , Pi, the number of partitions {P\, ... , Pt} of {\, ... , k}with card Pj = ps for 1 < J < / is1 f c !

\ \m2\ wk\p\\p2\ P[\ 'where m ; = card{s \ ps = / }. Thus, we can rewrite the sum as{2 )2k ^ 1 - ^ fc! ' '

Ps\(2ps)\

Bin

B lk mLm \ - mk\ \22\J m"\

I t is then enough to prove the following proposition.PROPOSITION 2.4. If Bn denotes the nth Bernoulli number, then fork > 1

B2k

Proof. Define

k = \

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280 M I C H A E L H O F F M ANIf we also let

/ i h / 2 \ f x \ \ x

then (x) = f{x) since (0) = / (0) = 0 andBnX" 1 ( x( 2 f c ) ! t ?

K, 1 lA1 1 1 ex 1 1ex- l x 2 ex- \ x 2 J v J'

( H e r e we have used the generating function for the Bernoulli num-bers.) Thus

__ sinhx/2 ,x^- l j! = JC/2 = e

l\ V 22! 44!Using the multinomial theorem, expand out the right- hand side toobtainfl 2k(2k) \

k=0 m+2m2+- +kmk=k ( ^ - G B2k \ kK2k{2k)\ 4

an d the conclusion follows by equating coefficients of x2k in (4).3. The sum and duality conjectures. We first state the sum conjec-

tu re , which is due to C. Moen [5].Sum conjecture. For positive integers k < n,

ix>\where the sum is extended over / - tuples i\, ... , ik of positive inte-gers with i\ > 1.

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MULTIPLE HARMONIC SERIES 281

Three remarks concerning this conjecture are in order. First, itimplies

as we pr ove below. Second, in the case k = 2 it says thatA(n - 1, 1) + A(n - 2, 2) + + A(l, n - 2) = (n),

or, using the relation between the A's an d S"s an d Theorem 2.1,n-2

2S{n - l, l) = ( + l)(n) - C(*)C(* - *).A : = 2

This was proved in Euler's paper [2] and has been rediscovered severaltimes, in particular by Williams [8]. Finally, C. Moen [5] has provedthe sum conjecture for k = 3 by lengthy but elementary arguments.F or the duality conjecture, we first define an involution on theset 6 of finite sequences of positive integers whose first element isgreater than 1. Let 3 be the set of strictly increasing finite sequencesof positive integers, and let : 6 3 be the function that sendsa sequence in to its sequence of partial sums. If 3

nis the set

of sequences in 3 whose last element is at most n, we have twocommuting involutions Rn and Cn on 3n defined by

andCn(a\9 ... , afi = complement of {a\, . . . , /}in {1,2, . . . ,} arranged in increasing order.

Then our definition of is (J) = - RnCn (I) = - CnRn (I)for / = (z'i, i2, . . . , ik) with i - \ h ik = n . (The reader mayverify that (7) is actually in 6 , has length n-k, an d its elementshave sum n .) For example,

( 3 , 4 ,1) = - 1 C 8 8 ( 3 , 7, 8)= - 1 ( 3 , 4 , 5 , 7 , 8 ) = ( 3 , 1 , 1 , 2 , 1 ) .

We shall say the sequences (i\9 ... , i^) an d (i\, . . . , i^) are dual toeach other, and refer to a sequence fixed by as self- dual.

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282 MICHAEL HOFFMANNow we can state our second conjecture.Duality conjecture.If {h\ , ... , hn_k) is dual to (i\ , . . . , ik), then

A(h 9 . . . , A ^ ) = ^ ( i i , . . . , / * ) .We include some remarks on how may be more easily computed.

T h e set 6 is a semigroup under the operation given by concatena-t ion , and the indecomposables are evidently sequences of the form(A + 1, 1, . . . , 1) with A > 1. I t is easily computed that(2) (A + 1, ^1,- : . ,1J = (k + 1, J , . ; . , 1>

k- \ h- \(In particular, theduality conjecture implies

-2for integer n>2. N ote that this is also the sum conjecture for the casek = n - 1. This case follows from Theorem 4.4 below.) Together withth e following proposition, (2)gives an effective method for computing ( / ) for any / e &.

PROPOSITION 3.1. For Ix,I2e , (IxI2) = {I2){Ix).Proof. By induction on the number of indecomposables in I 2 wecan reduce to thecase where I 2 is indecomposable. So leth = (h , , ik) > h = (A + 1, 1,..., 1),

n = ix H h / fc, m = n + h + l.Then

= ' ^ i ! , /i + i2, .. . , /i + + ik, n + A+ 1,rt + A+ 2, . . . , n + h + l)

, c2 - c\ , . . . , cn__k- cn_k_),

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MULTIPLE HARMONIC SERIES 283where CnRn (I\) = (c\ , C2, . . . , cn- k) * from which the conclusionfollows.

We close this section by proving that the sum conjecture implies(1). We first note that has a partial order given by refinement, e.g.(2, 1, 2) and (3 , 1, 1) both refine (3 , 2) . Further, S(i , .. . , ik) isthe sum of those A(j\9 . . . , j)) for which (i 9 . . . , ik) is a refinementof C/i9 9 Ji) Thus the sum

{- \

can be written as a sum of terms A(j\, . . . , j)), each of which appearswith multiplicity

card{(/ i, . . . , ik)\h > 1 and (i , .. . , ik) refines (jx, . . . , 7/)}

The equality can be seen combinatorially: think of an ordered sumi\ H h z = n that refines 7 H \ - J = n as defined by choosing/c - / division points (in addition to those defining the first sum) outof n - / - 1 possibilities. Thus

Assuming the sum conjecture, the latter sum is

4. Partial proof of the duality conjecture. We shall prove the dualityconjecture for sequences (i 9 1, . . . , 1) with i\ > 1. We use thefollowing theorem of L. J. Mordell [6].

THEOREM 4.1 (Mordell). For positive integer k and any a> k,y ! = fc,f (- ' (a- i\ > , i 2 *( i + + * + ) " j / ! ( / + 1 ) * + * \ i J't h ^ 1

F r o m this we deduce the following result.

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284 MICHAEL HOFFMANCOROLLARY 4.2. For integer h >1,

= k\A(k+, 1 , . . . , 1).

Proof. Differentiate Mordel s formula p times with respect to ato get

' ' ' + Hk

i=p ^ ' '

Now set a = 0 to obtain

from which theconclusion follows by setting h = p + 1.

O n theother hand, we can use thefollowing rearrangement lemmafor multiple series to rewrite theleft- hand side of 4.2 anotherway.

LEMMA 4.3. Let f be a symmetric function in k variables. Then

provided thesums converge.

Proof.We proceed by induction on k. The result is immediate for

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MULTIPLE HARMONIC SERIES 285k = 1. Now assume it for k and consider

(k + 1 ) !/ ( n i , . . . , nk+)- 1 ' **' ' A:

1 + + * + i = 1where / is symmetric in k + 1 variables. Since the function

^ (k+l)f(nlt...,nk+) _, "1+ + *+ !

is symmetric in / i , . . . 5 ^ , w e can apply the induction hypothesist o transform the right- hand side of (1) intof(n 9 ... , nk+ )

Let 5" be this sum divided by k + 1. Then

n,...,nk+i>l= 0 01 \ ^ f(n\,..., nk+)(n + + nk)

_ v^

By permuting the variables, we see that each of the k terms in the

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286 MICHAEL HOFFMANla t ter summation is just S . Thus, the sum is kS and we have

/ ( / i i , . . . , n k + { ) _ Un,, , n t + l> l { k + \ ) \ f{ nx, . . . ,nk + )

(i + + fc+ i)'Applying thelemma with f{n\,... , nk) = { \ H 1 nk)~h gives

fc- l , , , t > l1 ^ 1

P u t t i n g this together with Corollary 4.2, we haveTHEOREM 4.4. For integers h, k > 1,

A(h + 1, 1,. ; . , l j = : 1 - l

F r o m theremarks in 3, Theorem 4.4 is just theduality conjecturefor indecomposable sequences of 6 .5. Evidence for the conjectures. For the computations of thissec-t ion , thefollowing result will be useful.

THEOREM 5.1. Let i\, i2, . . . , ik be any sequence of positive inte-gers with i\ > 1. Thenk

i- 2 A(ii, .,i- i,i- J,J+l,i+i,. .,ik)

\

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MULTIPLE HARMONIC SERIES 287where we write sr forn\ + hnr. But the right- hand side of (1)can be written

LIan d by astandard partial- fractions identity we have

1 = 1 1l4 ' *+ l f 0^ + 1 4 ' 7= 0or

T h u s , (2) can be rewritten as1

i,-2 ,+ Y. - '^' 2^' Z lk-\ l2 7 + 1 Zl~^7 jj j _ j _ i _ 1 1 / ^ 1 /C rC~rlor, since the first sum isunchanged by permuting n^ an d n^+i ?

'+ l 7=0Now we use the partial- fractions expansion

to obtain1

n,...,nk+i>lS1S2 Sk_

- h k- i k+7 = 0

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288 MICHAEL HOFFMANContinuing in this way, we conclude that (2) equals

i x - j 9 j + i 9 i 2 9 m m . 9 i k )7 = 0

7 = 0

7 = 0

a n d since the last sum is A(i\, . . . , /^, 1) the conclusion follows bysubstitution for (2) on the right- hand side of (1) and appropriate can-cellation.

N o t e that by taking k = 1 in Theorem 5.1 we get

which is just the sum conjecture for two arguments. Recall that thesum conjecture for n - 1 arguments (where n is the sum of the argu-ments) follows from Theorem 4.4: using this together with Theorem5.1 applied to the sequence

-3

we get the sum conjecture for n - 2 arguments.Now we consider relations among the quantities A(i , ... , ik) with

n = i\ + - - + ik

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MULTIPLE HARMONIC SERIES 289so A(2, 3) = A(2, 1, 2) (aninstance of the duality conjecture). Now ( 3 , 1, 1) = A(4, 1) by Theorem 4.4, so A(2,2,l) = A(3, 2) from(3) and all instances of the duality conjecture hold in this case.

For n = 6, we get the sumconjecture immediately for k = 2, 4, 5.Theorem 5.1applied to thesequences ( 4, 1), ( 3 , 2 ) , and ( 2, 3) givesrespectively(4) A(5, l) +A(4,2) = A(4,l,l) +A(3,2, 1) +A(2,3, 1)(5) A(4,2) +A(3,3) = A(3, 1, 2) + A(2, 2, 2) +A{3, 2, 1)(6) A(3, 3) +A(2, 4) = A(2, 1, 3) +A(2, 3, ) + A(2,2,2).O n theothe r ha nd , 5.1a p p l i e d tothes e q u e nc e s ( 3 , 1, 1) , ( 2 , 2 , 1 ),a n d ( 2 , 1 , 2 ) g i v e(7) A(4,l,l) + A(3,2,l)+A(3,l ,2)

= A(3, 1, 1, l)+A(2,2, 1, 1)(8) A(3,2, l) +A(2,3,) + A(2,2,2)

= A(2,l ,2,l) +A(2,2,l, 1)(9) A(3,l,2)+A(2,2,2) + A(2,l,3)

= A(2, l,l,2)+A(2, 1 , 2 , 1 )respectively. Since we know the sumconjecture holds for k = 2 andk = 4 in this case, the sum of the left- hand sides of (4) and (6) isC(6), as is the sum of the right- hand sides of (7) and (9). Thus

or A(2, 3, 1) = A(3, 1, 2), an instance of the duality conjecture.( N o t e that all other sequences of length 3 are self-dual.) Using thisfact, we can add equations (4) and (6) to get the sumconjecture fork = 3. Also, we canconclude from (4) and (7) that

A(5, l) +A(4,2) = A(3, 1, 1, 1)+ A{2, 2, 1, 1).B ut A(5, 1) = A(3, 1, 1, 1) from Theorem 4.4, so this means A(4, 2)= A{2, 2, 1, 1). Now we can use (5) and (8) to conclude similarlythat A{3, 3) = A(2, 1, 2 , 1), and finally (6) and (9) to get A(2, 4) =A{2, 1, 1, 2). Thus all instances of both conjectures are true whenn = 6.

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290 MICH AEL HOFFM ANR E F E R E N C E S

[1] Bruce C. Berndt, Ramanujars Notebooks, Part I, Springer-Verlag, New York,1985.[2] L. Euler, Medtationes circa singulare serierum genus, Novi Comm. Acad. Sci.P e t r o p o l , 20 (1775), 140- 186. Reprin ted in "Opera Omnia", ser. I, vol. 15,B. G . Teubner, Berlin, 1927, pp . 217- 267.

[3] P. H. Fuss (ed.), Correspondence Mathematique et Physiquede quelques c lebresG ometres (Tome 1), St. Petersburg, 1843.

[4] K. Ireland and M. Rosen, A ClassicalIntroduction to Modern Number Theory,Springer-Verlag, New York, 1982. Chap. 15.

[5] C. Moen, Sums of multiple series, preprint.[6] L. J. Mordell, On the evaluation of some multiple series, J. London Math. Soc,

33 (1958), 368- 371.[7] R. Sita Ramachandra Rao and M. V. Subbarao, Transformation formulae for

multiple series, Pacific J. Math., 113 (1984), 471- 479.[8] G. T. Williams, A new method of evaluating (2n), Amer. Math. Monthly, 60

(1953), 19- 25.Received May 1, 1990 and in revised form March 29, 1991. The author was partiallysupported by a grant from the Naval Academy Research Council. He also thankshis colleague Courtney Moen for introducing him to the subject of this paper anddiscussing it with him.

U. S. NAVAL ACADEMYANNAPOLIS , MD 21402

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PACIFIC JOURNAL OF MATHEMATICSV. S. VARADARAJAN(Managing Editor)University of CaliforniaLos Angeles, CA 90024-1555-05HERBERT CLEMENSUniversity of UtahSalt Lake City, UT 84112F. MICHAEL CHRISTUniversity of CaliforniaLos Angeles, CA 90024-1555THOMAS ENRIGHTUniversity of California, San DiegoLa Jolla, CA 92093

EDITORSNICHOLAS ERCOLANIUniversity of ArizonaTucson, AZ 85721R. FINNStanford UniversityStanford, CA 94305VAUGHAN F. R. JONESUniversity of CaliforniaBerkeley, CA 94720STEVEN KERCKHOFFStanford UniversityStanford, CA 94305

C. C. MOOREUniversity of CaliforniaBerkeley, CA 94720MARTIN SCHARLEMANNUniversity of CaliforniaSanta Barbara, CA 93106HAROLD STARKUniversity of California, San DiegoLa Jolla, CA 92093

ASSOCIATE EDITORSR. ARENS E. F. BECKENBACH B. H. NEUM ANN F. WO LF(1906-1982) (1904-1989)SUPPORTING INSTITUTIONS

K. YOSHIDA

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PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATIONCopyright 1992 by Pacific Journal of Mathematics

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Pacific Journal of MathematicsVol. 152, No. 2 February, 1992

Edoardo Ballico, On the restrictions of the tangent bundle of the

Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Edward Burger, Homogeneous Diophantine approximation inS- i n t e g e r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 1

Jan Dijkstra, Jan van Mill and Jerzy Mogilski, The space of

infinite-dimensional compacta and other topological copies of(l2f) . . . . 255

Mike Hoffman, Multiple harmonic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

Wu Hsiung Huang, Superharmonicity of curvatures for surfaces of constant

mean curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

George Kempf, Pulling back bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

Kjeld Laursen, Operators with finite ascent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Andrew Solomon Lipson, Some more states models for link invariants . . . . . 337Xiang Yang Liu, Bloch functions of several complex variables . . . . . . . . . . . . . 347

Madabusi Santanam Raghunathan, A note on generators for arithmetic

subgroups of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Marko Tadic, Notes on representations of non-Archimedean SL(n) . . . . . . . . 375