Higher Mahler measures and zeta functions

Transcription

1 Higher Mahler measures and zeta functions N Kuroawa, M Lalín, and H Ochiai arxiv:987v [mathnt] 3 Aug 9 November 5, 8 Abstract: We consider a generalization of the Mahler measure of a multivariablepolynomialp astheintegraloflog P intheunittorus, asopposed to the classical definition with the integral of log P A zeta Mahler measure, involving the integral of P s, is also considered Specific examples are computed, yielding special values of zeta functions, Dirichlet L-functions, and polylogarithms Keywords: Mahler measure, zeta functions, Dirichlet L-functions, polylogarithms Mathematics Subject Classification: M6, R9 Introduction The logarithmic Mahler measure of a non-zero Laurent polynomial P C[x ±,,x± n ] is defined by mp log P e πiθ,,e πiθn dθ dθ n In this wor, we consider the following generalization: Definition The -higher Mahler measure of P is defined by m P : log P e πiθ,,e πiθn dθ dθ n In particular, notice that for we obtain the classical Mahler measure m P mp, and m P These terms are the coefficients in the Taylor expansion of Aatsua s zeta Mahler measure Zs,P P e πiθ,,e πiθn s dθ dθ n, SupportedbyUniversityofAlbertaFac Sci StartupGrantN36andNSERC Discovery Grant

2 that is, Zs,P m Ps! Aatsua [] computed the zeta Mahler measure Zs,x c for a constant c A natural generalization for the -higher Mahler measure is the multiple higher Mahler measure for more than one polynomial Definition Let P,,P l C[x ±,,x ± r ] be non-zero Laurent polynomials Their multiple higher Mahler measure is defined by : mp,,p l log P e πiθ,,e πiθr log Pl e πiθ,,e πiθr dθ dθ r This construction yields the higher Mahler measures of one polynomial as a special case: m P mp,,p }{{} Moreover, the above definition implies that mp mp l mp,,p l when the variables of P j s in the right-hand side are algebraically independent This identity leads us to speculate about a product structure for the logarithmic Mahler measure This would be a novel property, since the logarithmic Mahler measure is nown to be additive, but no multiplicative structure is nown This definition has a natural counterpart in the world of zeta Mahler measures, namely, the higher zeta Mahler measure defined by Zs,,s l ;P,,P l P e πiθ,,e πiθr s Pl e πiθ,,e πiθr s l dθ dθ r, Its Taylor coefficients are related to the multiple higher Mahler measure: l s s l Z,,;P,,P l mp,,p l In this wor, we compute the simplest examples of these heights and explore their basic properties In section we consider the case of higher

3 3 Mahler measure for one-variable polynomials More precisely, we consider linear polynomials in one variable In particular, we obtain m x π, m 3 x 3ζ3, m 4 x 9π4 4, m x, e πiα x π α α+ 6, α In section 3, we consider two examples of two-variable Mahler measure and we compute m Sections 4 and 5 deal with examples of zeta Mahler measures of linear polynomials and their applications to the computation of higher Mahler measure, recovering the results from section and giving an insight into them Finally, we explore harder examples of zeta and higher Mahler measures in Section 6 For example, m x+y + ζ, m 3 x+y logζ 4 ζ3, Zs,x+x +y +y +c c s s 3 F, s,, 6 c, c > 4 Higher Mahler measure of one-variable polynomials The case of x Our first example is given by the simplest possible polynomial, namely P x Theorem 3 m x b + +b h,b i where ζb,,b h denotes a multizeta value, ie, ζb,,b h! h ζb,,b h, l <<l h l b lb h The right-hand side of Theorem 3 can be re-written in terms of classical zeta values by using the following result h

4 4 Proposition 4 ζb σ,,b σh σ S h l h l e s! ζ e + +e l h s π b ζ π l b where the sum in the right is taen over all the possible unordered partitions of the set {,,h} into l subsets π,,π l with e,,e l elements respectively PROOF of Theorem 3 First observe that x varies in the unit circle Therefore, we can choose the principal branch for the logarithm We proceed to write the function in terms of integrals of rational functions We have log x Relog x log x+log x dt t x + dt j dt j dt t x j t x t x Now observe that dt j j dt t x t x j! j! j dt dt t x }{{ t x } j dt t x dt }{{ t x } j We have just used the iterated integral notation of hyperlogarithms Combining the previous equalities gives m x log x dx πi x x! dt dt dt πi j x t x }{{ t x } t x dt }{{ t x } j j If we now set s xt in the first j-fold integral and s t in the second x j-fold integral, the above becomes! j x πi x ds s ds x s ds s ds s We proceed to compute the integrals in terms of multiple polylogarithms: m x! x l j m j dx πi x l l j m m j x j <l <<l j < <m <<m j < dx x dx x

5 5! j <l <<l j <u<,<m <<m j <u< Now we need to analyze each term of the form <l <<l j <u<,<m <<m j <u< l l j m m j u For an h-tuple a,,a h such that a ++a h h, we set a ah a ++a h h d a,,a h e ++e h j h e ++e h j h e Then the term is equal to min{j, j } h e h d a,,a h ζ{} a,,,{} ah, Note that each term ζ{} a,,,{} ah, comes from choosing h of the l s and h of the m s and maing them equal in pairs Once this process has been done, one can choose the way the other l s and m s are ordered All these choices give rise to the coefficients d a,,a h The total sum is given by m x c a,,a h ζ{} a,,,{} ah,, h l l j m m j u where c a,,a h! j h! h! j h h On the other hand, ζ{} a,,,{} ah, ζa h +,,a + To see this well-nown fact, observe that the term in the left is h dt t dt dt }{{ t } t a + Maing the change t t gives h dt t dt t dt t } {{ } a h + dt t dt }{{ t } a h + dt t dt t dt t dt t } {{ } a +,

6 6 which corresponds to the term in the right Thus, the total sum is m x b ++b h,b i! h ζb,,b h We show a proof of Proposition 4 for completeness PROOF Proposition 4 We first show that we can write σ S h ζb σ,,b σh e ++e l h re,,e l ζ π b ζ π l b wherethefunctionre,,e l satisfiessomerecurrencerelationships Here, as in the statement, the sum in the right is taen over all the possible unorderedpartitionsoftheset{,,h}intolsubsetsπ,,π l withe,,e l elements respectively Notice that r is a function that is invariant under any permutation of its arguments We proceed by induction on h It is clear that r Also ζa,b+ζb,a ζaζb ζa+b, from where r,, r Assume that the case of h is settled Now, we multiply everything by ζb h+, ζb σ,,b σh ζb h+ σ S h re,,e l ζ ζ ζb h+ e ++e l h Observe that σ S h ζb σ,,b σh ζb h+ + h j where b j b j +b h+ Hence, e ++e l h π b π l b σ S h+ ζb σ,,b σh+ σ S h ζb σ,,b σ j,,b σh, σ S h+ ζb σ,,b σh+ re,,e l ζ π b ζ π l b ζb h+

7 7 h re,,e l ζ j e ++e l h π b ζ b h+ + πf b ζ From the above equation, we deduce the following identities: re,,e f,,e f+,,e l re,,e f,e f+,,e l, re,,e f +,,e l e f re,,e f,,e l π l b ζb h+ Now it is very easy to conclude that l re,,e l h l e s! s Examples 5 Theorem 3 enables us to compute m x Here are the first few examples for,3,,6 m x ζ, ζ3 m 3 x 6 3ζ3 4, ζ4 m 4 x ζ, 6 6ζ4+ 3ζ ζ4 3ζ +ζ4, 4 4 ζ5 m 5 x 4 + ζ,3+ζ3, 6 3ζ5 5ζζ3 ζ5 5ζζ3+45ζ5, ζ6 m 6 x ζ3,3 + ζ,4+ζ4, + ζ,, ζ6+ 45ζ3 ζ6 +45ζζ4 ζ6 + 45ζ6 3ζζ4+ζ ζ6+8ζ3 +35ζζ4+5ζ 3 8 Remar 6 Ohno and Zagier [3] prove a result that generalizes Proposition 4 Following their notation from Theorem, [3], and setting y, z x, so that s n we have 4 hζb,,b h x exp b ++b h,b i t ζt x t t t

8 8 This identity also explains the relationship between the result in the statement of Theorem 3 and the result that is re obtained in Section 4 Higher Mahler measure for several linear polynomials As before, the simplest case to consider involves linear polynomials in one variable Theorem 7 For α m x, e πiα x π In particular, one obtains the following examples: Examples 8 PROOF By definition, m x, x π, m x,+x π 4, m x,±ix π 96, α α+ 6 m x, e πiα x α 3± 3 6 m x, e πiα x,l Relog e πiθ Relog e πiθ+α dθ cosπθ l l l cosπθ+α cosπθ cosπlθ + α dθ dθ On the other hand, cosπθcosπlθ+α dθ By putting everything together we conclude that m x, e πiα x cosπα cosπα if l, otherwise π α α+ 6

9 9 Remar 9 The same calculation shows that ReLi α β m αx, βx ReLi αβ α if α, β, if α, β, ReLi α β +log α log β if α, β αβ From this, one sees that for P C[x ± ], m P is a combination of dilogarithms and products of logarithms In fact, for Px cx s r j α j x, we have m P mp,p log c +log c r log + α j + j r m α j x, α x j, The formula above plays an analogous role to Jensen s formula Remar The previous computations may be extended to multiple higher Mahler measures involving more than two linear polynomials For example, m x, e πiα x, e πiβ x 4 4 4,l,m l,m cosπ +lβ lα l +l cosπ +mα mβ m +m cosπlα+mβ lml +m 3 Higher Mahler measure of two-variable polynomials In this section we are going to consider examples of higher Mahler measures of polynomials in two variables In particular, we will focus on the computation of m using the formula from Remar 9, analogously to the way Jensen s formula for computing the classical Mahler measure of multivariable polynomials The two polynomials that we consider were among the first examples of multivariable polynomials to be computed in terms of Mahler measure by Smyth [6]

10 3 m x+y + Theorem PROOF We have, by definition, m x+y + πi m x+y + 5π 54 y x log x+y + dx x We apply the result from Remar 9 respect to the variable y, m x+y + πi x, x+ Li +x dx x + πi x, x+ Li +x Recalling the functional identity for the dilogarithm, Li z Li z log z π 6 for z,, we obtain m x+y + πi + πi πi π Notice that cos n θ dθ tanθ In particular, 3 cos n θ dθ π 3 π 3 n x, x+ x, x+ x, x+ 4π 3 π 3 Li +x dx x ReLi dy y dx +log +x x +x + π 6 Li +x dx x + π 9 θ Li 4cos dθ + π 9 n n n l + n n n n n n l n l l l l θ l+ l l + n cos l θ dx x n π n 3 Now we use the identity for the sum of the inverses of Catalan numbers, π 3 9 l l+ l l,

11 in order to get π 3 π 3 cos n θ dθ 3 n n n ln l+ l l Note that l!l! l +! Bl+,l+ s l s l ds Thus the above sum may be written as π ln s l s l ds Putting everything together yields 4π 3 π 3 θ Li 4cos 3 π dθ+ π 3 9 π n At this point, we need the following Lemma For t, we have 4 t Li s n s n s s ds n n n s n s n n n s s 4t log PROOF of Lemma We start from the series t + convergent for t 4 By integration, we have n n s n s n s s ds+π 9, ds+ π t 4 4t + 4t t log + 4t +log By integration again, we obtain the result Now, if we set t s s, we obtain 4t s Then the quantity 3 becomes 3 π Li s log ds s s s + π 9

12 3 ds π s s s s But ds s ds 3 s+s π s s s s+s 3i s ω, s ω where ω + 3i Thus, the above equals i ds ds π s s s ω s ω + i π s s s s s s ds ds s s s ω s ω ds ds ds ds s s s+s +π 9 ds+ π 9 i π Li,ω, ω Li, ω,ω Li,,,ω, ω+li,,, ω,ω+ π 9 where we have written the result in terms of polylogarithms Now Li,,, ω,ω Li,,,ω, ω 5iπ3 8, and Li, ω,ω Li, ω, ω 7iπ3 6 see for example [], and note that 7π 6 5π 8 + π 9 5π 54 The result should be compared to Smyth s formula mx+y π Lχ 3, L χ 3, 3 m +x+y x Theorem 3 m +x+y x 4i π Li, i, i Li, i,i+ 6i π Li, i,i+li, i, i + i π Li,,i+Li,, i 7ζ 6 + log π Lχ 4, PROOF In order to apply the formula from Remar 9 for the variable y we need to have a rational function that is monic in y Therefore, we divide by the factor +x: x m x+y+x m +x +y +m x +y,+x +m +x +x 5

13 3 For the first term, we have x m +y +x πi y x x log +x +y dx x dy y By applying Remar 9, this equals πi x, x +x Li x dx +x x + πi + log x πi x, x +x +x x Li dx πi x, x +x +x x + πi x, x +x dx x x, x +x Li +x dx x x log x +x For the second term in equation 5 we obtain x m +y,+x log x +x πi +y dx y x +x log +x x By Jensen s formula respect to the variable y, this equals log + x dx πi +x log +x x log x πi +x x Then 5 becomes m x+y+x πi + πi x, x +x x, x +x For the first term on the right-hand side, π πi π 4 π 4 x, x +x Li tan θ dθ 4 π x, x +x x Li dx +x x dx x dy y log +x dx x log x log +x dx x + ζ 6 x Li dx +x x π 4 After the change of variables y tanθ, this becomes 8 π dy Li y+li y y + 4 π π 4 Li tanθ+li tanθ dθ Li y+li y +iy + iy 4 π ili,i, i+ili, i, i ili, i,i ili, i,i dy

14 4 For the second term in 6, we have log x log +x dx πi x x, x +x 3 4 +l cosπθ cosπlθ dθ l,l 4 +l i +l+ +l πl +l,l i +l i +l i +l i +l π l π +ll + i π,l >l,l +l i l i ll π >l + i l+ l l +l i l i π Li ili i Li ili i Li,,i+Li,, i + i π ζli i Li i Li, i,i+li, i, i i π iloglχ 4, πi 6 ζ Li,,i+Li,, i + i π ζπi Li, i,i+li, i, i Putting everything together in 6, we obtain the final result l m x+y+x 4i π Li, i, i Li, i,i+ 6i π Li, i,i+li, i, i + i π Li,,i+Li,, i 7ζ 6 + log π Lχ 4, The previous result should be compared to see [6] m x+y+x π Lχ 4, 4 Zeta Mahler measures In this section, we consider zeta Mahler measures We compute some examples and apply them to the computation of higher Mahler measures 4 Zs,x As usual, we start with the linear polynomial x

15 5 Theorem 4 around s Zs,x sinπθ s dθ ζ exp s This result is a particular case of a formula obtained by Aatsua [] PROOF First we show that Zs,x Γs+ s! s Γ s + s,! s/ where s! Γs+ In fact, / Zs,x s+ sinπθ s dθ After the change of variables t sin πθ this becomes s π t s t / dt So, we have obtained the Beta function: Zs,x s s+ π B, s π Γ s+ Γ Γ s + s Hence, by using s+ Γ Γs Γ s π Γs+ s Γ π s + s Γ s+ π Γ s + we conclude that Zs,x Γs+ Γ s + 7 On the other hand, the product expression yields Zs,x Γs+ e γs n exp + s n + s n n n + s e s n n { log + s log + s } n n,

16 6 { exp }s n n n exp ζ s ζ exp s An analogous idea for evaluating Zs,P appears in [5] 4 m x We can now use the evaluation of Zs,x to re obtain the formula for m x From Theorem 4, ζ Zs,x exp 4 s ζ3 4 s3 + 7ζ4 3 s4 + + ζ 4 s ζ3 4 s3 + On the other hand, by construction, 7ζ4 3 + ζ 3 s 4 + Zs,x +m x s+ m x s + 6 m 3x s m 4x s 4 + Putting both identities together, we recover the result from Theorem 3 In particular, m x, m x ζ π, m 3 x 3ζ3, m 4 x 3 4 7ζ4+ζ 9π4 4, 5 A computation of higher zeta Mahler measure We compute the simplest example of a higher zeta Mahler measure and apply it to multiple higher Mahler measures

17 7 Theorem 5 i Zs,t;x,x+ sinπθ s cosπθ t dθ Γs+Γt+ Γ s + Γ t + Γ s+t + s!t! s! t! s+t! + s n + t n + s+t n + s n + t n n ii Zs,t;x,x+ exp ζ { s +t s+t } Q[π,ζ3,ζ5,][[s,t]] around s t iii mx,,x,x+,,x+ }{{}}{{} l log sinπθ log cosπθ l dθ belongs to Q[π,ζ3,ζ5,ζ7,] for integers,l PROOF i By definition, Zs,t;x,x+ s+t sinπθ s cosπθ t dθ / s+t+ sinπθ s cosπθ t dθ By the change of variables u sin πθ, Zs,t;x,x+ s+t π s+t π B s+t π Γ s+ u s t u du s+, t+ Γ t+ Γ s+t +

18 8 We now use again the identity z + Γ z π Γz + Γ z +, to get Zs,t;x,x+ Γs+Γt+ Γ s + Γ t + Γ s+t + + s n + t n + s+t n + s n + t n n ii The above expression yields Zs,t;x,x+ { exp log + s n n log +log + t n + s n log +log + s+t n + t n } { s t exp + + n n n exp ζ { s + t + s+t s t } exp ζ { s + t s+t } s+t s } t n n n This power series belongs to Q[π,ζ3,ζ5,ζ7,][[s,t]] iii From ii, we see that +l s t lz,;x,x+ Q[π,ζ3,ζ5,ζ7,], which is simply mx,,x,x+,,x+ }{{}}{{} l log sinπθ log cosπθ l dθ Example 6 In order to compute examples, we compare the terms of lowest degrees in the two expressions of Zs,t;x,x+ On the one hand, we

19 9 have Zs,t;x,x+ ζ 3 exp 4 s +t 4 s+t ζ s3 +t 3 8 s+t3 +degree 4 ζ exp s +t st ζ3 s 3 +t 3 s t st +degree On the other hand, Zs,t;x,x+ + mx,x s + mx+,x+t +mx+,x st + 6 mx,x,x s3 + 6 mx+,x+,x+t3 + mx,x,x+s t+ mx,x+,x+st We obtain: +degree 4 mx,x+ mx,x,x+ mx,x+,x+ Note that the calculation log sinπθ log cosπθ dθ ζ 4 log sinπθ log cosπθ dθ ζ3 8 log sinπθ log cosπθ dθ ζ3 8 Zs,;x,x+ Zs,x yields m x again We also remar that we have another relation s s/ Zs,s;x,x+ Zs,x Zs,x+ 6 Further examples 6 The case P x+x +y +y +c Theorem 7 For c > 4, Zs,x+x +y +y +c c s j s j c j c s 3F s j j, s,, 6, c π 4, ζ3 4, ζ3 4

20 where the generalized hypergeometric series 3 F is defined by 3F a,a,a 3 b,b z j a j a j a 3 j z j, b j b j j! with the Pochhammer symbol defined by a j aa+ a+j PROOF We first write x + x + y + y x+x + c c +y+y + c Since c 4, x+x +y+y + is a positive number in the unit torus Hence, c we may omit the absolute value in the computation of the zeta function Therefore we may write Zs,x+x +y +y +c x+x +y +y +c s dx dy πi y x x y c s + x+x +y +y s dx dy πi y x c x y s x+x c s +y +y dx πi y x c x s j c s j c j j j dy y The last equality is the result of the following observation The number x+x +y +y dx dy πi x y y x is the constant coefficient of x+x +y +y This idea was observed by Rodriguez-Villegas [4] who studied this specific example as part of the computation of the classical Mahler measure for this family of polynomials The expression in terms of the generalized hypergeometric function is derived by s j j! j s j s j and j! j jj! Note that the series 3 F z converges in z <, which is compatible with the condition c > 4 in the statement of the Theorem 6 Properties of zeta Mahler measures The proof of Theorem 7 may also be achieved by combining the following elementary properties of zeta Mahler measures: Lemma 8 i For a positive constant λ, we have Zs,λP λ s Zs,P

21 ii Let P C[x ±,,x ± n ] be a Laurent polynomial such that it taes non-negative real values in the unit torus Then we have the following series expansion on λ /maxp, where maxp is the maximum of P on the unit torus: Zs,+λP m+λp s Z,Pλ, Z,Pλ More generally, m j +λp j! j j Z j,pλ j j < << j iii Zs,P Z s,p P, where we put P αāαx α for P α a αx α Note that P P is real-valued on the torus Therefore, in principle, the nowledge of m + λp yields enough information to determine Zs,+λP PROOF i and iii are obvious For ii, we may use the Taylor expansions in λ; +λp s In particular, we may write s λ P, log+λp λ P Zs,+λP m +λp s! Z,Pλ ss s +! In other words, the coefficients with respect to the monomial basis are the -logarithmic Mahler measures m +λp, while the coefficients with respect to the shifted monomial basis are the special values of zeta Mahler measures Z,Pλ Combining these observations, we obtain the three equalities 63 The case P x+y +c Now we apply these ideas to P x+y +c with c Theorem 9 Let c Then

22 i s/ Zs,x+y +c c s j j c j j, j ii m x+y+c log c+ c, iii m 3 x+y+c log 3 c+ 3 logc c 3 c j j iv In particular, we obtain the special values m x+y + ζ, v m 3 x+y + 9 logζ 5 4 ζ3 PROOF i In this case, the polynomial is not reciprocal, so we first need to consider x+y +cx +y +c Then, Zs,x+y +c Zs/,x+y +cx +y +c x+y +cx +y +c πi dy y x x y c s + x+y s/ + x +y s/ dx dy πi y x c c x y j s/ s/ x+y x c s +y j πi j y x c c s/ c s j j c j j j s/ dx dx x dy y The last identity was obtained, as in the case of x+x +y +y +c, by computing the constant coefficient of the product of powers of polynomials in the integrand

23 3 Formulas ii and iii are consequence of i and Lemma 8 If we set t /4in theequation of Lemma, we obtainζ log Combining this with ii, we get the result of iv For the last formula v, it is enough to prove the following identity: 4 j 4 3 log3 ζlog+ 5 ζ3 j We have t 4t Li log + 4t Now we turn the left-hand side into a double series: t t x j x x j 4xt + 4xt Li log xx 4t + 4t Li log x In particular, by evaluating at t, we obtain 4 x j 4 j x + x Li log xx ζ log x Integrating fromto, weobtainthedoubleseries thatwewishtoevaluate: I : 4 j j x + x Li log xx ζ log dx x We just need to perform the integration For that, we consider the change of variables y x I : Li y log y 4 y y y 4 ζ log dy y dy

24 4 We write the expression in terms of iterated integrals, so that we can relate the result to multiple polylogarithms: We have I Li y log y dt dt t t + t t y t t y t t t t y t t y dt dt t t dt dt 4 t t y y dy y dt dt 4 t t y y dy y dt dt dt dt + t t t t t t After some rearranging we get dt dt I t t y t t y + dy y dt dt + t t y + y +8 t t y y,t t dy dt y t dt t +8 dy y,t t 4 dy y dy dt dt y t t We mae the change of variables s i t i, z y Then ds ds I s s z s s z + dz z ds ds + s s z s s z + dz z dz ds ds dz ds ds z s s z s s z,s s z,s s Now we mae another change of variables u i s i, w z to get I w u u w+ + dw du du w u u + w u u w+ + dw du du w u +u + du du dw 4 u u + w 4 du du dw u +u + w u w,u u w,u We may now express all the terms as hyperlogarithms, and then as multiple polylogarithms evaluated in ± I I,,,,, I,,,,, I,,,,, I,,,,,

25 5 4I,,, 4I,,, 4I,,, 4I,,, Li,,,, +Li,,,, +Li,,,, +Li,,,, 4Li,, 4Li,, 4Li,, 4Li,, The terms involving multiple polylogarithms of length greater than may be expressed as terms involving ordinary polylogarithmsof length First, we reduce the multiple polylogarithms from length 3 to length and using the following identities: Li,,,, 3 Li Li,, Li,, Li,,, Li,,,, 6Li Li,, Li Li,, Li,, Li,, 6Li,, 6Li,,, Li,,,, Li 3, 6 Li,,,, 6 Li Li,, +Li,, +Li,, Incorporating these identities in the expression for I, we get I 3 Li Li,, 3 Li,, 3 Li,, +Li Li,, 3 Li Li,, 6 Li,, 6 Li,, Li,, Li,,+ 3 Li Li Li,, + 3 Li,, + 3 Li,, 4Li,, 4Li,, 4Li,, 4Li,, Li Li,, 9 Li,, Li,, +Li Li,, 5Li,, Li,, + 3 Li 3 4Li,, 4Li,, Now we consider identities of multiple polylogarithms of length in terms of classical polylogarithms Li,, Li Li, Li,, 4 Li Li +Li 3, Li,, 3Li Li +Li 3, Li,, Li, Li,, 8 8Li Li +5Li 3,

26 6 Li,, Li 3, Li,, 8 Li Li 3Li 3, Li,, Li 3 8 Applying the previous identities to the expression for I gives I Li 3 Li Li Li Li Li Li Li Li 3+ Li 3 5Li Li 5 8 Li 3 Li Li 3 +6Li Li + 3 Li 3 Li Li 3 +Li Li + 5 Li 3 We may now write the expression in terms of values of the zeta function and logarithms I 4 3 log3 ζlog+ 5 ζ3 This shows the required identity for the formula 5 The previous Theorem may be completed with the trivial statement mx+y + log In fact, the motivation for setting c is that this is the precise point where the family of polynomials x+y+c reaches the unit torus singularly In classical Mahler measure, those polynomials are among the simplest to compute the Mahler measure, and the same is true in higher Mahler measures 64 A family related with Dyson integrals Consider the following family of polynomials P N x,,x N x h x j h j N h<j NN h<j x h x j x j x h sin πθ h θ j, x h e πiθ h Then we have the following result due to Dyson: Z,P N P N e πiθ,,e πiθ N dθ dθ N N!! N

27 7 Incorporating this identity into the formula for the zeta Mahler measure we obtain Zs,+λP N +λp N s dθ dθ N s s N! Z,P N λ! N λ s, N F, N,, N N N N λ,, N N As always, we may use the expression of zeta to compute higher Mahler measures By Lemma 8 ii, m+λp N m +λp N Z,P N λ N!! N λ, + + Z,P N λ + + N!! N λ In particular, for N, m+λp m +λp λ, + + λ These correspond to the higher Mahler measures of +λx+x +y+y Acnowledgements: We would lie to than Fernando Rodriguez-Villegas for helpful discussions References [] H Aatsua, Zeta Mahler measures, in: Mahler measure conference, Toyo Institute of Technology, December 7 [] D J Broadhurst, Massive 3-loop Feynman diagrams reducible to SC primitives of algebras of the sixth root of unity, Eur Phys J C Part Fields 8 999, no, [3] Y Ohno, D Zagier, Multiple zeta values of fixed weight, depth, and height, Indag Math NS, no 4,

28 8 [4] F Rodriguez-Villegas, Modular Mahler measures I, Topics in number theory University Par, PA, 997, 7 48, Math Appl, 467, Kluwer Acad Publ, Dordrecht, 999 [5] F Rodriguez-Villegas, Personal communication, August 7 [6] C J Smyth, On measures of polynomials in several variables, Bull Austral Math Soc Ser A 3 98, Corrigendum with G Myerson: Bull Austral Math Soc 6 98, Nobushige Kuroawa Department of Mathematics, Toyo Institute of Technology -- Oh-Oayama, Meguro, Toyo, 5-855, Japan uroawa@mathtitechacjp Matilde Lalín Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G G, Canada mlalin@mathualbertaca Hiroyui Ochiai Department of Mathematics, Nagoya University Furo, Chiusa, Nagoya , Japan ochiai@mathnagoya-uacjp