Higher Mahler measures and zeta functions

Transcription

1 Higher Mahler measures and zeta functions N Kuroawa, M Lalín, and H Ochiai September 4, 8 Abstract: We consider a generalization of the Mahler measure of a multivariable polynomial P as the integral of log P in the unit torus, as opposed to the classical definition with the integral of log P A zeta Mahler measure, involving the integral of P s, is also considered Specific eamples 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[ ±,, ± ] is defined by n 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 Supported by University of Alberta Fac Sci Startup Grant N36 and NSERC Discovery Grant

2 These terms are the coefficients in the Taylor epansion of Aatsua s zeta Mahler measure Zs, P P e πiθ,, e πiθn s dθ dθ n Zs, P m P s! Aatsua [] computed the zeta Mahler measure Zs, 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[ ±,, ± 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

3 3 In this wor, we compute the simplest eamples of these heights and eplore their basic properties In section we consider the case of higher Mahler measure for one-variable polynomials More precisely, we consider linear polynomials in one variable In particular, we obtain m π, m 3 3ζ3, m 4 9π4 4, m, e πiα π α α + 6, α In section 3, we consider two eamples of two-variable Mahler measure and we compute m Sections 4 and 5 deal with eamples 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 eplore harder eamples of zeta and higher Mahler measures in Section 6 For eample, m + y + ζ, m 3 + y log ζ 4 ζ3, Zs, + + y + y + c c s s 3F, s, 6,, c > 4 c Higher Mahler measure of one-variable polynomials The case of Our first eample is given by the simplest possible polynomial, namely P Theorem 3 m b + +b h, b i where ζb,, b h denotes a multizeta value, ie,! h ζb,, b h, ζb,, b h l <<l h l b l b h h

4 4 The right-hand side of Theorem 3 can be re-written in terms of classical zeta values by using the following result Proposition 4 σ S h ζb σ,, b σh e + +e l h h l l e s! ζ 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 moves 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 log Re log log + log dt t + dt dt t j t Now observe that dt j dt j t t j! j! j dt t dt } {{ t } j j dt t dt }{{ t } j j dt t We have just used the iterated integral notation of hyperlogarithms Combining the previous equalities, m log d πi! dt πi j t dt dt } {{ t } t dt }{{ t } j j d If we now set s t in the first j-fold integral and s t j-fold integral,! j πi ds s ds s ds s ds s in the second d

5 5 We proceed to compute the integrals in terms of multiple polylogarithms! l j m j d πi l l j m m j! j <l <<l j < <m <<m j < 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 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 e + + e h j h e ++e h j h Then the term is equal to min{j, j } 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 c a,,a h ζ{} a,,, {} ah,, h 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 + dt t dt }{{ t } a h + dt t

6 6 Maing the change t t, h dt t dt t dt t } {{ } a h + which corresponds to the term in the right Thus, the total sum is m b ++b h, b i dt t dt t! h ζb,, b h We show a proof of Proposition 4 for completeness PROOF Proposition 4 We first show that we can write ζb σ,, b σh σ S h e ++e l h re,, e l ζ π b ζ dt t } {{ } a + π l b where the function re,, e l satisfies some recurrence relationships Here, as in the statement, 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 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+, σ S h ζb σ,, b σh ζb h+ e ++e l h re,, e l ζ π b ζ π l b ζb h+ Observe that we have the following relation ζb σ,, b σh ζb h+ ζb σ,, b σh+ σ S h σ S h+ + h j σ S h ζb σ,, b σ j,, b σh,,

7 7 where b j b j + b h+ Hence, h e ++e l h j e ++e l h ζb σ,, b σh+ σ S h+ re,, e l ζ ζ re,, e l ζ π b π b π l b ζ b h+ + πf b ζb h+ 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 Eamples 5 Theorem 3 enables us to compute m Here are the first few eamples for, 3,, 6 m ζ ζ3 m 3 6 3ζ3 4 ζ4 ζ, m ζ4 + 3ζ ζ4 3ζ + ζ4 4 4 ζ5 ζ, 3 + ζ3, m ζζ3 ζ5 5ζζ3 + 45ζ5 3ζ5 ζ6 ζ3, 3 ζ, 4 + ζ4, ζ,, m ζ6 + 45ζ3 ζ6 + 45ζζ4 ζ6 + 45ζ6 3ζζ4 + ζ ζ6 + 8ζ3 + 35ζζ4 + 5ζ 3 8

8 8 Remar 6 Ohno and Zagier [3] prove a result that generalizes Proposition 4 Following their notation from Theorem, [3], and setting y, z, so that s n we have 4 b ++b h, b i h ζb,, b h ep t ζt t t t This identity also eplains 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, e πiα π In particular, one obtains the following eamples: Eamples 8 α α + 6 PROOF m, π, m, + π 4, m, ± i π 96, m, e πiα α 3 ± 3 6 By definition, m, e πiα,l R log e πiθ R log e πiθ+α dθ cos πθ l l cos πθ + α l cosπθ cosπlθ + α dθ dθ On the other hand, cosπθ cosπlθ + α dθ cosπα if l, otherwise

9 9 By putting everything together we conclude, m, e πiα π Remar 9 The same calculation shows that α β m α, β RLi RLi RLi αβ α α β cosπα α α + 6 if α, β, if α, β, αβ + log α log β if α, β From this, one sees that for P C[ ± ], m P is a combination of dilogarithms and products of logarithms In fact, for P c s r j α j, we have m P mp, P log c +log c r log + α j + j r m α j, α j, The formula above plays an analogous role to Jensen s formula Remar The previous computations may be etended to multiple higher Mahler measures involving more than two linear polynomials For eample, m, e πiα, e πiβ 4 4 4,l,m l,m cos π + lβ lα l + l cos π + mα mβ m + m cos πlα + mβ lml + m 3 Higher Mahler measure of two-variable polynomials In this section we are going to consider eamples 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 in an analogous way as the

10 usual use of Jensen s formula for computing the classical Mahler measure of multivariable polynomials The two polynomials that we consider were among the first eamples of multivariable polynomials to be computed in terms of Mahler measure by Smyth [6] 3 m + y + Theorem PROOF We have, by definition, m + y + πi m + y + 5π 54 y log + y + d dy y We apply the result from Remar 9 respect to the variable y, πi, + Li + d + d πi Li + log + +, + Recall the functional identity for the dilogarithm, Li z Li z log z π 6 for z, Thus, we obtain πi + πi πi π, +, +, + 4π 3 π 3 Notice that cos n θ dθ tan θ Li + d ReLi + + π 6 Li + d + π 9 θ Li 4 cos dθ + π 9 n n n l + n n n n l l l l θ d cos l θ

11 In particular, π 3 3 cos n θ dθ π 3 n n n n l l + l l + n n π n 3 Now we use the identity for the sum of the inverses of Catalan numbers in order to get Note that π 3 π 3 π 3 9 cos n θ dθ l!l! l +! Then the sum may be written as π ln 3 n l l + l l n n ln Bl +, l + s l s l ds Putting everything together, 4π 3 π 3 θ Li 4 cos 3 π dθ+ π 9 n 3 At this point, we need the following Lemma For t, we have 4 PROOF t Li π, l + l l s l s l ds s n s n s s ds n n n s n s n n n s s 4t log of Lemma We start from the series t + n n s n s n s s ds+π 9, ds + π t 4 4t + 4t convergent for t 4 By integration, we have t log + 4t + log

12 By integration again, we conclude the result Now, if we set t s s, we obtain 4t s Then equation 3 becomes 3 Li s log ds s π s s + π 9 3 π But s s s where ω + 3i Thus, i π + i π ds s ds s ds 3 s + s π s s s s + s 3i s ω, s ω s s s s s s ds s ds ds s s ds s s ω s ω s ω s ω ds ds ds s s s + s +π 9 ds 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 eample [] Then we obtain 7π 6 5π 8 + π 9 5π 54 The result should be compared to Smyth s formula m + y π Lχ 3, L χ 3, 3 m + + y Theorem 3 m ++y 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,

13 3 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 + : m +y+ m + y +m + y, + +m For the first term, we have m + y + πi By applying Remar 9, πi, + Li y d + log + + πi + log πi, + + Li d πi, πi y + y, + d, + d Li dy y log + + d For the second term in equation 5 we obtain m + y, + log + πi + y d + log + By Jensen s formula respect to the variable y, log + d πi + log + πi Then equation 5 becomes m + y + πi For the first term, π πi π 4 π 4, + + πi, + Li tan θ dθ 4 π, +, + Li d + d dy y log d + log + log log + d + ζ 6 Li d + π 4 π 4 Li tan θ + Li tan θ dθ

14 4 Now we mae the change of variables y tan θ 8 π 4 π dy Li y + Li y y + Li y + Li y + iy + iy 4 π ili,i, i + ili, i, i ili, i, i ili, i, i For the second term in 6, πi, + dy log log + d 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 i log Lχ 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 + y + 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 + y + π Lχ 4, 4 Zeta Mahler measures In this section, we consider zeta Mahler measures We compute some eamples and apply them to the computation of higher Mahler measures

15 5 4 Zs, As usual, we start with the linear polynomial Theorem 4 around s Zs, sin πθ s dθ ζ ep s This result is a particular case of a formula obtained by Aatsua [] PROOF First we show that Γs + s! s Zs, + s,! s/ where s! Γs + In fact, Γ s / Zs, s+ sin πθ s dθ We consider the change of variables t sin πθ: s π Thus, be obtain the Beta function s π B s π s t s t / dt s +, Γ Γ s+ Γ s + Γ s+ π Γ s + Hence, by using s + Γ Γs Γ s π Γs + s Γ s + π s we conclude Γs + Zs, Γ s + 7 On the other hand, the product epression Γs + e γs n + s e s n n,

16 6 yields + s n Zs, + s n n { ep log + s n n { ep n ep ζ s ζ ep s log + s } n } s n n An analogous idea for evaluating Zs, P appears in [5] 4 m We can now use the evaluation of Zs, to re obtain the formula for m From Theorem 4 ζ Zs, ep 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, +m s+ m s + 6 m 3 s m 4 s 4 + Putting both identities together, we recover the result from Theorem 3 In particular, m, m ζ π, m 3 3ζ3, m ζ4 + ζ 9π4 4,

17 7 5 A computation of higher zeta Mahler measure We compute the simplest eample of a higher zeta Mahler measure and apply it to multiple higher Mahler measures Theorem 5 Zs, t;, + sin πθ s cos πθ 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 Zs, t;, + ep ζ { s + t s + t } Q[π, ζ3, ζ5, ][[s, t]] around s t 3 m,,, +,, + }{{}}{{} l log sin πθ log cos πθ l dθ belongs to Q[π, ζ3, ζ5, ζ7, ] for integers, l PROOF By definition, Zs, t;, + s+t sin πθ s cos πθ t dθ / s+t+ sin πθ s cos πθ t dθ Now we mae the change of variables u sin πθ, s+t π u s u t du,

18 8 thus obtaining the beta function s+t s + π B, t + s+t Γ s+ Γ t+ π Γ s+t + We now use the identity z + Γ z π Γ z + Γ z + Replacing into the epression for zeta, Zs, t;, + The above epression yields Zs, t;, + { ep log + s n n log Γs + Γt + Γ s + Γ t + Γ s+t + + s n + t n + s+t n + s n + t n n + log + t n + s n log + log + s + t n + t n } { s t ep + + n n n ep ζ { s + t + s + t s t } ep ζ { s + t s + t } This power series belongs to 3 From, we see Q[π, ζ3, ζ5, ζ7, ][[s, t]] +l s t Z, ;, + l Q[π, ζ3, ζ5, ζ7, ], which is simply m,,, +,, + }{{}}{{} l s + t s } t n n n log sin πθ log cos πθ l dθ

19 9 Eample 6 In order to compute eamples, we compare the terms of lowest degrees in the two epressions of Zs, t;, + On the one hand we have Zs, t;, + ζ 3 ep ζ ep 4 On the other hand, 4 s + t s + t 4 s + t st ζ3 8 ζ s3 + t 3 s + t3 8 s 3 + t 3 s t st + degree 4 Zs, t;, + + m, s + m +, + t + m +, st + 6 m,, s3 + m +, +, + t3 6 + m,, + s t + m, +, + st We obtain: +degree 4 m, + m,, + m, +, + Note that the calculation log sin πθ log cos πθ dθ ζ 4 log sin πθ log cos πθ dθ ζ3 8 log sin πθ log cos πθ dθ ζ3 8 Zs, ;, + Zs, yields m again We also remar that we have another relation s s/ Zs, s;, + Zs, Zs, + 6 Further eamples 6 The case P + + y + y + c Theorem 7 For c > 4, Zs, + + y + y + c c s j s j c j j j π 4, ζ3 4, ζ3 4 + degree 4

20 c s 3F s, s,, 6, c where the generalized hypergeometric series 3 F is defined by a, a, a 3 3F b, b z 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 + + y + y + + c c +y+y + c Since c 4, + +y+y + is a positive number in the unit torus Hence, c we may dismiss the absolute value in the computation of the zeta function Therefore we may write j Zs, + + y + y + c + + y + y + c s d dy πi y y c s y + y s d dy πi y c y s + c s + y + y d πi y c s j c s j c j j j dy y The last equality is the result of the following observation The number + + y + y d dy πi y y is the constant coefficient of + + y + y This idea was observed by Rodriguez-Villegas [4] who studied this specific eample as part of the computation of the classical Mahler measure for this family of polynomials The epression in terms of 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:

21 Lemma 8 For a positive constant λ, we have Zs, λp λ s Zs, P Let P C[ ±,, ± n ] be a Laurent polynomial such that it taes non-negative real values in the unit torus Then we have the following series epansion on λ / map, where map is the maimum of P on the unit torus: Zs, + λp m + λp s Z, P λ, Z, P λ More generally, m j + λp j! j j Z j, P λ j j < << j 3 Zs, P Z s, P P, where we put P α āα α for P α a α 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 and 3 are obvious For, we may use the Taylor epansions in λ; + λp s log + λp In particular, we may write s λ P, λ P Zs, + λp m + λp s! ss s + Z, P λ! 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, P λ Combining these observations, we obtain the three equalities

22 63 The case P + y + c Now we apply these ideas to P + y + c with c Theorem 9 Let c s/ Zs, + y + c c s j j c j j j m + y + c log c + c 3 m 3 +y+c log 3 c+ 3 log c c 3 c j j 4 In particular, we obtain the special values m + y + ζ, 5 m 3 + y + 9 log ζ 5 4 ζ3 PROOF In this case, the polynomial is not reciprocal, so we first need to consider + y + c + y + c Then, Zs, + y + c Zs/, + y + c + y + c πi + y + c + y s/ d dy + c y y c s + + y s/ + + y s/ d dy πi y c c y j s/ s/ + y c s + y j πi j y c c s/ c s j j c j j j d dy y

23 3 The last identity was obtained, as in the case of + + y + y + c, by computing the constant coefficient of the product of powers of polynomials inside the integral sign Formulas and 3 are consequence of and Lemma 8 If we set t /4 in the equation of Lemma, we obtain ζ log Combining this with, we get the result of 4 For the last formula 5, it is enough to prove the following identity: 4 j We have t 4t Li j 4 3 log3 ζ log + 5 ζ3 log + 4t Now we turn the left-hand side into a double series t t j j 4t + 4t Li log 4t + 4t Li log In particular, evaluating at t, we obtain 4 4 j j Li log ζ log + Integrating between and, we reach the double series that we wish to evaluate I : 4 j j + Li log ζ log d

24 4 We just need to perform the integration For that, we consider the change of variables y I Li y log y 4 y y y 4 ζ log dy y We write the epression 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 dy 4 dy y After some rearranging, dt dt t t y t t y + dy y dt dt + t t y t t y + dy y dy dt dt y t t y,t t y,t t 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 Then I w + + dw du du w u u + w u u

25 5 4 w u u u w,u w + + dw du w u + du du dw u u + w 4 u w,u du u + du du u + u + dw w We may now epress all the terms as hyperlogarithms, and then as multiple polylogarithms evaluated in ± I,,,,, I,,,,, I,,,,, I,,,,, 4I,,, 4I,,, 4I,,, 4I,,, Li,,,, + Li,,,, + Li,,,, + Li,,,, 4Li,, 4Li,, 4Li,, 4Li,, The terms involving multiple polylogarithms of length greater than may be epressed as terms involving ordinary polylogarithms of 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 epression 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,,

26 6 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, Li,, Li 3, Li,, 8 Li Li 3Li 3, Li,, Li 3 8 Applying the previous identities to the epression for I, 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 epression in terms of values of zeta functions I 4 3 log3 ζ log + 5 ζ3 This shows the required identity for the formula 5 The previous Theorem may be completed with the trivial statement m + y + log In fact, the motivation for setting c is that this is the precise point where the family of polynomials + 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

27 7 64 A family related with Dyson integrals Consider the following family of polynomials P N,, N h j h j N h j j h h<j NN h<j sin πθ h θ j, h e πiθ h Then we have the following result Z, P N N!! N P N e πiθ,, e πiθ N dθ dθ N due to Dyson Incorporating this identity into the formula for the zeta Mahler measure we obtain Zs, + λp N + λp N s dθ dθ N s Z, P N λ s N!! N λ s, N F,,, N N N N N λ,, N N As always, we may use the epression of zeta to compute higher Mahler measures By Lemma 8, m + λp N m + λp N Z, P N λ N!! N λ, Z, P N λ N!! N λ

28 8 In particular, for N, m + λp m + λp λ, + + λ Which correspond to the higher Mahler measures of +λ+ +y +y Acnowledgements: We would lie to than Fernando Rodriguez-Villegas for helpful discussions References [] H Aatsua, Zeta Mahler measures, December 7, Mahler measure conference, Toyo Institute of Technology [] D J Broadhurst, Massive 3-loop Feynman diagrams reducible to SC primitives of algebras of the sith root of unity, Eur Phys J C Part Fields 8 999, no, [3] Y Ohno, D Zagier, Multiple zeta values of fied weight, depth, and height, Indag Math NS, no 4, [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

29 Hiroyui Ochiai Department of Mathematics, Nagoya University Furo, Chiusa, Nagoya , Japan 9