Some examples of Mahler measures as multiple polylogarithms

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1 Some exmples of Mhler mesures s multiple polylogrithms Mtilde N. Llín, University of Texs t Austin. Deprtment of Mthemtics. University Sttion C. Austin, TX 787, USA Abstrct The Mhler mesures of certin polynomils of up to five vribles re given in terms of multiple polylogrithms. Ech formul is homogeneous nd its weight coincides with the number of vribles of the corresponding polynomil. Key words: Mhler mesure, L-functions, polylogrithms, hyperlogrithms, polynomils, Jensen s formul Introduction Ltely there hs been some interest in finding explicit formule for the Mhler mesure of polynomils. The (logrithmic) Mhler mesure of polynomil P C[x,...,x n ] is defined s m(p) = log P(x (πi) n,...,x n ) dx,..., dx n x x n T n where T n = {(z,...,z n ) C n z =... = z n = } is the n-torus. For the one-vrible cse, Jensen s formul π π log e iθ α dθ = log + α E-mil ddress: mllin@mth.utexs.edu Supported by Hrrington Fellowship Preprint submitted to Elsevier Science 5 October 5

2 (where log + x = log x if x nd zero otherwise), provides simple expression of the Mhler mesure s function on the roots of the polynomil: Given P(x) = j (x α j ), then m(p) = log + j log + α j The two-vrible cse is much more complicted. Severl exmples with explicit formule hve been produced. Boyd [,3] nd Smyth [9] hve computed severl exmples nd expressed some of them in terms of specil vlues of L-functions of qudrtic chrcters. Also Boyd [3] nd Rodriguez Villegs [8] hve obtined nlogous results with L-functions of certin elliptic curves. Further, Boyd nd Rodriguez Villegs [4], Millot [7], nd Vndervelde [] hve produced exmples where the Mhler mesure is expressed s combintions of dilogrithms. There re only few exmples for three vribles. Smyth [] relted the mesure of +bx +cy +(+bx+cy)z to combintions of trilogrithms nd dilogrithms. Vndervelde [] obtined the mesure of + x + z(x y) s combintions of trilogrithms. In this pper, we express the Mhler mesure of some prticulr cses of up to five vrible polynomils s combintions of multiple polylogrithms. More precisely, we determine the Mhler mesure of ( + w )...( + w n ) + ( w )...( w n )y in C[w,...,w n,y] for n =,,, 3. We will refer to these s exmples of the first kind. We lso consider ( + w )...( + w n )( + x) + ( w )...( w n )(y + z) in C[w,...,w n,x,y,z] for n =,, (exmples of the second kind). In ddition to these, we use the sme method strting from Millot s exmple [7], in order to compute the Mhler mesure of ( + w)( + y) + ( w)(x y)

3 Summry of the results for the cse = In order to be concrete nd for future reference, we summrize the results obtined for the prticulr cse of = : πm(( + x) + ( x)y) L(χ 4, ) π m(( + w)( + x) + ( w)( x)y) π 3 m(( + v)( + w)( + x) + ( v)( w)( x)y) 7ζ(3) 7πζ(3) + 4 j<k ( ) j (j+) k π m(( + x) + (y + z)) 7 ζ(3) π 3 m(( + w)( + x) + ( w)(y + z)) π 4 m(( + v)( + w)( + x) + ( v)( w)(y + z)) π L(χ 4, ) + 8 j<k 93ζ(5) ( ) j+k+ (j+) 3 k π 7 π m(( + w)( + y) + ( w)(x y)) ζ(3) + log The third nd fifth formule cn be lso written s π 3 m(( + v)( + w)( + x) + ( v)( w)( x)y) = 7πζ(3) + 6(L(χ 4,χ ;, ) L(χ 4,χ 4;, )) () π 3 m(( + w)( + x) + ( w)(y + z)) = 7 π ζ(3) 6 log L(χ 4, 3) + 6(L(χ,χ 4 ;, 3) L(χ 4,χ 4 ;, 3)) () Here χ is the principl chrcter nd χ 4 is the rel odd chrcter of conductor 4, i.e. if n mod 4 χ 4 (n) = if n mod 4 otherwise The L functions re defined by χ (k )χ (k )...χ m (k m ) L(χ,...,χ m ;n,...,n m ) := <k <k <...<k m k n k n...km nm This series is bsolutely convergent if Re(n m ) > nd Re(n i ) for i < m. 3

4 We would like to point out tht ll these formule re new, except for m(( + x) + ( x)y), proved in [9], nd m( + x + y + z). 3 Ide of the procedure nd some technicl steps Let P α C[x,...,x n ], polynomil where the coefficients depend polynomilly on prmeter α C. We replce α by α w. A polynomil +w P α C[x,...,x n,w] is obtined. The Mhler mesure of the new polynomil is certin integrl of the Mhler mesure of the former polynomil. More precisely, Proposition Let P α C[x,...,x n ] s bove, then, m( P α ) = πi ( m T P α w +w ) dw w (3) Moreover, if the Mhler mesure of P α depends only on α, then m( P α ) = π m(p x ) α dx x + α (4) PROOF. Equlity (3) is direct consequence of the definition of Mhler mesure. In order to prove equlity (4), write w = e iθ. Observe tht s long s w goes through the unit circle in the complex plne, w goes through the +w imginry xis ir, indeed, w = i tn ( ) θ +w. The integrl becomes, m( P α ) = π π m ( ) P α tn( θ ) dθ = π π ( ) m P α tn( θ ) dθ Now mke x = α tn ( ) θ, then dθ = α dx, x + α m( P α ) = π m (P x ) α dx x + α Our ide is to integrte the Mhler mesure of some polynomils in order to get the Mhler mesure of more complex polynomils. We will need the following: 4

5 Proposition Let P with > be polynomil s before, (its Mhler mesure depends only on α ) such tht F() if m(p ) = G() if > Then m( P ) = π F(x) dx x + + ( ) dx G π x x + (5) PROOF. The Proof is the sme s for eqution (4) in Proposition, with n dditionl chnge of vribles x in the integrl on the right. x Recll the definition for polylogrithms which cn be found, for instnce, in Gonchrov s ppers, [5,6]: Definition 3 Multiple polylogrithms re defined s the power series k x x k...x km m Li n,...,n m (x,...,x m ) := <k <k <...<k m k n k n...km nm which re convergent for x i <. The weight of polylogrithm function is the number w = n n m. Definition 4 Hyperlogrithms re defined s the iterted integrls I n,...,n m ( :... : m : m+ ) := m+ dt dt dt dt... dt dt dt dt t t t t }{{} t t t }{{} m t n n... dt t } {{ } n m where n i re integers, i re complex numbers, nd b k+ dt t b... dt t b k = t... t k b k+ dt t b... dt k t k b k The vlue of the integrl bove only depends on the homotopy clss of the pth connecting nd m+ on C\{,..., m }. To be concrete, when possible, we will integrte over the rel line. 5

6 It is esy to see (for instnce, in [6]) tht, ( I n,...,n m ( :... : m : m+ ) = ( ) m Li n,...,n m, 3 m,...,, ) m+ m m Li n,...,n m (x,...,x m ) = ( ) m I n,...,n m ((x...x m ) :... : x m : ) which gives n nlytic continution to multiple polylogrithms. For instnce, with the bove convention bout integrting over rel segment, simple polylogrithms hve n nlytic continution to C \ [, ). There re modified versions of these functions which re nlytic in lrger sets, like the Bloch-Wigner dilogrithm, D(z) := Im(Li (z)) + log z rg( z) z C \ [, ) (6) which cn be extended s rel nlytic function in C \ {, } nd continuous in C. We will eventully use some properties of D(z): D( z) = D(z) ( D R ) (7) θ log sin t dt = D(e iθ ) = n= sin(nθ) n (8) More bout D(z) cn be found in []. Often we will write polylogrithms evluted in rguments of modulo greter thn, mening n nlytic continution given by the integrl. Although the vlue of these multivlued functions my not be uniquely defined, we will lwys get liner combintions of these functions which re one-vlued, since they represent Mhler mesures of certin polynomils. Now recll eqution (4). If the Mhler mesure of P α is( liner combintion α of multiple polylogrithms, nd if we write = i ) x + α x+i α x i α, then it is likely tht the Mhler mesure of P α will be lso liner combintion of multiple polylogrithms. This will be the bsis of our work. In order to express the results more clerly, we will estblish some nottion. Definition 5 Let G := σ, σ, τ ( = Z/Z Z/Z Z/Z) 6

7 n belin group generted by the following ctions in the set (R ) : σ : (,b) (,b) σ : (,b) (, b) ( τ : (,b), b) Also consider the following multiplictive chrcter: χ : G {, } χ(σ ) = χ(σ ) = χ(τ) = Definition 6 Given (,b) R,, define, log(,b) := log Definition 7 Let R, x,y C, L r(x) := Li r (x) Li r ( x) L r:(x) := log (Li r (x) Li r ( x)) L r,s(x,y) := ( χ(σ)li r,s ((x,y), ) σ ) σ G L r,s:(x,y) := σ Gχ(σ) ( log, ) σ ( ( Li r,s (x,y), ) σ ) where (x,y ) (x,y ) = (x x,y y ) is the component-wise product. Observtion 8 Let R, x,y C, then, L r,s(x,y) = L r,s(x, y) = L r,s( x,y) nd nlogously with L r,s: 7

8 Observe lso tht the weight of ny of the functions bove is equl to the sum of its subindexes. We will need some technicl Propositions: Proposition 9 Given R > L w +w n T (i) dw w = 4nL n+() + 4L n:() (9) PROOF. By definition, L w +w n T (i) dw w = T = i ( w (Li n i + w (Li n (ix) Li n ( ix)) ) ( )) w Li n i dw + w w dx x + () (which cn be proved in the sme wy s eqution (4) in Proposition ). Recll tht ( ) Li n (ix) = I n ix : = dt... dt t t + i dt t x }{{} n using this nd the fct tht t+ i t i x x = ix, the integrl in () becomes t x + 8 t t... t n x dt t x + dt t... dt n t n dx x + () But x dx (t x + )(x + ) = = ( x + t (t ) log = t t ) x dx x + t ( x + t ) x + ds s + log t () 8

9 The integrl in () becomes 8 t t... t n t ds s dt t dt t... dt n t n 8 log t t... t n dt t dt t... dt n t n Although the sum of these two integrls is well defined, ech of them is not defined for > if we choose the rel segment [, ] s the integrtion pth. We choose different integrtion pth, such s γ(θ) = eiθ + for π θ. By using t = t t, nd ds + t = t s t ds ds s t n s we get ( ) 4n (I n+ : I n+ ( )) ( ) : 4 log (I n : I n ( )) : = 4n(Li n+ () Li n+ ( )) + 4 log (Li n () Li n ( )) = 4nL n+() + 4L n:() Observe tht we will only use the bove Proposition for the cse n =. Proposition For z = e iθ, R >, L x n(z) x + dx + L x n(z) x + dx = i L n,(iz, i) (3) PROOF. By definition, the sum of the integrls is equl to ( (Li n (zx) Li n ( zx)) x + + ) dx (4) x + Using tht dt Li n (zx) = t zx dt t... dt t = t... t n x dt t z dt t... dt n t n The term in (4) with t... t n x ( is equl to x + t + z ) dt t dt t z... dt n t n x + dx 9

10 Writing ( = i ) x + x+i x i, we get i (I n, ( ) ( ) ( ) z : i : I n, z : i : + I n, z : i : I n, ( )) z : i : = i ( (Li n, iz, i ) ( Li n, iz, i ) ( + Li n, iz, i ) ( Li n, iz, i )) The other integrl cn be computed in similr wy, (or tking dvntge of the symmetry ) ( ) i iz (Li n,, i Li n, ( iz ) ( ) iz, i + Li n,, i Li n, ( iz )), i Adding both lines, we get the result. Proposition For z = e iθ, R >, T v L x +v dv n(z) x + v dx v + +v T L x n(z) v +v x +v + v dx dv v = i(l n,(z, ) + L n,:(z, )) (5) PROOF. Consider the first integrl. By definition, this is T v +v dv (Li n (zx) Li n ( zx)) x + v dx v +v We do the sme chnge of vribles s in the Proof of eqution (4) in Proposition nd we get 4i (Li n (zx) Li n ( zx)) y dx x + y dy y + (6) In the sme wy s in (), we hve: y dy (x + y )(y + ) = x log + x ds s

11 Integrl (6) becomes: 4i t... t n x ( t + z ) dt t dt... t z dt n t n log + x ds dx s x This integrl decomposes into two summnds, one with x ds nd the other s with log. But, s before, when we do this, ech summnd not longer converges if we integrte on the rel intervl [, ] nd if. So, we will chnge the pth of integrtion s we did before, to γ(θ) = eiθ + for π θ. We first compute the integrl with we get x ds s. By using tht ( = ) x x x+, ( ) i (I n, z : : I n, ( ) z : : + I n, ( ) ( )) z : : I n, z : : ( = i (Li n, z, ) ( Li n, z, ) ( + Li n, z, ) ( Li n, z, )) The term with log yields ( ) i log (I n, z : : I n, ( ) z : : + I n, ( ) ( )) z : : I n, z : : ( = i log (Li n, z, ) ( Li n, z, ) ( + Li n, z, ) ( Li n, z, )) The other integrl is bsolutely nlogous, except tht we use y dy (y x + )(y + ) = x log x ds s (we cn lso compute it using the symmetry ). 4 Exmples of the first kind The Mhler mesure of the polynomils tht we study in this section depends only on the bsolute vlue of the prmeter α, hence, we will write the formule with = α in order to simplify nottion.

12 We strt with the simple polynomil + y, whose Mhler mesure is m( + y) = π π log + e iθ dθ = log + (7) The first ppliction of our procedure yields: Theorem For R >, πm(( + x) + ( x)y) = il (i) (8) PROOF. By eqution (4) in Proposition, ( πm(( + x) + ( x)y) =πm( + x) + πm + x ) + x y (we mde z = w ). = log + z dz z + = = i w ds s ( w + i w i log w ) dw dw w + (9) = i (I ( i ) ( )) i : I : = i (Li (i) Li ( i)) = il (i) Recll tht we men the nlytic continution of Li. If we wnt to void this nd work with the series, the formul should be stted in the following wy: il πm(( + x) + ( x)y) = (i) if π log il (i) if > () The cse is cler. For the > cse, m(( + x) + ( x)y) =m(( x)y + ( + x)) ( ) = log + m ( x)y + ( + x) which proves the formul ().

13 Now we pply the procedure gin: Theorem 3 For R >, π m(( + w)( + x) + ( w)( x)y) = 4L 3() L :() () PROOF. By Proposition 9, π m(( + w)( + x) + ( w)( x)y) = L w +w T (i) dw w = 4L 3() L :() As before, we cn express this with the following formul: π 4L m(( + w)( + x) + ( w)( x)y) = 3() L :() if π log + 4L3 () L : () if > Note tht we could compute the sme Mhler mesure using the formul () for the Mhler mesure of m(( + x) + ( x)y). By doing this, we obtin different formul for the Mhler mesure of the polynomil considered in Theorem 3: Theorem 4 For R >, π m(( + w)( + x) + ( w)( x)y) = iπl (i) L,(, i) PROOF. We integrte formul () nd use Proposition π m(( + w)( + x) + ( w)( x)y) = i L x (i) x + dx +π ( log x) x + dx i L x (i) x + dx () The sum of the first nd third integrls is L,(, i) = L,(, i) becuse of Proposition with n = nd z = i. The second integrl is the sme s the one tht occurs in eqution (9) nd it yields iπl (i). 3

14 Adding the three terms together we prove the sttement. If we compre the two formule tht we hve got for m(( + w)( + x) + ( w)( x)y), we hve proved the following equlity between multiple polylogrithms: Corollry 5 For R >, 4L 3() L :() = iπl (i) L,(, i) (3) PROOF. This Corollry is obtined from the two formule for the Mhler mesure of ( + w)( + x) + ( w)( x)y. See the Appendix for direct proof. Let us do the process of integrtion one more time. Theorem 6 For R >, π 3 m(( + v)( + w)( + x) + ( v)( w)( x)y) = 4πL 3() πl :() i(l,(i, ) + L,:(i, )) (4) PROOF. We will integrte eqution () of Theorem 4. The term of higher weight corresponds to T L x (i) x + v +v v +v dv dx v T L x (i) v +v x +v + v dx dv v We solve this prt by Proposition, setting z = i nd n =. The third term is: π L v +v T (i) dv v = 4πL 3() πl :() by Proposition 9. 4

15 5 Exmples of the second kind In this section we still hve tht ll the Mhler mesures only depend on = α. We strt with different polynomil, + x + y + z. Theorem 7 For R >, π L m( + x + y + z) = F() = 3() if π log + L3 () if (5) PROOF. This ws proved by Vndervelde []. It is lso possible to dpt some of the proofs of m(+x+y+z) = 7 ζ(3). For instnce, following Smyth π [], For V = m( + x + y + z) = m(( + x + y + z) V ) =m( + x z + xy + xyz) = m(x + y + (x + y)z) = π (Li 3() Li 3 ( )) = π L 3() for. Another possibility is to dpt the elementry proof given in Boyd []. For : ( ) + y π m( + x + y + z) = π m( + y + x( + w)) = π m + w + x = π π π = log + + e it + e is ds dt = (π t) log + e it dt t s π π log + e it log + e is ds dt π s log + e is ds = (here we hve used tht, nd formul (7)). t log + e it dt 5

16 Now use tht log + e it ( ) n = Re n n= n e int = n= ( ) n cos(nt) n (6) n nd pply integrtion by prts, π m( + x + y + z) = π + n= n= ( ) n sin(nt) n n dt = 4 ( ) n sin(nt) n n=(odd) n t π n n 3 = (Li 3() Li 3 ( )) When, use tht ( m( + x + y + z) = log + m + x ) + y + z If we compre with formul (), for exmple, we my wonder whether the formul in the second cse of (5) is vlue of L 3(): F(x)? = L x 3(), x > Mening, s lwys, some brnch of the nlytic continution of Li 3. We will see now tht this is flse. We should hve for x >, ( ( ) ( π log x + Li 3 Li 3? (Li3 (x) Li 3 ( x)) x x)) Differentiting, nd using tht x Li 3(x) = Li (x) π x + ( (Li ) ( ))? Li x x x x (Li (x) Li ( x)) Multiplying by x, nd differentiting gin, Li ( x ) ( Li ) = Li (x) Li ( x) x Since x >, the left term is log ( ) x + x 6

17 (using tht the principl brnch of Li is equl to log( x)). The term in the right is equl to t x t + x dt = lim α π α i e iθ dθ e iθ + x + log(x + ) (we integrted on the pth γ(θ) = eiθ + for π θ ). ( ) x = log x = log ( ) x + iπ x ( cosβ e iβ + lim log x β π x ) + log(x + ) γ(θ) = e iθ + for π θ represents the other homotopy clss nd in this cse, the integrl is = log ( ) x + + iπ x Hence both functions re not equl, which implies tht F(x) cn not be expressed s L x 3(). Theorem 8 For R >, π 3 m(( + w)( + x) + ( w)(y + z)) = iπ L (i) + il 3,(i, i) (7) PROOF. Applying Proposition to formul (5), π 3 m(( + w)( + x) + ( w)(y + z)) = 4 L x 3() x + dx +π ( log x) x + dx + 4 L x 3() x + dx (8) The sum of the first nd third integrls is il 3,(i, i) becuse of Proposition with n = 3 nd z =. The second term is the sme s the one in integrl (9) in Theorem, equl to iπ L (i). Adding the three terms together we prove the sttement. 7

18 Finlly our result with the mximum number of vribles: Theorem 9 For R >, π 4 m(( + v)( + w)( + x) + ( v)( w)(y + z)) = 4π L 3() π L :() + 4 (L 3,(, ) + L 3,:(, )) (9) PROOF. We will integrte eqution (8) of the bove Theorem s lwys. The highest weight term corresponds to i T v L x +v dv 3() x + v dx v + i +v T L x 3() v +v x +v + v dx dv v which cn be evluted using Proposition setting z =, n = 3. Finlly, π T L v +v (i) dv v = 4π L 3() π L :() by Proposition 9. 6 Integrtion of Millot s formul So fr we hve been considering the cses of + y nd ( + x) + (y + z) nd integrted them severl times. We my wonder wht hppens with n intermedite cse, nmely + x + y. This cse is not so esy to hndle, so we will consider the following vrint of the two vrible cse: +x+( )y, with C. This time the Mhler mesure will depend on the rgument of s well. According to Millot [7], α log + β log b + γ log c + D ( e iγ) b πm( + bx + cy) = π log mx{, b, c } not (3) 8

19 c α b π / θ / β γ = - i tn ( θ / ) A) B) C) Fig.. A) Reltion mong the prmeters in Millot s formul. B) Tringle for the generl cse of + + ( )y. C) Tringle for the cse = i tn ( θ ). Where stnds for the sttement tht, b, nd c re the lengths of the sides of tringle; nd α, β, nd γ re the ngles opposite to the sides of lengths, b nd c respectively (Figure.A). In our prticulr cse, πm( + x + ( )y) = rg log + rg( ) log D() if Im() + D(ā) if Im() < (3) since, nd re the lengths of the sides of the tringle whose vertices re, nd in the complex plne (Figure.B). For the rgument in the dilogrithm, γ = rg(), then we hve to tke or ā so γ is lwys positive. We will integrte s lwys. We replce by w +w. Theorem We hve the following: π m(( + w)( + y) + ( w)(x y)) = L 3() + π log (3) PROOF. We will pply Proposition to eqution (3) nd then the chnge of vribles w = e iθ, which implies w = i tn ( ) θ +w. With tht chnge, will be lwys pure imginry, so rg = π (Figure.C). π m(( + w)( + y) + ( w)(x y)) = = π ( π π log cos ( ) θ + ( ) ( ( ))) θ log θ θ tn + D i tn dθ 9

20 Using the definition (6) of Bloch Wigner dilogrithm, = π ( π ( ) ( ( )))) θ θ log cos + Im Li (i tn π dθ For the first term, mke τ = π θ, this prt becomes, π π by eqution (8). log sin τ dτ = π π (log sin τ log ) dτ = π 4 D(eiπ ) + π log = π log For the second term, mke x = tn ( ) θ, then dθ = dx, the integrl becomes, x + dx Im(Li (i x )) x + = i dx (Li (ix) Li ( ix)) x + = L 3() The lst equlity is prticulr cse of the vlue computed for expression () in Proposition 9 Adding both terms we obtin the result. 7 Concluding remrks To conclude, let us observe tht ll the presented formule shre common feture. Let us ssign weight to ny Mhler mesure, to π nd to ny logrithmic function. Then ll the formule re homogeneous, mening ll the monomils hve the sme weight, nd this weight is equl to the number of vribles of the corresponding polynomil. Appendix : Some dditionl remrks bout the cse = In this ppendix, we would like to give some dditionl detils bout the computtion of the specific formule tht occur in the tble of results for =. All but two of these formule cn be directly deduced from the Theorems we

21 hve proved. These exceptions re: the formul with the term ζ(5) nd formul (). For the formul with the term ζ(5): Theorem We hve the following, π 4 m(( + v)( + w)( + x) + ( v)( w)(y + z)) = 93ζ(5) (33) PROOF. By Theorem 9 we hve to prove tht 4π L 3() + 4 L 3,(, ) = 93ζ(5) (34) i.e., 7π ζ(3) + 8(Li 3, (, ) Li 3, (, ) + Li 3, (, ) Li 3, (, )) = 93ζ(5) Now we use formul (75) of [], which in this prticulr cse, sttes tht Li 3, (x,y) = Li 5(xy) + Li 3 (x) Li (y) + 3Li 5 (x) + Li 5 (y) Li (xy)(li 3 (x) + Li 3 (y)) for x,y = ±. Tking into ccount tht Li k () = ζ(k) nd Li k ( ) = ( k ) ζ(k) (35) we get Li 3, (, ) Li 3, (, ) + Li 3, (, ) Li 3, (, ) = 93 ζ()ζ(3) ζ(5) We obtin the result by using tht ζ() = π 6 For formul () we hve the following Proposition We hve j<k ( ) j+k+ (j + ) 3 k = 7 4 ζ(3)l(χ 4, ) 3 ζ()l(χ 4, ) log L(χ 4, 3) + <m<n (m even) χ 4 (n) m n 3

22 PROOF. Writing l = j + nd l + n = k, the left side is equl to j<k ( ) j+k+ (j + ) 3 k = <l,n (l,n odd) ( ) (l )+(l+n) + l 3 (l + n) = <l,n (l odd) χ 4 (n) l 3 (l + n) Now write: l 3 (l + n) = l 3 n l n + l n 3 n 3 (l + n) Using formule (35), we get the first two terms of the sttement. We need to look t the lst two terms together in order to ensure convergence, χ 4 (n) ( n 3 l ) = χ 4 (n) n log + l + n <n n 3 m <n <l (l odd) <m (m even) nd the sttement follows. Formul () cn be obtined from the bove Proposition nd the fct tht L(χ 4, ) = π 4. Appendix : A direct proof for Corollry 5 Recll the sttement: Corollry 5 For R >, 4L 3() L :() = iπl (i) L,(, i) PROOF. First observe tht fter the chnge the equlity (3) remins the sme with the cnceltion of term of the form π log in ech side. Then, it is enough to prove eqution (3) for <. Eqution (3) is equivlent to 8 n=(odd) n n 3 4 log dt + 4 t + dt t n=(odd) n n? = iπ n=(odd) ( ) dt t + + dt t + i n n n

23 = π +4 n=(odd) rctn s s ( ) n n n ( π rctn ( s ) ) rctn(s) ds (36) Our strtegy will be s follows: we will prove the equlity for the derivtives nd for the prticulr cse =. In order to prove the equlity for the derivtives, we will do the sme, i.e., we will exmine the cse = nd differentite gin nd compre the second derivtives. Let us strt with =. The term in (36) becomes = π n=(odd) ( ) n n + π rctn s s ds 8 rctn s s ds Mke s = tnx: rctn s s ds = π 4 x sin x cos x dx = x log(tnx) π 4 π 4 x(log( sinx) log( cosx)) dx The first term is zero. For the second term mke y = π x = = π 4 π x log( sinx) dx + s log( sins) ds + π π π 4 π π 4 (π y) log( siny) dy log( sins) ds Using properties (7) nd (8), of the Bloch Wigner dilogrithm, = s D(e is ) π π n= sin(ns) n ds π D(eπis ) π π 4 3

24 = n=(odd) n 3 + π n=(odd) ( ) n n Using the power series of +s nd integrting, it is esy to see tht rctn s s ds = n=(odd) ( ) n n Putting ll of this together in eqution (36), we conclude: iπl (i) L,(, i) = 8 n=(odd) n 3 = 4L 3() s we expected. We differentite the originl eqution (36) nd multiply it by : 4 n=(odd) n n 4 log n=(odd) n n? = π n=(odd) i n n n + 4 ( rctn s s + s + ) ds (37) Set =, we get 4 3 ζ() lim 4 log (log( + ) log( )) =? π (log( + i) log( i)) i This is n equlity, becuse the first term in the left nd the term in the right re equl to π nd the other term is zero. Apply integrtion by prts on the lst term of (37): 4 ( rctn s s + s + ( ( ) = π rctn rctn() ) ) ds 4 ( ( ) s ds rctn rctn(s) ) s + 4

25 Now we differentite (37), 4 log +4 ( n=(odd) n? = π n=(odd) ) s + + s ds s + s + (i) n π + And this is n equlity indeed, which cn be seen from the fct: 4 log = 4 ( ) s + s ds s + Acknowledgements I m deeply grteful to Fernndo Rodriguez Villegs for his constnt guidnce nd support nd for shring severl ides tht hve enriched this work. I would lso like to thnk Sm Vndervelde for severl helpful discussions, nd the Referee, whose suggestions led to severl improvements. References [] J. M. Borwein, D. M. Brdley, D. J. Brodhurst, Evlutions of k-fold Euler/Zgier sums: compendium of results for rbitrry k, Electronic J. Combin. 4 (997), no., #R5. [] D. W. Boyd, Specultions concerning the rnge of Mhler s mesure, Cnd. Mth. Bull. 4 (98), [3] D. W. Boyd, Mhler s mesure nd specil vlues of L-functions, Experiment. Mth. 7 (998), [4] D. W. Boyd, F. Rodriguez Villegs, Mhler s mesure nd the dilogrithm (I), Cnd. J. Mth. 54 (), [5] A. B. Gonchrov, Polylogrithms in rithmetic nd geometry, Proc. ICM-94 Zurich (995), [6] A. B. Gonchrov, Multiple polylogrithms nd mixed Tte motives, (Preprint). 5

26 [7] V. Millot, Géométrie d Arkelov des vriétés toriques et fibrés en droites intégrbles. Mém. Soc. Mth. Fr. (N.S.) 8 () 9pp. [8] F. Rodriguez Villegs, Modulr Mhler mesures I, in Topics in number theory (University Prk, PA 997), Mth. Appl., 467, Kluwer Acd. Publ. Dordrecht, 999, pp [9] C. J. Smyth, On mesures of polynomils in severl vribles, Bull. Austrl. Mth. Soc. Ser. A 3 (98), Corrigendum (with G. Myerson): Bull. Austrl. Mth. Soc. 6 (98), [] C. J. Smyth, An explicit formul for the Mhler mesure of fmily of 3-vrible polynomils, J. Nombres Bordeux 4 (), [] S. Vndervelde, A formul for the Mhler mesure of xy + bx + cy + d. J. Number Theory (3), no., 84. [] D. Zgier, The Dilogrithm function in Geometry nd Number Theory, Number Theory nd relted topics, Tt Inst. Fund. Res. Stud. Mth. Bomby (988),