For k a positive integer, the k-higher Mahler measure of a non-zero, n-variable, rational function P (x 1,..., x n ) C(x 1,..., x n ) is given by

Transcription

1 HIGHER MAHLER MEASURE OF AN n-variable FAMILY MATILDE N LALÍN AND JEAN-SÉBASTIEN LECHASSEUR Abstract We prove formulas for the k-higher Mahler measure of a family of rational functions with an arbitrary number of variables Our formulas reveal relations with multiple polylogarithms evaluated at certain roots of unity Introduction For k a positive integer, the k-higher Mahler measure of a non-zero, n-variable, rational function P x,, x n Cx,, x n is given by m k P x,, x n log k P e πiθ,, e πiθn dθ dθ n We observe that the case k recovers the formula for the classical Mahler measure This function, originally defined as a height on polynomials, has attracted considerable interest in the last decades due to its connection to special values of the Riemann zeta function, and of L-functions associated to obects of arithmetic significance such as elliptic curves as well as special values of polylogarithms and other special functions Part of such phenomena has been explained in terms of Beilinson s conectures via relationships with regulators by Deninger [Den97] see also the crutial articles by Boyd [Boy97] and Rodriguez-Villegas [R-V97] Higher and multiple Mahler measures were originally defined in [KLO8] and subsequently studied by several authors [Sas, BS, LS, BBSW, BS, Sas, Bis4, BM4] A related obect, the Zeta Mahler measure, was first studied by Akatsuka in [Aka9] As remarked by Deninger, higher Mahler measures are expected to yield different regulators than the ones that appear in the case of the usual Mahler measure and they may reveal a more complicated structure at the level of the periods see [Lal] for more details In order to continue this program of understanding periods that arise from higher Mahler measure, an essential component is to generate examples Mathematics Subect Classification Primary R6; Secondary M6, R9 Key words and phrases Mahler measure, Higher Mahler measure, special values of ζs and Dirichlet L-functions, polylogarithms

2 M N LALÍN AND J-S LECHASSEUR of formulas of higher Mahler measure involving special functions that can be easily expressed as periods, such as polylogarithms In the present work we consider the family of rational functions in Cx,, x m, z given by x R m x,, x m, z : z + + x xm + x m Let ζs be the Riemann zeta function, and let Lχ 4, s be the Dirichlet L-function in the character of conductor 4, defined, for Res >, as Lχ 4, s : n ζs : n n s, χ 4 n, χ n s 4 n n if n odd, if n even We also consider the functions L n,,n m w,, w m rm Li n,,n m r w,, rm w m, r,,r m {,} m given by combinations of multiple polylogarithms of length m, defined for n i positive integers by Li n,,n m w,, w m w n < < < m wm m nm The series above is absolutely convergent for w i and n m > Finally, for a, a m C, consider if l, s l a,, a m i < <i l a i a il if < l m, if m < l In the present article, we prove the following result m Theorem We have, for n, the following formula m k R n n h s n h, 4,, n n! h A k h, π

3 where HIGHER MAHLER MEASURE OF A FAMILY 3 A k h :h + k! + k k! + k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n k k+ k!! ɛ,,ɛ n {,} n #{i:ɛ i }k n h+k k n ζh + k h!l ɛ,,ɛ n,,h,, k n k k n h +!L ɛ,,ɛ n,,h+,, We have, for n, the following formula n s n h, 3,, n m k R n+ n! where h B k h :h + k!lχ 4, h + k + + k+ k! + ɛ,,ɛ n {,} n #{i:ɛ i }k n k k n k k++ k!! ɛ,,ɛ n {,} n #{i:ɛ i }k n k n h+ B k h π ih!l ɛ,,ɛ n,,h+,,, i, i k n k k n ih +!L ɛ,,ɛ n,,h++,,, i, i For the sake of clarity, we record here the case of k A h :h +! ζh + + h!l,h, h+ B h :h +!Lχ 4, h + 3 ih!l,h+ i, i

4 4 M N LALÍN AND J-S LECHASSEUR The case of k clearly yields formulas that only depend on ζs, Lχ 4, s and powers of π This is equivalent to saying that all the terms can be expressed in terms of polylogarithms of length one There is another case in which we can prove a similar formula Corollary The previous result includes the following particular case m R 3π log ζ3+ π π Li π log log π, where all the terms are product of polylogarithms of length one Our method of proof for Theorem relies in the ideas of [Lal6a] combined with key properties of the Zeta Mahler function constructed by Akatsuka [Aka9] The same method yields a formula for a simpler polynomial Let x xm Q m x,, x m : x + x m + We can express the higher Mahler measure of this family in terms of rational combinations of powers of π More precisely, we obtain the following result Proposition 3 We have, for n, the following formula m k Q n π k n h s n h, 4,, n k+h+ n! k + h h k+h B k+h We have, for n, the following formula π k n m k Q n+ h In addition, for k and m, s n h, 3,, n k+h E k+h n! m k+ Q m In the above expressions, B m and E m are the Bernoulli and Euler numbers respectively Formulas for m k Q where recovered in [BBSW] It is not necessary to use the method of [Lal6a] to find formulas for m k Q m By using simple properties for the higher Mahler measure of a product of polynomials with different variables, we recover alternative expressions for the same formulas By comparing with the results of Proposition 3, we obtain the following identities between Bernoulli and Euler numbers which generalize some results from the Appendix in [Lal6a]

5 HIGHER MAHLER MEASURE OF A FAMILY 5 Corollary 4 n s n h, 4,, n h+ n! k + h k+h k+h B k+h h k E E n,,, n and + + n k n s n h, 3,, n h E k+h n! h k E E n+,, n+ + + n+ k The present article is organized as follows In Section we recall previous results on the classical Mahler measure of R n, as well as similar results for other rational functions that were obtained in [Lal6a] We outline the general method of proof in Section 3 In Section 4 we discuss some properties of the Zeta Mahler measure Sections 5 and 6 treat certain technical simplifications of the involved integrals We discuss properties of polylogatithms in Section 7, and we present the final details of the proofs in Section 8 Some technical results were already part of [Lal6a] but we include them in this article for the sake of completeness Description of similar results for Mahler measure In this section we present the previous results that were obtained with this method for the classical Mahler measure Theorem [Lal6a] We have the following identities For n, n s n h, 4,, n h m R n A h, n! π where where For n, m R n+ h A h : h! ζh + h+ n h s n h, 3,, n n! B h : h +!Lχ 4, h + h+ B h, π

6 6 M N LALÍN AND J-S LECHASSEUR In addition, similar results were proved in [Lal6a] for other rational functions by the same method Let S m x,, x m, x, y, z x xm : + xz + + x + x m T m x,, x m, x, y x xm : + + x + x m h x + + y, x + x xm + x m y Theorem [Lal6a] We have the following identities For n, n s n h, 4,, n h+ m S n C h, n! π where C h : where h l For n, D h : m S n+ For n, h h l B h l π h l l +! ζl + 3 l 4h l+3 h l n h s n h, 3, n n! h+3 D h, π h + h l l + h + B h lπ h l l + 3!Lχ 4, l + 4 where m T n log E h : h! For n, + n h ζh + + h+ B h l π h l l! m T n+ log + n h s n h, 4,, n n! l+ h E h, π h h h l h l+ l h l ζl + s n h, 4,, n n +! h+ F h π

7 where HIGHER MAHLER MEASURE OF A FAMILY 7 h +! F h : + π n h! + nn + h+3 l ζh + 3 h+ B h l π h+ l l! ζh + h h h l h l+ l 4h ζl + l+ We remark that the presentation of some of the above formulas differ from those in [Lal6a] as they have been simplified by recent results by the authors [LL] 3 The general method Here we describe the general structure of the proof of Theorem, based on the ideas of [Lal6a] Let P a Cz be such that its coefficients are rational functions in a parameter a C By making the change of variables a x x + x n x n+ we can see the rational function P a as a new rational function in n + variables, namely P Cx,, x n, z Thus, the k-higher Mahler measure of P is given as follows m k P log k πi P n n+ T x x n+ x n log k πi n T πi P n T x x P πi n x x + xn xn+ T n m k x n x n n x n, where the term m k P x inside the integral is a function on the x + xn xn+ variables x,, x n By making the change of variables x e iθ, followed by y tanθ /, the integral above equals π π m k P i n tan θ tan θn dθ dθ n π n π n π π m k P i n y y n dy y + dy n yn + Assume that the function on the parameter a given by m k P a is even, namely, that it verifies m k P a m k P a this happens if, for instance,

8 8 M N LALÍN AND J-S LECHASSEUR m k P a only depends on a Then the same can be said for the integrand and the previous expression equals n π n n π n m k P i n y y n m k P in ˆx n ˆx dˆx ˆx + dy y + dy n yn + ˆx dˆx ˆx + ˆx ˆx n dˆx n ˆx n + ˆx n dˆx n, ˆx n + ˆx n where we have set ˆx i y i This change of variables is motivated by the goal of recovering a term of the form m k P x inside the integral Indeed, if we further assume that m k P a depends only on a, we have that m k P in ˆx n m k Pˆxn Now choose P a z + a Then P R n As we will see in Section 4, m k P a is a function of a This implies 3 m k R n n π n m k z + ˆx n ˆx dˆx ˆx + ˆx dˆx ˆx + ˆx ˆx n dˆx n ˆx n + ˆx n dˆx n ˆx n + ˆx n Thus, if we have a good expression for m k z + x, then, under favorable circumstances we may obtain a good expression for m k R n 4 Zeta Mahler measure In this section we discuss the Zeta Mahler measure, an obect that is closely related to higher Mahler measure and that willl allow us to compute m k z + a for any a C Definition 4 Let P Cx,, x n be a non-zero, n-variable, rational function Its Zeta Mahler measure is defined by Zs, P : P x πi n,, x n s n T x n x n Remark 4 It is not hard to see that d k Zs, P ds k log k P x s πi n,, x n T x n n x n m k P The simplest possible case of Zeta Mahler measure was computed by Akatsuka Theorem 43 Akatsuka, [Aka9] Let a C such that a We then have s F, s; ; a if a <, Zs, z + a a s s F, s; ; a if a >,

9 where, for t <, HIGHER MAHLER MEASURE OF A FAMILY 9 F α, β; γ; t n α n β n γ n denotes the hypergeometric series, and { n, α n : αα + α + n n, is the Pochhammer symbol Formulas for m k z + a can be derived from Zs, z + a by means of Remark 4 We proceed to compute some derivatives of Zs, z + a Lemma 44 Let t C such that t < and s G t s F, s ; ; t Then n G t, G t n t n n! s n t n n! Proof Indeed, setting s we obtain that s unless n and in n that case s This implies that G t Differentiating G t s, we obtain s s G t n ts n n n! If we now set s, we see that each term in the sum equals and therefore G t Akatsuka [Aka9] found a formula for m k z +a for a < This formula can be easily adapted to the general case of a C Theorem 45 Let L n,,n mw : We have, for a and k, m k z + a k k! k n k nm wm n < < < m k n m ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n, a

10 M N LALÍN AND J-S LECHASSEUR For a and k, k m k z + a log k a + ɛ,,ɛ n {,} n #{i:ɛ i }k n k k k! k n k log a L ɛ,,ɛ n, a k n Proof The case of a < is Theorem 7 in [Aka9] It is proved by observing that Zs, z + a G a s and m k z + a dk Zs, z + a ds k G k a s Akatsuka applies the eighth formula in page 485 of [OZ] for α β and x in order to conclude the result For a >, we have, by Theorem 43, Zs, z + a a s G a s Thus we get, by Lemma 44, m k z + a dk Zs, z + a ds k s k k log a G k a k k log k a + log a k k! k n k k n ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n, a and the result follows Finally, the case a is considered in the third formula in page 73 of [KLO8] It is not hard to see that both formulas given above remain true for a 5 Integral simplification In this section we discuss how to simplify the integral in Equation 3 In order to achieve this, we define certain polynomials and prove a recurrence relation for them We then use these polynomials to compute certain family of integrals

11 HIGHER MAHLER MEASURE OF A FAMILY Definition 5 Let A m x Q[x] be defined by RT ; x ext sin T m A m x T m m! Thus, A x x, A x x, A x x3 3 + x 3, etc Lemma 5 The polynomials A m x satisfy the following recurrence 5 A m x xm+ m + + m+ + m + A m+ x m + > odd Proof By writing sin T eit e it, we obtain, i e e xt it e it RT ; x i In other words, x m T m m m! > odd T! l A l x T l l! The result is obtained by comparing the coefficient of T m+ in both sides of the above equality Remark 53 More properties of the polynomials A m x can be found in the Appendix to [Lal6a] For instance, one can prove that A m x m m + B h h i h x m+ h, m + h h where the B n are the Bernoulli numbers We will eventually compute certain integral For this, we need the following auxiliary result Lemma 54 For < β <, we have the following integral evaluation: 5 x β x + a x + b πaβ b β cos πβ b a Proof We decompose the integrand by means of partial fractions in order to obtain the following x β 53 x + a x + b x + a x β x + b b a By integrating over a well-chosen contour see page 59, Section 53 of [Ahl79], we obtain, x β x + a e πi { } x β Res πaβ πiβ x + a cos πβ x

12 M N LALÍN AND J-S LECHASSEUR By replacing this in 53, the result follows We will use the polynomials A m x to compute certain integrals Proposition 55 For m, we have the following equation Proof Let x log m x x + a x + b π fβ : m+ A m log a π x β x + a x + b log b Am π a b As the integral converges for < β < 3, the function is well-defined and continuous in this interval We differentiate m times and obtain f m Lemma 54 implies, for < β <, x log m x x + a x + b fβ cos πβ πaβ b β b a By differentiating m times, we obtain m m f m β cos πβ π b a aβ log m a b β log m b We now take the limit of β in order to obtain m + m f m odd π πlog m a log m b b a Changing m to m +, isolating the term f m, and dividing by π m +, f m m+ m + > odd + m + + logm+ a log m+ b m + a b For m the above equation becomes f f logm a log m b a b π π f m+ A log a π log b A π a b The rest of the proof proceeds by induction, using the recurrence for A m x that was proved in Lemma 5

13 HIGHER MAHLER MEASURE OF A FAMILY 3 By Proposition 55, we can write the integral in Formula 3 as a sum of simpler integrals For instance, for n, we have m k z + ˆx ˆx dˆx dˆx ˆx + ˆx + ˆx ˆx dˆx m k z + ˆx ˆx + ˆx ˆx + π log ˆx A π m k z + ˆx ˆx π log ˆx π m k z + ˆx dˆx ˆx m k z + x log x x Analogously, for n 3, we obtain π 8 dˆx A log π dˆx m k z + ˆx 3 ˆx dˆx ˆx dˆx dˆx 3 ˆx + ˆx + ˆx ˆx 3 + ˆx m k z + x x + + m k z + x log x x + More generally, we can always reduce the computation to a sum of integrals of the form m k z + x log h+ x for n even, x 54 m k z + x log h x for n odd x + 6 Coefficient formulas In this section, we find the coefficients that allow us to express the integral from 3 as a linear combination of the integrals from 54 Let φa be a function that depends on a Eventually we will have φa m k P a, where P a is a rational function such that m k P a m k P a for instance, we could take P a z + a In what follows, it is assumed that φa is such that all the integrals converge Definition 6 For n and h n, let a n,h Q be defined by 6 φx x n n x n n x n + x n + x n x + x n π n h a n,h φx log h x x h

14 4 M N LALÍN AND J-S LECHASSEUR For n and h n, let b n,h Q be defined by 6 φx x n+ n+ x n n x n+ + x n + x n+ x + x n π n h b n,h φx log h x x + h Lemma 6 We have the following identities: n n 63 b n,h x h a n,h A h x A h i, 64 h n+ a n+,h x h h h n b n,h A h x, h where the A m x are the polynomials given in Definition 5 Proof By making the change of variables x i y i x n+ for i,, n, we have, φx x n+ n+ x n n x n+ + x n + x n+ x + x φy x n+ n+ y n dy n y n dy n x n+ + yn + yn + yn dy y + y We now ignore the variable x n+ and apply Equation 6 where φy x n+ n+ is the function depending on y x The above equals n+ + n π n h a n,h h φy x n+ n+ x n+ + logh y dy y We set x y x n+ and rename y y We obtain n π n h y a n,h φx x + y logh y dy y h By applying Equation 6, we conclude n π n h 65 b n,h φx log h x h x + n π n h a n,h φxy log h y dy y x + y h By Proposition 55 with a x and b i, we obtain y log h ydy π h y + x y A log x h π Ah i x +

15 Thus, 65 equals n π n a n,h h HIGHER MAHLER MEASURE OF A FAMILY 5 log x φx A h A h i π x + This equality is true for any choice of φx and therefore 63 must hold Analogously, we can prove that 66 n+ π n+ h a n+,h φx log h x h x n π n h b n,h φxy log h y dy y + x + y h By Proposition 55 with a x and b, y log h ydy π h+ y + x y + A log x h π Ah x Thus, 66 becomes n π n+ b n,h h log x φxa h π x By comparing the above equation with 66, we obtain Equation 64 We now prove a result that will be key in finding formulas for a n,h and b n,h Lemma 63 We have the following identitites: n l s n l, 4,, n n h h s n h, 3,, n l hl n + l s n l, 3,, n n h + h s n h, 4,, n l hl Proof We multiply by x l in both sides of 67 and we sum for l, n n n s n l, 4,, n l x l l n l n h h s n h, 3,, n x l l hl By identifying the left-hand side with the corresponding polynomial and reversing the sums on the right-hand side, we conclude that it suffices to

16 6 M N LALÍN AND J-S LECHASSEUR prove that x n n n h s n h, 3,, n h The right-hand side equals h l h x l l n h s n h, 3,, n x x + h x h h x n n x + x x n n + x x + x x By inspecting the common zeros in both products, we obtain that the line above equals xn + x xn x x n x n n x This concludes the proof for Equation 67 For Equation 68 we proceed analogously by multiplying each side by x l+ and summing over l,, n n + n l hl n s n l, 3,, n l x l+ l Then, it suffices to prove that n + x n h + h x l+ l n x n h s n h, 4,, n h h h + x l+ l l

17 HIGHER MAHLER MEASURE OF A FAMILY 7 The right-hand side equals n h s n h, 4,, n x x + h+ x h+ h x n n x + x + x x x n x x x n x + x + x n + + x + n + x x n n + x x n x and this concludes the proof for Equation 68 Remark 64 Mathew Rogers has remarked that the sequences under consideration are related with Stirling numbers of the first class in the following way s n h, 4,, n n h h m h m S m n S n h m We are now ready to compute the coefficients a n,h and b n,h from Definition 6 Theorem 65 We have the following identities For n, h n a n,h x h x + x + n, n! h and for n, n b n,h x h x + x + 3 x + n n! In other words, 69 6 a n,h s n h, 4,, n, n! b n,h s n h, 3,, n n!

18 8 M N LALÍN AND J-S LECHASSEUR Proof We proceed by recurrence By definition, when n +, we have that n and φx x + b, φx x + Therefore, b, Analogously, when n we have that n We have seen that φx ydy y + x + y a, log x φx x log x φx x Thus a, and the result holds for the first two cases Now assume that for a fixed n and all h n we have that a n,h s n h, 4,, n n! We will then prove that for all h n, b n,h s n h, 3,, n n! By Lemma 6, it suffices to show that n s n h, 3,, n x h h n n s n h,, n A h x A h i h Taking m h in Equation 5, we obtain h h x h A h x + Multiplying by s n h, 3,, n and summing over h, we get n s n h, 3,, n x h h n h h s n h, 3,, n A h x + h + s n, 3,, n

19 HIGHER MAHLER MEASURE OF A FAMILY 9 Evaluating this equation at x i, we obtain, n s n h, 3,, n h h n h h s n h, 3,, n A h i + h + s n, 3,, n Since n s n h, 3,, n h x + x + n x, h we deduce that n s n h, 3,, n x h h n s n h, 3,, n h h h A h x A h i + By setting l h, the right-hand side becomes n h h s n h, 3,, n h l A l x A l i l h l n n h h s n h, 3,, n l l hl l A l x A l i Lemma 63 then implies 6 Now suppose that for fixed n and all h n, we have We wish to show that all h n b n,h s n h, 3,, n n! a n+,h s n h, 4,, n n +! By Lemma 6, it suffices to show that n s n h, 4,, n x h+ h n + n s n h, 3,, n A h x h

20 M N LALÍN AND J-S LECHASSEUR By setting m h in Equation 5, we obtain h h + x h+ A h x + Therefore, n s n h, 4,, n x h+ h n s n h, 4,, n h Setting l h, we obtain n s n h, 4,, n h h n h + h l l hl h l h h + A h x + h + h l A l x l s n h, 4,, n Thus, Equation 69 follows from Lemma 63 l A l x 7 Polylogarithms and hyperlogarithms In order to complete the proof of Theorem, we need to compute the integrals of the form m k z + x log x x ± These integrals are related to polylogarithms Definition 7 Let w,, w m be complex variables and n,, n m be positive integers Define the multiple polylogarithm by the power series Li n n m w,, w m : w n < < < m wm m nm We say that the above series has length m and weight ω n + + n m It is absolutely convergent for w i and n m > Remark 7 We remark that Akatsuka s polylogarithm from Theorem 45 is a particular case of multiple polylogarithms L n,,n mw : wm n Li < < < m m nm n,,n m,,, w Multiple polylogarithms have meromorphic continuations to the complex plane m

21 HIGHER MAHLER MEASURE OF A FAMILY Definition 73 Hyperlogarithms are defined by the following iterated integral I n n m a : : a m+ : am+ where bh+ dt dt t a t dt dt dt t t a }{{} t dt dt dt t t a }{{} m t dt t }{{} n n n m dt dt t b t b h t t h b h+ dt dt h t b t h b h Remark 74 The path of integration should be interpreted as any path connecting and b h+ in C \ {b,, b h } The integral depends on the homotopy class of this path For our purposes, we will always integrate in the real line Multiple polylogarithms and hyperlogarithms are related by the following identities see [Gon95] Lemma 75 I n n m a : : a m+ m a Li n n m, a 3, a m+ a a a m Li n n m w,, w m m I n n m w w m : : wm : The following example will be useful later Example 76 The second equation in Lemma 75 implies, for ɛ,, ɛ n {, } n, and ɛ + + ɛ n k, L ɛ,,ɛ n,w n+ dt t w dt t w dt t, where there is a term of the form dt after each dt term corresponding to t t w ɛ i and there is no term dt after each dt term corresponding to ɛ t t i, w dt and the last two terms dt correspond to the subindex t w t By setting s w t, the above equals w n+ sds s sds s ds s w n+ k n s + s + s + ds s + s

22 M N LALÍN AND J-S LECHASSEUR In the previous formula, each ds has contributed a factor of to the s leading coefficient In order to express our results more clearly, we recall the notation from Equation Definition 77 We will work with certain combination of polylogarithms given by L n,,n m w,, w m rm Li n,,n m r w,, rm w m, r,,r m {,} m The following result relates the integrals that we need to evaluate in terms of polylogarithms Lemma 78 We have the following identitites log x x +! + log x x +!Lχ 4, + ζ + For ɛ,, ɛ n {, } n, and ɛ + + ɛ n k, we have the following identities 7 L ɛ,,ɛ n,x log x x + k n!l ɛ,,ɛ n,,+,, 7 L ɛ,,ɛ n,x log x x + i + k n!l ɛ,,ɛ n,,+,,, i, i Proof The first two identities are proven in Lemma 9 of [Lal6a] By applying Example 76, L ɛ,,ɛ n,x log x x x n+ k n log x x s + s + s + ds s + s

23 We now use the fact that x n++ k n! x x + n++ k n! HIGHER MAHLER MEASURE OF A FAMILY 3 dt t log x and obtain s + s + dt t dt }{{ t} r,,r n+,r {,} n+ r I ɛ,,ɛ n,,+ r : : rn : r n+ : r : + k n! r,,r n+,r {,} n+ r s + ds s + s Li ɛ,,ɛ n,,+ r +r,, rn+r n+, r n++r, r + k n!l ɛ,,ɛ n,,+,, This yields Equation 7 We proceed similarly to prove Equation 7 L ɛ,,ɛ n,x log x x + x n+ k n log x x + i n+ k n! x i x + i i n+ k n! s + s + s + ds s + s s + s + s + ds s + s dt t dt }{{ t} r,,r n+,r {,} n+ r I ɛ,,ɛ n,,+ r : : rn : r n+ : r i : i k n! r,,r n+,r {,} n+ r Li ɛ,,ɛ n,,+ r +r,, rn+r n+, r n++r i, r+ i i + k n!l ɛ,,ɛ n,,+,,, i, i

24 4 M N LALÍN AND J-S LECHASSEUR 8 The final steps in the proof of Theorem By combining Equation 3, Definition 6, and Theorem 65, we obtain π n m k R n n s n h, 4,, n h π n h m k z + x log h x n! x h and π n+ m k R n+ n s n h,, n h+ π n h m k z + x log h x n! x + h We are now ready to express these two integrals in terms of polylogarithms For the case of h, we have, by Theorem 45, m k z + x log h x x k k! k n + k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n k + log h+k x x k k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n,x log h x x k n k log xl ɛ,,ɛ n, k n x log h x x

25 HIGHER MAHLER MEASURE OF A FAMILY 5 By making the change of variables y in the integrals over x, and x then replacing y by x again, the above expression becomes k k! + k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n k + k n k log h+k x x k k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n,x log h x x k n k L ɛ,,ɛ n, k n x log h+ x x Finally, by Lemma 78, m k z + x log h x x h + k! ζh + k + k k! k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n k + k h+k k n h!l ɛ,,ɛ n,,h,, k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n k n k k n h +!L ɛ,,ɛ n,,h+,,

26 6 M N LALÍN AND J-S LECHASSEUR The case of h proceeds similarly First, by Theorem 45, m k z + x log h x x + k k! k n + k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n k + log h+k x x + k k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n,x log h x x + k n k log xl ɛ,,ɛ n, k n x log h x x + Then, by making the change of variables y x and then replacing y by x again, in the integrals over x, k k! k n k ɛ,,ɛ n {,} n #{i:ɛ i }k n + k + k n k log h+k x x + k k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n L ɛ,,ɛ n,x log h x x + k n k L ɛ,,ɛ n, k n x log h+ x x +

27 Finally, by Lemma 78, HIGHER MAHLER MEASURE OF A FAMILY 7 m k z + x log h x x + h + k!lχ 4, h + k + + k+ k! ɛ,,ɛ n {,} n #{i:ɛ i }k n k + k k n k k n ih!l ɛ,,ɛ n,,h+,,, i, i k k! ɛ,,ɛ n {,} n #{i:ɛ i }k n k n k k n i + h +!L ɛ,,ɛ n,,h++,,, i, i This concludes the proof of Theorem 8 The reduction to Corollary We now proceed to deduce Corollary from Theorem Recall that Theorem implies in particular that m R π π L,, By using the following formulas that can be found in [BJOPL], Li,, 3 ζ, Li,, 8 ζ Li,3,, Li,, 4 ζ + Li,3,, Li,, 3 4 ζ, one can reduce L,, in order to obtain m R 89π π Li,3, By combining the following identity from [BJOPL] Li,3, ζ log ζ3 Li,3, with this result from [BBG95] Li,3, ζ4 Li log log π,

28 8 M N LALÍN AND J-S LECHASSEUR one finally obtains Equation 8 The simpler case of Proposition 3 In order to find formulas for x m k Q m, we start by taking P a az Indeed, replacing a by +x x m +x m yields zq m which has the same higher Mahler measures as Q m This computation is particularly easy to do because m k az log k a Therefore we obtain and π n m k Q n n s n h, 4,, n h π n h log k+h x n! x h π n+ m k Q n+ n s n h,, n h+ π n h log k+h x n! x + h Proposition 3 follows with the same arguments that we used in the final steps of the proof of Theorem and the combination of the well-known formulas ζ + B π! and Lχ 4, + E π + +! Finally, Corollary 4 follows from the simple observation that k x m k Q m m,, m + x + + mk and the fact that x m + x m Q m m xm + x m { / π E even, odd 9 Concluding remarks We have proved exact formulas for m k of a particular family of rational functions with an arbitrary number of variables Much like in the case of the classical Mahler measure for this family, we have obtained formulas involving multiple polylogarithms evaluated in roots of the unity It is expected that many, if not all, of the formulas from Theorem could be reduced to expressions solely involving terms of length For example, L,h+, can be reduced to combinations of products of ζn by means of a result in [BBB97] that generalizes a result of Euler in the reduction,

29 HIGHER MAHLER MEASURE OF A FAMILY 9 of multizeta values of length Additional efforts in this direction may be found in [Lal6b, LL], but they are currently insuffient to reduce all the terms involved in such expressions Another direction for future exploration is the search for formulas for m k S m and m k T m Acknowledgements The authors would like to thank Andrew Granville and Mathew Rogers for interesting discussions and useful observations In particular, they are grateful to Granville for the suggestion to use the generating function for the polynomials A k x and to Rogers for remarking that s n h, 4,, n is related to Stirling numbers of the first kind The authors would like to express their gratitude to the anonymous referee for several corrections and helpful suggestions This research was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant ] and the Fonds de recherche du Québec - Nature et technologies [Établissement de nouveaux chercheurs to ML, Bourse de doctorat en recherche B to JSL] References [Ahl79] Ahlfors, Lars V Complex analysis An introduction to the theory of analytic functions of one complex variable Third edition International Series in Pure and Applied Mathematics McGraw-Hill Book Co, New York, 978 xi+33 [Aka9] Akatsuka, Hirotaka Zeta Mahler measures J Number Theory 9 9, no, [BJOPL] Bigotte, M; Jacob, G; Oussous, N E; Petitot, M Lyndon words and shuffle algebras for generating the coloured multiple zeta values relations tables WORDS Rouen, 999 Theoret Comput Sci 73, no -, 7 8 [Bis4] Biswas, Arunabha Asymptotic nature of higher Mahler measure Acta Arith 66 4, no, 5 [BM4] Biswas, Arunabha; Monico, Chris Limiting value of higher Mahler measure J Number Theory 43 4, [BBG95] Borwein, David; Borwein, Jonathan M; Girgensohn, Roland Explicit evaluation of Euler sums Proc Edinburgh Math Soc , no, [BBSW] D Borwein, J Borwein, A Straub, J Wan, Log-sine evaluations of Mahler measures, II Integers, no 6, 79 [BBB97] Borwein, J M; Bradley, D M; Broadhurst, D J Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k The Wilf Festschrift Philadelphia, PA, 996 Electron J Combin 4 997, no, Research Paper 5, approx pp [BS] J Borwein, Jonathan, A Straub, Special values of generalized log-sine integrals ISSAC Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 43 5, ACM, New York, [BS] J Borwein, A Straub, Log-sine evaluations of Mahler measures J Aust Math Soc 9, no, 5 36 [Boy97] Boyd, David W Mahler s measure and special values of L-functions Experiment Math 7 998, no, 37 8

30 3 M N LALÍN AND J-S LECHASSEUR [Den97] Deninger, Christopher Deligne periods of mixed motives, K-theory and the entropy of certain Z n -actions J Amer Math Soc 997, no, 59 8 [Gon95] Goncharov, Alexander B Polylogarithms in arithmetic and geometry Proceedings of the International Congress of Mathematicians, Vol, Zürich, 994, , Birkhäuser, Basel, 995 [KLO8] Kurokawa, N; Lalín, M; Ochiai, H Higher Mahler measures and zeta functions Acta Arith 35 8, no 3, [Lal6a] Lalín, Matilde N Mahler measure of some n-variable polynomial families J Number Theory 6 6, no, 39 [Lal6b] Lalín, Matilde N On certain combination of colored multizeta values J Ramanuan Math Soc 6, no, 7 [Lal] Lalín, M Higher Mahler measure as a Massey product in Deligne Cohomology Low-dimensional Topology and Number Theory Abstracts from the workshop held August 5-August, Organized by Paul E Gunnells, Walter Neumann, Adam S Sikora and Don Zagier Oberwolfach Reports Vol 7, no 3 Oberwolfach Rep 7, no 3, 63 [LS] Lalín, Matilde; Sinha, Kaneenika Higher Mahler measure for cyclotomic polynomials and Lehmer s question Ramanuan J 6, no, [LL] Lalín, Matilde N; Lechasseur, Jean-Sébastien On certain multizeta values twisted by roots of unity Preprint 6 [OZ] Ohno, Yasuo; Zagier, Don Multiple zeta values of fixed weight, depth, and height Indag Math NS, no 4, [R-V97] Rodriguez-Villegas, F Modular Mahler measures I Topics in number theory University Park, PA, 997, 7 48, Math Appl, 467, Kluwer Acad Publ, Dordrecht, 999 [Sas] Sasaki, Yoshitaka On multiple higher Mahler measures and multiple L values Acta Arith 44, no, [Sas] Sasaki, Yoshitaka On multiple higher Mahler measures and Witten zeta values associated with semisimple Lie algebras Commun Number Theory Phys 6, no 4, Département de mathématiques et de statistique, Université de Montréal, CP 68, succ Centre-ville, Montreal, QC H3C 3J7, Canada address: mlalin@dmsumontrealca Département de mathématiques et de statistique, Université de Montréal, CP 68, succ Centre-ville, Montreal, QC H3C 3J7, Canada address: eansebl777@gmailcom