APPLICATIONS OF MULTIZETA VALUES TO MAHLER MEASURE

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1 APPLICATIONS OF MULTIZETA VALUES TO MAHLER MEASURE MATILDE LALÍN Abstract. These notes correspond to a mini-course taught by the author during the program PIMS-SFU undergraduate summer school on multiple zeta values: combinatorics, number theory and quantum field theory. Please send any comments or corrections to the author at mlalin@dms.umontreal.ca.. Primes, Mahler Measure, and Lehmer s question We start our study by discussing prime numbers. First consider the sequence of numbers M n n for n natural. We may ask, when is M n prime? We can write rs r s r + s r + + r +. This implies that M r M rs and we need n prime in order for M n to be prime. Notice that M 3, M 3 7, M 5 3, M 7 7, but M Therefore the converse is not true. The primes of the form M p are called Mersenne primes. It is unknown if there are infinitely many of such primes, or if there are infinitely many M p composite with p prime. The largest Mersenne prime known to date is with p 57, 885, 6 it has 7, 45, 7 digits. The search for large Mersenne primes is being carried by the Great Internet Mersenne Prime Search : Exercise. Let a, p be natural numbers such that a p is prime, then show that either a or p Exercise. Let p be an odd prime. Show that every prime q that divides p must be of the form q pk + with k integer. Looking for large primes, Pierce [Pi7] proposed the following construction in 97. Consider P Z[x] monic, and write P x x α i i then, we look at n i α n i. The α i are algebraic integers. By applying Galois theory, it is easy to see that n Z. Note that if P x, we get the sequence n n, the Mersenne numbers. The idea is to look for primes among the factors of n. The prime divisors of such integers must satify some congruence conditions that are quite restrictive, hence they are easier to factorize than a randomly given number. Exercise 3. Prove that n is a divisibility sequence, namely, if n m, then n m. Then we may look at the numbers p, p prime. Pierce and Lehmer observed that the only possible factors of n are given by prime powers p e of the form nk + for some integer k and e deg P. It is then natural to look for P that generate sequences that grow slowly so that they have a small chance of having factors. Lehmer [Le33] studied n+ n, observed that α n+ { α if α >, lim n α n if α <, and suggested the following definition: Key words and phrases. Mahler measure, higher Mahler measure, polylogarithms.

2 Definition.. Given P C[x], such that P x a i x α i define the Mahler measure of P as MP a i max{, α i }. The logarithmic Mahler measure is defined as mp log MP log a + i log + α i, where log + α log max{, α }. Exercise 4. Prove that for any n Z, mp x mp x n. Exercise 5. Prove that if P x a d x d + + a, then a i d i MP. As MP measures the growth of the sequence n+ n, it is natural to ask about the sequences that do not grow: When does MP for P Z[x]? We have Lemma.. Kronecker, [Kr57] Let P i x α i Z[x], if α i, then the α i are zero or roots of the unity. Proof. Consider the polynomial P n x d x αi n. i The coefficients of P n x are symmetric functions in the algebraic integers αi n, so they are elements of Z all the conjugates of each α i are present as roots of P x, since the coefficients are rational. Each of the coefficients is uniformly bounded as n varies, because α j and the set {P n } n N must be finite. In other words, there are n n for which That means, P n P n. {α n,..., αn d } {αn,..., αn d }. Thus, there is a permutation σ S d such that If σ has order k, we get, and α i is a root of α n i α n σi. α nk i α nk i, x nk x n k nk. This shows that each α i is either zero or a root of unity. case. The name Mahler came about 3 years later after the person who successfully extended this definition to the several-variable

3 Exercise 6. a Give examples to show that in general a polynomial in Z[x] may have zeros of absolute value one that are not roots of the unity. b Show that a monic example of a can only occur in degree at least 4. By Kronecker s Lemma, P Z[x], P, then MP if and only if P is the product of powers of x and cyclotomic polynomials. This statement characterizes integral polynomials whose Mahler measure is. Lehmer found the example mx + x 9 x 7 x 6 x 5 x 4 x 3 + x + log and asked the following Lehmer s question, 933: Is there a constant C > such that for every polynomial P Z[x] with MP >, then MP C? Lehmer s question remains open nowadays. His -degree polynomial remains the best possible result. Exercise 7. With the help of a computer find the Mahler measures of the following polynomials a x + x 9 x 7 x 6 x 5 x 4 x 3 + x +, b x x 6 + x 5 x 4 +, c x 4 + x x x 7 x 4 + x 3 +, d x 4 x + x 7 x +. Examples of polynomials with small Mahler measure may be found with the search engine from Mossinghoff s website [M]: Exercise 8. With the help of a computer investigate the sequences n for x +x 9 x 7 x 6 x 5 x 4 x 3 +x+ and for x 3 x. Challenge: Find 495 for the first polynomial and 33 for the second polynomial and check that they are prime numbers. Hint: it may be necessary to verify that the n satisfy a recurrence sequence as proved in [Le33], section 8. The following definiton will be used often in the notes. Definition.3. Let P x C[x] be a nonzero polynomial of degree d. We set P x : x d P x the polynomial with the same coefficients as P, but in reverse order and conjugated. reciprocal if P ±P. Otherwise, we say that P is non-reciprocal. We list here some important results in the direction of solving Lehmer s question. We say that P is Theorem.4. Breusch [Br5], Smyth [Sm7] If P Z[x] is monic, irreducible, non-reciprocal, then MP Mx 3 x θ Corollary.5. If P Z[x] is monic, irreducible, and of odd degree, then MP θ. The most general result with a bound involving the degree is given by Theorem.6. Dobrowolski [Do79] If P Z[x] is monic, irreducible and noncyclotomic of degree d, then 3 log log d MP + c log d where c is an absolute positive constant. Theorem.7. Schinzel If P Z[x] is monic of degree d having all real roots and satisfies P P and P, then + d 5 MP with equality iff P is a power of x x. For the proof, we need the following 3

4 Exercise 9. For any d and y,..., y d > be real numbers, prove that Hint: { This is a} special case { of Mahler s inequality. y,..., y d y d and to y,..., y d }. y Proof. Theorem.7 Consider y y d y y d /d d. E Apply the arithmetic geometric mean inequality to d αi. Remark that E since P is monic and α i ±. Note that MP α i α α i < i. Thus, we may rewrite E α i < α i α i > α i > i α i We apply Exercise 9 in order to obtain E MP Since E, α i < MP α i MP MP 4/d α i > d MP /d MP /d d. α i <. MP /d MP /d. α i α i /d α i > d α i. Since MP >, this implies that MP /d + 5 and we obtain the desired result. Exercise. Prove the last assertion of Theorem.7. Exercise. What happens if we relax the condition of P? Theorem.8. Bombieri, Vaaler [BV87] Let P Z[x] with MP <, then P divides a polynomial Q Z[x] whose coefficients belong to {,, }. The last result and Theorem.4 suggest that if we want to beat Lehmer s -degree polynomial, one should search for reciprocal polynomials having coefficients in {,, }. The most general result in terms of families is given by Theorem.9. Borwein, Dobrowolski, Mossinghoff [BDM7] Let D m denote the set of polynomials whose coefficients are all congruent to modulo m. Si f D m has degree d and no cyclotomic factors, then mf c m, d + where c log 5/4 and c m log m + / for m >. As a final comment, we remark that Mahler measure is related to classical heights. Generally speaking, a height is a function that measures the size of a mathematical object. For example, the absolute value measures the size of complex numbers. Another less immediate example is given by the canonical height defined in Q as follows. If a b is a rational number written in lowest terms in other words, a, b, then the height of a b is defined as max{ a, b }. This definition is what allows us to disntiguish between the 4

5 number and the number ,999,999,999,,, : while the first has height, the second one has height 9,999,999,999. For a polynomial P x a d x d + + a x + a C[x] define the height and the length of P. Exercise. Prove the following inequalities: a LP d MP. b HP d MP. HP max i d { a i }, LP d a i, i Mahler [Ma6] considered this construction because he was looking for inequalities of the classical polynomial heights such as LP or HP between the height of a product of polynomials and the heights of the factors. These kinds of inequalities are useful in transcendence theory. MP is multiplicative that is, MP Q MP MQ and comparable to the typical heights, and that makes it possible to deduce such inequalities. While investigating this in several variables, he discovered the generalization that we explore in the next section.. Mahler Measure in several variables We will be concerned mostly with the Mahler measure of multivariable polynomials. Definition.. For P C[x ±,..., x± n ], the logarithmic Mahler measure is defined by mp : The Mahler measure is defined by... πi n log P e πiθ,..., e πiθn dθ... dθ n log P x,..., x n dx T x n MP : e mp.... dx n x n. It is possible to prove that this integral is not singular and that mp always exists. appeared for the first time in the work of Mahler [Ma6]. The relationship between Definitions. and. is given by the following. This definition Theorem.. Jensen s formula Let α C. Then where log + x log max{, x} for x R. log e πiθ α dθ log + α Proof. First assume that α <. We have that log e πiθ α dθ 5 log αe πiθ dθ log αe πiτ dτ

6 where we have done τ θ. We have log αe πiτ dτ Re log αe πiτ dτ α n Re dτ Re. n n eπiτn n α n n e πiτn dτ We have exchanged the integral and the infinite sum, but this is justified since the sum is absolutely convergent. Now assume that α >. Then, by the previous argument, log e πiθ α dθ log α + log α e πiθ dθ log α. Finally, we are left with the hardest case, α. After multiplying by α, we may assume that α. Writing e πiθ sinπθ for θ, we have. Let I We write sinπθ sin πθ log e πiθ dθ log sinπθdθ + log. log sinπθdθ. The integral exists since sinπθ πθ for small θ. cos πθ. Thus, I log + log sin πθ dθ + log cos πθ dθ. By making the change τ θ/ in the first integral and τ / θ/ in the second one, we obtain, I log + 4 / log sinπτdτ log + log sinπτdτ log + I. From this, I log and we obtain the desired result by combining with equation.. The multivariable Mahler measure is still multiplicative, meaning that for P, Q C[x,..., x n ], we have mp Q mp + mq. Proposition.3. Let P C[x,..., x n ] such that a i,...,i n d i in x i. Then a b c where a i,...,i n d i... is the coefficient of x i... xin n dn i n MP MP LP d+ +dn MP d + d n + / MP HP d+ +dn n MP HP max { a i,...,i n }, LP i j d j,j,...,n 6 n d j j i j a i,...,i n. and P has degree

7 Proof. [c, lower bound] By an inequality of Hardy Littlewood Pólya, / MP... P e πiθ,..., e πiθn dθ... dθ n. Parseval s formula implies... P e πiθ,..., e πiθn dθ... dθ n i j a i,...,i n d +... d n + HP. Exercise 3. Prove the rest of Proposition.3 Exercise 4. If P C[x, x ] has a constant coefficient a that in absolute value exceeds the sum of the absolute values of all the other coefficients, prove that mp log a. Exercise 5. Let P x m c m x m C[x,..., x n ], where x m x m x mn n. Let A be an n n integer matrix with non-zero determinant, and define P A x : c m x Am. m Prove that mp m P A. It is also true that mp if P has integral coefficients. In addition, an analogous of Kronecker s lemma is true. Theorem.4. Smyth [Sm8] For any primitive polynomial i.e., the coefficients have no nontrivial common factor P Z[x ±,..., x± n ], mp is zero if and only if P is a monomial times a product of cyclotomic polynomials evaluated on monomials. Let us also mention the following result: Theorem.5. Boyd [Bo8], Lawton [La83] For P C[x,..., x n ] lim... lim mp x,,..., x k k xk kn mp x,..., x n. n It should be noted that the limit has to be taken independently for each variable. Writing this properly would take half a page. Exercise 6. Explore the limit of Theorem.5 in the case of + x + y. Namely, compute several values of m + x + x n and compare them with the value of m + x + y Because of the above theorem, Lehmer s question in the several-variable case reduces to the one-variable case. In addition, this theorem shows us that we are working with the right generalization of the original definition for one-variable polynomials. The formula for the one-variable case tells us some information about the nature of the values that Mahler measure can reach. For instance, the Mahler measure of a polynomial in one variable with integer coefficients must be an algebraic number. It is natural, then, to wonder what happens with the several-variable case. Is there any simple formula, besides the integral? Unfortunately, this case is much more complicated and we only have some particular examples. On the other hand, the values are very interesting. If it were that easy this area would not be so interesting! 7

8 3. Examples We show some examples of formulas for Mahler measures of multivariable polynomials. Smyth [Sm8] 3. mx + y π Lχ 3, L χ 3,, where Lχ 3, s n is a Dirichlet L-function. Smyth [Bo8] χ 3 n n s and χ 3 n 3. mx + y + z + 7 π ζ3, where ζs n s is the Riemann zeta function. Boyd & L. 5 L. [La3] n if n mod 3 if n mod 3 if n mod 3 mx + + x + y + x z π Lχ 4, + 8π ζ3. m + x + x x + x + x In fact, there are known formulas [La6] for x xn m + x x + x n Rogers & Zudilim [RZ] m x + x + y + y + + yz 93 π 4 ζ5. + yz. 5 4π LE 5, L E 5, where E 5 is an elliptic curve of conductor 5 that happens to be the algebraic closure of the zero set of the polynomial. Roughly speaking, an ellitic curve is the zero set of a polynomial of the form Y X 3 + AX + B where 4A 3 + 7B. The polynomial x + x + y + y + k generally corresponds to an elliptic curve under the following transformation kx Y kx + Y x y XX XX. k Y X X + 4 X +. It is then easy to eliminate the term of X by completing the cube. The L-function of an elliptic curve is a function similar to the Riemann zeta function with coefficients that encode the number of points of the curve over finite fields. Boyd 5 mz + x + y +? L E 5,. The question mark stands for an identity that has been verified up to decimal places, but for which no proof is known. 8

9 Examples with K3 surfaces, mostly due to Bertin. These includes polynomials in the family x + x + y + y + z + z + k. How do we get such formulas? Some of them are very difficult to prove. To be concrete, we are going to show the proof of the first example by Smyth from [Bo8]: Proof. Equation 3. By Jensen s formula, m+x+y π π π log +e it +e is dtds π log max{ +e it, }dt π/3 log +e it dt. π π π π π π/3 Now we write log + e it Re π/3 n e int. n The series does not converge absolutely but it converges uniformly in t [ π/3, π/3], since we are far from the singularity at t ±π. It follows from π/3 e int dt nπ 3 sin n 3 n χ 3n that 3.3 m + x + y 3 π n n n χ 3 n 3 χ 3 n n π n n and use that χ 3 n χ 3 χ 3 n χ 3 n to obtain the initial formula. Exercise 7. Prove Smyth s formula 3.. Hint: m + x + y + z m + x + z + y n χ 3 n n log max { + e πiθ, + e πiθ } dθ dθ. Some of the formulas explored in this section can be proved and better understood by using polylogarithms. 4. Polylogarithms Many examples should be understood in the context of polylogarithms. Definition 4.. The kth polylogarithm is the function defined by the power series x n Li k x : n k x C, x <. n This function can be continued analytically to C \ [,. In order to avoid discontinuities, and to extend polylogarithms to the whole complex plane, several modifications have been proposed. Zagier [Z9] considers the following version: k L k x : Re k j B j log x j Li k j x, j! j where B j is the jth Bernoulli number, Li x and Re k denotes Re or Im depending on whether k is odd or even. This function is one-valued, real analytic in P C \ {,, } and continuous in P C. Moreover, L k satisfies very clean functional equations. The simplest ones are L k k L k x L k x k L k x. x There are also lots of functional equations which depend on the index k. For instance, for k, we have the Bloch Wigner dilogarithm, Dx : Im Li x + arg x log x 9

10 which satisfies the well-known five-term relation Dx + D xy + Dy + D The dilogarithm can be also recovered in terms of an integral. θ log sin t dt De iθ More generally, recall the definition for polylogarithms. y x + D. xy xy n sinnθ n Definition 4.. Multiple polylogarithms are defined as the power series Li n,...,n m x,..., x m : x k n <k <k < <k kn m k xk... xkm m m... knm which are convergent for x i <. The weight of a polylogarithm function is the number w n + + n m. When n m > the above series converges for x i. We can then find multizeta values by setting x i : Li n,...,n m,..., ζn,..., n m. Exercise 8. a Express Li n in terms of zeta functions. b Express Li n e πi/3 Li n e πi/3 in terms of Lχ 3, n. c What is Li n i Li n i? Definition 4.3. Hyperlogarithms are defined as the iterated integrals I n,...,n m a : : a m : a m+ : am+ 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 where n i are integers, a i are complex numbers, and bk+ dt dt dt... t b t b k t t k b k+ t b dt k t k b k The value of the integral above only depends on the homotopy class of the path connecting and a m+ on C \ {a,..., a m }. To be concrete, when possible, we will integrate over the real line. It is easy to see that, I n,...,n m a : : a m : a m+ m Li n,...,n m a a, a 3 a,..., a m, a m+ a m a m Li n,...,n m x,..., x m m I n,...,n m x... x m : : x m : which gives an analytic continuation to multiple polylogarithms. For instance, with the above convention about integrating over a real segment, simple polylogarithms have an analytic continuation to C \ [,. Exercise 9. Prove the previous equations. Exercise. Prove that ζn,..., n m ζ,...,,,...,,...,,. }{{}}{{} n m n

11 4.. Applications of polylogarithms to Mahler measure. Many of the Mahler measure formulas can be proved via polylogarithms. For instance, formula 3.3 may be rewritten as m + x + y π Deπi/3 De πi/3 πi Li e πi/3 Li e πi/3. Similarly, the Riemann zeta funcion arises as a special value of the polylogarithm: ζn Li n and this identity appears also in Mahler measure formulas. Here is a more detailed example. Theorem 4.4. For a R >, m + x + a xy i π Li ia Li ia. Proof. We apply multiplicativity of the Mahler measure and then make the change of variable x e iθ and notice that x +x i tan θ. + a x Now make w, then dθ a tan θ πm + x + a xy πm + x + πm log + i a x dx T + x x π θ log + a tan dθ. a dw +a w. The previous equation equals log w a dw w a + ds i w s w + i a w i dw a i I ia ia : I : + x y i Li ia Li ia. Exercise. Prove that for a C, m + x + ay + az π Li 3 a Li 3 a a, log a + π Li 3 a Li 3 a a. Exercise. Prove that m + x + x + x x y 7 π ζ3. Polylogarithms play a crutial role in these type of formulas, relating Mahler measure to special values of zeta functions and L-functions. These relationships provide examples of very general conjectures Beilinson s conjectures which play a central role in number theory. These statements include, for example, the Birch Swinnerton-Dyer conjecture, one of the seven Millenium Prize Problems posed by the Clay Institute with a prize of one million dollars for the solution of each. It is natural to wonder if these applications of polylogarithms to Mahler measure can be generalized. The key ingredient here is the relationship that can be stablished from the integral to hyperlogarithms. Inspired

12 by this, we will explore other types of integrals which should also fit into the statement of Beilinson s conjectures. 5. Higher Mahler measure Definition 5.. The k-higher Mahler measure of P is defined by and m k P :... log k P e πiθ,..., e πiθn dθ dθ n. In particular, notice that for k we obtain the classical Mahler measure m P mp, m P. The simplest example of higher Mahler measure is with the polynomial P x [KLO8]: Theorem 5.. [KLO8] We have, m x ζ π, m 3 x 3ζ3, m 4 x 3ζ + ζ4 9π4 4 4, 5ζζ3 + 45ζ5 m 5 x, m 6 x 93ζ6 + 8ζ3 + 35ζζ4 + 5ζ ζ π6, and similar equations for higher indexes. Before proceeding into the proof of Theorem 5. we recall some properties of the Gamma function Γs : t s e t dt. Exercise 3. Prove Γ and that Γs + sγs. Deduce that Γn + n! for n positive integer. We notice that Γ t / e t dt e s ds π, where we have made the change t s and used the Gaussian integral. Let the Beta function. Br, s t r t s dt, Exercise 4. Prove that, whenever the integral for the Beta function converges, Now, Bs, s t s t s dt Br, s ΓrΓs Γr + s. s s + u u du, The relationship between higher Mahler measure and Beilinson s conjectures is yet to be stablished. One of the motivations to find examples of higher Mahler measure formulas is precisely the search of a precise formulation of this relationship.

13 where we have set t +u. This yields, By setting v u, this equals Thus, s s u s du Γs s Γ ΓsΓ s v s v / dv s B s + The above is called the duplication formula for Γ. Finally, we state the Weierstrass product without proof. Γs e γs s n u s du., s. s ΓsΓ s +. π + s s e n, n n where γ lim n k k log n is the Euler Mascheroni constant. Proof. [Theorem 5.] To prove the above equalities one uses a construction by Akatsuka [A9]. We consider the integral Zs; x e πiθ s dθ. This is called the Zeta Mahler measure. We first make the change t sin πθ: / Zs, x s+ sin πθ s dθ s π t s t / dt. Thus, we obtain the Beta function s s + π B, s Γ s+ Γ π Γ s + s Γ s+ π Γ s +, where Γs t s e t dt is the Gamma function. Hence, by using s + Γ Γs Γ s s π Γs + Γ s + π s, and Γs + e γs n 3 + s e s n, n

14 we obtain Zs, x n exp + s n + s n n k { log + s n log + s } n { k exp k k n k exp ζk k s k k k k k ζk exp s k. k } k n n k s k The Zeta Mahler measure yields a Taylor series whose coefficients are given by m k P. m k x s k Zs, x k! k k k ζk exp s k. k Exercise 5. Prove that where ζb,..., b h denotes Hint: m k x k b + +b h k, b i k k! h ζb,..., b h, ζb,..., b h l < <l h l b.... lb h h log k x Re log x k log x + log x dt k k t x + dt k k t x k j j k dt j t x We know how to answer Lehmer s question in the case of the higher Mahler measure. Theorem 5.3. [LS] If P x Z[x] is not a monomial, then for any h, h π, if P x is reciprocal, m h P h π 48, if P x is non-reciprocal. Let P n x xn x. For h fixed, lim Moreover, this sequence is nonconstant. m h+p n. n k j dt. t x A natural generalization for the k-higher Mahler measure is the multiple higher Mahler measure for more than one polynomial. 4

15 Definition 5.4. Let P,..., P k C[x ±,..., x± n ] be non-zero Laurent polynomials. Their multiple higher Mahler measure is defined by : mp,..., P k log P e πiθ,, e πiθn log Pk e πiθ,, e πiθn dθ dθ n. This construction yields the higher Mahler measure of one polynomial as a special case: Moreover, the above definition implies that m k P mp,..., P. }{{} k mp mp k mp,..., P k when the variables of P j s are algebraically independent. The simplest example is this regard is, again, given by linear polynomials. Theorem 5.5. [KLO8] For α m x, e πiα x π α α +. 6 Proof. Let A ε [ε, α ε] [α + ε, ε]. By definition, m x, e πiα x k,l + Re log e πiθ Re log e πiθ+α dθ cos πkθ cos πlθ + α k l kl k [,]\A ε k l cosπkθ cosπlθ + αdθ A ε cos πkθ cos πlθ + α dθ. k l Because log e πiθ log sin πθ log πθ for θ near zero, and ε logkxdx as ε, the last term approaches zero. Notice that Oε if l k, and cosπkθ cosπlθ + αdθ [,]\A ε cosπkθ cosπlθ + αdθ By putting everything together we conclude that m x, e πiα x k 5 cosπkα k Oε k l l otherwise, cosπkα if l k, otherwise. π α α +. 6 dθ

16 More generally, 5. m αx, βx Re Li α β Re Li αβ α if α, β, if α, β, Re Li α β αβ + log α log β if α, β. yields the equivalent of a Jensen s formula for multiple Mahler measure: if α, m αx log α if α. Exercise 6. Prove Equation 5.. Exercise 7. Prove that m x, e πiα x, e πiβ x k,l k,m l,m cos πk + lβ lα klk + l cos πk + mα mβ kmk + m cos πlα + mβ. lml + m Multiple higher Mahler measure has applications to the computation of the higher Mahler measure. For example, Theorem 5.3 is proven as a consequence of the following result. Theorem 5.6. [LS]If P x Z[x] is reciprocal, then m P π. This inequality is sharp, with equality for x. Proof. Theorem 5.3 from Theorem 5.6 First suppose that h and P x non-reciprocal. Let d deg P, and consider P x x d P x. Thus P xp x Z[x] is reciprocal. Moreover, m P m P mp, P, thus, m P P m P + mp, P + m P 4m P. We obtain the desired bound by applying Theorem 5.6 to P P. The rest of the proof for h > is left as an exercise. Exercise 8. Complete the proof of Theorem 5.3 by proving a m h P m P h, b m h P mp h. Open Question. Is it possible to improve the bounds in Theorem 5.3? Exercise 9. With the help of a computer explore the values of the higher Mahler measure in polynomials. For example, compute m for the polynomials in Exercise 7. 6

17 Open Question. Can you find P Z[x] non-reciprocal with m P < m x 3 +x ? Notice that π Theorem 5.7. [KLO8] Proof. m x + y + x 4i π Li, i, i Li, i, i + 6i π Li,i, i Li, i, i First notice that 5. m x + y + x m x + x + i π Li,, i Li,, i 7ζ 6 + log π Lχ 4, + y + m x + y, + x + m + x. + x For the first term, we have x m + y x + x πi log + y y x + x By applying 5., the above line becomes πi x, x +x Li x dx + x x + πi + log x dx πi x, x +x + x x x Li dx πi + x x + πi x, x +x For the second term in equation 5. we obtain x m + y, + x + x πi y By Jensen s formula respect to the variable y, log + x πi + x log + x dx x πi x Then equation 5. becomes 5.3 m x + y + x For the first term in 5.3, set x e iθ, πi π π 4 π 4 πi + πi x x, x +x + ζ. x, x +x Li tan θ dθ 4 π x, x +x x, x +x dx dy x y. Li log x + x log x + y + x log + x dx x x, x +x x Li dx + x x + x dx x x dx x. dy y. log x + x log + x dx x. x, x +x log x log + x dx x x Li dx + x x π 4 π 4 7 Li tan θ + Li tan θdθ.

18 Now we make the change of variables y tan θ. 8 π 4 π dy Li y + Li y y + Li y + Li y + iy + iy dy 4 π ili,i, i + ili, i, i ili, i, i ili, i, i. For the second term in 5.3, πi x, x +x log x log + x dx x 3 4 k+l cosπkθ cosπlθdθ kl k,l 4 k+l i k+l+ k+l πkl k + l k,l k,l k,l + ik l+ k l k l i k+l i k+l π kl i k+l i k+l π k + ll + i π k>l k+l i k l k ll i π k>l k+l i k 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 5.3, we obtain the final result m x + y + x 4i π Li, i, i Li, i, i + 6i π Li, i, i + Li, i, i This result may be contrasted to Smyth s kl + i π Li,, i + Li,, i 7ζ 6 + log π Lχ 4,. m x + y + x π Lχ 4,. Exercise 3. Prove the above formula from Theorem 4.4. Naturally, we have the following generalization of the Zeta Mahler measure. Definition 5.8. Let P,..., P k C[x ±,..., x± n ] be non-zero Laurent polynomials. Their higher zeta Mahler measure is defined by Zs,..., s l ; P,..., P l P e πiθ,, e πiθr s Pl e πiθ,, e πiθr s l dθ dθ r, 8

19 Its Taylor coefficients are related to the multiple higher Mahler measure: Similarly to the case of x, one gets Setting u sin πθ, around s t. Thus, l s s l Z,..., ; P,..., P l mp,..., P l. Zs, t; x, x + Zs, t; x, x + s+t π s+t π n sin πθ s cos πθ t dθ / s+t+ sin πθ s cos πθ t dθ. u s t u du Γ s+ Γ t+ Γ s+t + Γs + Γt + Γ s + Γ t + Γ s+t + + s n + t n + s+t n + s n + t n k exp ζk { k s k + t k k s + t k} k k mx,..., x, x +,..., x + }{{}}{{} k l belongs to Q[π, ζ3, ζ5, ζ7,... ] for integers k, l. We obtain, for instance, the following equalities. mx, x + mx, x, x + mx, x +, x + log sin πθ k log cos πθ l dθ 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 π 4, ζ3 4, ζ3 4. Exercise 3. Prove the following. a For a positive constant λ, we have Zs, λp λ s Zs, P. b Let P C[x ±,..., x± n ] be a Laurent polynomial such that it takes 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 s Zk, P λ k, k k m + λp k Zk, P λ k. k k 9

20 More generally, m j + λp j! kj j Zk j, P λ kj. k... k j <k < <k j c Zs, P Z s, P P, where we put P α āαx α for P α a αx α. Note that P P is real-valued on the torus. Theorem 5.9. [KLO8] Let c. s/ Zs, x + y + c c s j j c j c s 3F s, s,, j j 4 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. m x + y + c log c + k k k c k. k m 3 x + y + c log 3 c + 3 log c k k k c k 3 k k k k c k j. j In particular, we obtain the special values k j m x + y + ζ, m 3 x + y log ζ 4 ζ3. 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. k 6. Log-sine integrals and Mahler measure We consider the works of Borwein, Straub and collaborators [BS, BBSW, BS]. Definition 6.. For n a positive integer and k a non-negative integer, the generalized log-sine integral is defined by σ Ls k n σ : θ k log n k sin θ dθ. This integral was studied by Lewin who gave several evaluations [Le8]. We use the notation Ls n σ for the case k. Some special values are expressed in terms of polylogarithms, such as π m Ls m+ π λm m! Γ + λ Γ + λ/,

21 which implies the recurrence n n! n Ls n+ π π n k ζn + + k +! +k n ζn kls k+ π, and yields, starting from Ls π, k Ls 3 π π3, Ls 4 π 3π ζ3, Ls 5 π 9 4 π5, Ls 4 π 45π ζ π3 ζ3. More generally, we have the following result from [BS]. Theorem 6.. [BS] For τ π, and nonnegative integers n, k such that n k, k ζ{} k iτ j, n k Li j! {},+k j{} n k, e iτ n k ik+ n n! j n k r r m n k, m, r m i r π r m Ls k+m n r m τ. Here are some results that were proved using properties of log-sine integrals. Let m k + x + y : m + x + y, + x + y,..., + x + y k. Then, Sasaki [Sa] proved In particular, [BS] proved m k + x + y π Ls k+ m + x + y π 54, and similar formulas por k 5, 6. Let π 3 π Ls k+ π. m 3 + x + y 9 π Im Li 4e πi/3, m 4 + x + y 6 π Im Li,4, e πi/3 π4 486, m k + x + y + z : m + x + y + z, + x + y + z,..., + x + y k + z k. Then, in [BS]: In particular, m k + x + y + z π k+ π m + x + y + z 7 π ζ3, θ log sin θ k + De iθ dθ. m + x + y + z 4 π Li,3, + 7π 36,

22 Other results from [BS] include m + x, + x + y + z π λ 4 m + x, + x, + x + y + z 4 3π λ π, 3 3 ζ π ζ5, where n λ n x : n! k k k! Li n k x log k x + n n log n x. Examples involving higher Mahler measure were computed in [BBSW] m + x + y 3 π π Ls 3 + π 3 4, m 3 + x + y? 6 π π Ls π Im Li 4e πi/3 π 4 Deπi/3 3 ζ3. Exercise 3. Find + x + y m. + x + z Open Question 3. Is it possible to express Ls π n 3 in terms of polylogarithms so that the above formulas only contain polylogarithms? Perhaps using Theorem 6.? Open Question 4. Investigate identities between log-sine integrals and polylogarithms with the help of the program developed by Borwein and Straub in [BS]. References [A9] H. Akatsuka, Zeta Mahler measures. J. Number Theory 9 9, no., [BV87] E. Bombieri, J.D. Vaaler, Polynomials with low height and prescribed vanishing, Analytic Number Theory and Diophantine Problems, Progress in Mathematics, vol 7, Birkhauser, 987, [BDM7] P. Borwein, E. Dobrowolski, M. J. Mossinghoff, Lehmer s problem for polynomials with odd coefficients. Ann. of Math. 66 7, no., [BBSW] D. Borwein, J. Borwein, A. Straub, J. Wan, Log-sine evaluations of Mahler measures, II. Integers, no. 6, 79. [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., [Bo8] D. W. Boyd, Speculations concerning the range of Mahler s measure. Canad. Math. Bull. 4 98, no. 4, [Bo98] D. W. Boyd, Mahler s measure and special values of L-functions, Experiment. Math , [Bo] D. W. Boyd, Mahler s measure and invariants of hyperbolic manifolds, in Number Theory for the Millennium, I Urbana, IL,, A K Peters, Natick, MA,, [BRV] D. W. Boyd, F. Rodriguez Villegas, Mahler s measure and the dilogarithm I, Canad. J. Math. 54, [BRV3] D. W. Boyd, F. Rodriguez-Villegas, with an appendix by N. M. Dunfield, Mahler s measure and the dilogarithm II. Preprint, July 3. [Br5] R. Breusch, On the distribution of the roots of a polynomial with integral coefficients. Proc. Amer. Math. Soc. 95, [Do79] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith , [EW99] G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Universitex UTX, Springer-Verlag London 999. [Kr57] L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, Crelle, Ouvres I [KLO8] N. Kurokawa, M. Lalín and H. Ochiai, Higher Mahler measure and zeta functions, Acta Arith. 35 8, no. 3, [La3] M. N. Lalín, Some examples of Mahler measures as multiple polylogarithms, J. Number Theory [La6] M. N. Lalín, Mahler measure of some n-variable polynomial families, J. Number Theory [La7] M. N. Lalín, An algebraic integration for Mahler measure, Duke Math. J. 38 7, no. 3, 39 4.

23 [LS] M. Lalín, K. Sinha; Higher Mahler measure for cyclotomic polynomials and Lehmer s question. Ramanujan J. 6, no., [La83] W. M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 6 983, no. 3, [Le33] D. H. Lehmer, Factorization of certain cyclotomic functions, Annals of Math. vol [Le8] L. Lewin, Polylogarithms and associated functions.north-holland Publishing Co., New York-Amsterdam, 98 xvii+359 pp. [LSW9] D. Lind, K. Schmidt, T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math. 99, no. 3, [Ma6] K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc [Pi7] T. Pierce, The numerical factors of the arithmetic funtions n i ± α i, Ann. of Math [RV97] F. Rodriguez Villegas, Modular Mahler measures I, Topics in number theory University Park, PA 997, 7-48, Math. Appl., 467, Kluwer Acad. Publ. Dordrecht 999. [RZ] M. Rogers, W. Zudilin, On the Mahler measures of + X + /X + Y + /Y. Preprint, March. To appear in Int. Math. Res. Not. IMRN [Sa] Y. Sasaki, On multiple higher Mahler measures and multiple L values, Acta Arith. 44, no., [Sm7] C. J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. Lond. Math. Soc. 3 97, [Sm8] 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, [Z9] D. Zagier, Polylogarithms, Dedekind Zeta functions, and the Algebraic K-theory of Fields, Arithmetic algebraic geometry Texel, 989, 39 43, Progr. Math., 89, Birkhauser Boston, Boston, MA, 99. [M] M. J. Mossinghoff, Lehmer s Problem web page. ~mjm/lehmer/lc.html 3

24 Specify that the polynomials are over C: sage: complexpoly.<x>polynomialringcc sage: load "mahler.sage" 7. Some simple SAGE routines 7.. Mahler measure for one-variable polynomials. def mahlerp: m for rm in p.roots: rtrm[] mulrm[] if absrt>: mm+logabsrt*mul returnm 7.. Higher Mahler measure of order for one-variable polynomials. def mahlerp: m for rm in p.roots: rtrm[] mulrm[] for rm in p.roots: rtrm[] mulrm[] if absrt< and absrt<: mm+realdilogrt*conjugatert/*mul*mul if absrt< and absrt>: mm+realdilogrt/conjugatert/*mul*mul if absrt> and absrt<: mm+realdilogrt/conjugatert/*mul*mul if absrt> and absrt>: mm+realdilog/rt/conjugatert/+logabsrt*logabsrt*mul*mul returnm 7.3. General higher Mahler measure for one-variable polynomials. This sometimes works. sage: xvar x sage: logabs-expi*x//pi.nintegralx,,*pi 7.4. n. def deltan,p: pr for r in p.roots: rtr[] mulr[] prpr*rt^n-^mul return introundreal_partpr 7.5. Log-sine integrals. The package logsine.sage is available at 4

25 Suggestion. By 8. Suggestions for the exercises a rs a r a s r + a s r + + a r +, we have that a is a nontrivial divisor of a p unless a, in which case a, or a p a, in which case p. Suggestion. By Fermat s little theorem, q mod q. We have, by hypothesis, p mod q. Since p is prime, we must have that p is the order of modulo q. Therefore, p q. Since q is also odd, we can write q pk +. Suggestion 3. If n divides m, then m kn for some k and kn P n P d i + α n i + + α k n i is a symmetric function of the roots of P, and so is an integer. Suggestion 4. Fix ξ n an nth root of unity. If the roots of P x are α i, then the roots of P x n are α /n i ξn k for k,..., n and α /n i an nth root of α i. In addition, α /n i ξn k α i /n, so α /n i ξn k and α i are both either > or at the same time. We have MP x a max{, α i } a max{, α i /n } n a max{, α i ξn k /n } MP x n. i i i Suggestion 5. If P x a d x d + + a a d i x α k, then a i /a d d i s d i α k where s j α k α k α kj k < <k j are the elementary symmetric polynomials. Observe that k α k α kj k max{, α k }. It is then clear that a i a d s d i α k d MP d i d MP. i Suggestion 6. a The polynomial 5x 6x + 5 has 3+4i 5 as a root. b If a degree three integer polynomial has roots α, α, α 3 with α and α α, then α α α 3 α α α 3 α 3 Z. Thus α is a root of a monic integer quadratic which is impossible unless α is a root of unity. For degree 4, an example is given by whose root x 4 + 4x 3 x + 4x + + i has absolute value. Suggestion 7. These values can be computed with mahler.sage a b c d Suggestion 8. Some of these values can be computed with delta.sage. However, for larger values it is better to use a recurrence of the n as proved in [Le33], section 8. Suggestion 9. We have /d y yd y y d 5 y y + + y d y d d

26 and y y d /d y + + y d. d Summing both inequalities and multiplying by y y d /d, y y d /d + y y d /d. Suggestion. We see from Equation. and Exercise 9 that the condition for equality is MP + d/ 5 that occurs when all the αi with α i > are equal and all the α i with α i < are equal. That means that P x x αx β d/ + with α >, β <, and MP d/. 5 Thus, α + 5, and we must have β 5 and P x x d/. Suggestion. If we relax the condition P, say that P c. Then MP c α i < αi and E c MP /d MP /d d. Thus, the conclusion is Thus, MP c ± d/. c 4 +4 Suggestion. Using exercise 5, we have a LP i MP /d MP /d c. a i i d MP d MP, i b d HP max a i max MP d MP. i i i In the last step, we have used d i d for d which can be proven by induction. Suggestion 3. a Write Thus, and By Exercise 5, we get a k,...,k n MP k,...,k n P k,...,k m x m+,..., x n i j P x,..., x n P k,...,k m x m,..., x n dm k n MP k,...,k n d k d m k m a k,...,k m,i m+,...,i n x im+ m+ xin n P k x,..., x n x k P k,...,k m x m+,..., x n x km. dm dm k n k n MP k,...,k n d k dm k n MP. b The upper bound is proved in a similar way as Exercise a. For the lower bound, just notice that P e πiθ,..., e πiθn LP. c The upper bound is proved in a similar way as Exercise b. Suggestion 4. Write P x, x a + Qx,x a. Then mp log a + m + Qx,x a. Since Qx,x a <, log + Qx,x a can be expanded as a Taylor series uniformly convergent in x, x. The integral of this is zero, since the individual terms vanish. Taking the real part of the logarithm, m + Qx,x a. Suggestion 5. Using Euclidean algortithm, A can be written as a product of integer matrices which are of the following three types: upper triangular with s in the diagonal, lower triangular with s in the 6

27 diagonal or 3 diagonal. It is easy to see that the Mahler measure is invariant by types and and for the type 3, we have, by setting k i θ i τ i, k kn... log P e πikθ,..., e πiknθn dθ... dθ n... log P e πiτ,..., e πiτn dτ... dτ n k k n Suggestion 6. This can be achieved by, for instance, sage: for n in range,:...: mahlerx^n+x+... Suggestion 7. By Jensen s formula, it is enough to evaluate I 4π π π log max { + e πiθ, + e πiθ } dθ dθ π π π π π log π + e it log P e πiτ,..., e πiτn dτ... dτ n. log max { + e it, + e it } dt dt log max { + e it, + e it } dt dt t dt dt π π log + e it π t dt π π log + e it t dt. In t < π, we have Suggestion 8. a We have π n e itn I Re π t dt n n n π Re π te itn dt n n n n π π Re + n n in n n n π n 3 n n 3 n π n 3 + n n 3 n 4 π ζ3 4 π 7 π ζ3. Li n + Li n k m m 3 + k k n 7 k k n n Li n

28 Therefore, b In this case we have Suggestion. We have Li n n Li n n ζn. Li n e πi/3 Li n e πi/3 e kπi/3 e kπi/3 k k 3 k n sin kπ 3 k k n χ 3 k k n 3Lχ 3, n. ζn,..., n m m I n,...,n m : :. Making the change of variable t t for each variable in the integral, we obtain m dt t dt t dt dt t t dt t dt t }{{}}{{} n n m m+ n k dt t dt t dt dt t t dt t dt t }{{}}{{} n m n m+ n k I,...,,,...,,..., }{{}}{{} nm n ζ,...,,,...,,...,,. }{{}}{{} n m n, : : Suggestion. First notice that this Mahler measure depends only on a since y and z can absorb number of absolute value in the integration. For a < this proof is done exactly in the same way as Exercise 7. For a >, we have, + x m + x + ay + az log a + m + y + z log a + m + x + y + z a a since we can exchange the variables x, y, z and the constant term in the integral by symmetry. Then apply the formula for a <. Suggestion 5. Observe that j!k j! j dt t x k j dt t x dt t x dt t x }{{ } j Combining the previous equalities gives m k x log k x dx πi x x k! k dt k πi j x t x dt t x }{{ } j 8 dt t x dt. t x }{{} k j dt t x dt t x }{{} k j dx x.

29 If we now set s xt in the first j-fold integral and s t x becomes k! k x ds k πi x s ds x s j We proceed to compute the integrals in terms of multiple polylogarithms: m k x k k! k k πi x k k! k j k Now we need to analyze each term of the form 8. in the second k j-fold integral, the above ds s ds dx s x. <l < <l j< <m < <m k j < j <l < <l j <u<, <m < <m k j <u< <l < <l j <u<, <m < <m k j <u< x lj m k j l... l j m... m k j u. For an h-tuple a,..., a h such that a + + a h k h, we set a ah a + + a h d a,...,a h... e e h e + + e h Then the term 8. is equal to e + +e h j h min{j,k j } h d a,...,a h ζ{} a,,..., {} ah,. dx l... l j m... m k j x l... l j m... m k j u. k h. j h Note that each term ζ{} a,,..., {} ah, comes from choosing h of the l s and h of the m s and making 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 where On the other hand, Thus, the total sum is Suggestion 6. being k m k x c a,...,a h ζ{} a,,..., {} ah,, c a,...,a h k k! k h k j k h k k! j h k k h k k! h. ζ{} a,,..., {} ah, ζa h +,..., a +. m k x b + +b h k, b i k k! h ζb,..., b h. First assume that α, β. Then the proof is similar to Theorem 5.5, the final step m αx, βx where e πiτ β α β α. 9 k cosπkτ αβ k k,

30 If α > and β, we have that m βx and m αx, βx mα, βx + mα x, βx log α m βx + m α x, βx m α x, βx, and we are in the first case. For α > and β >, the previous identity becomes m αx, βx log α log β + m α x, βx and we apply the formula for the case β > and α <. Suggestion 7. Following a similar proof to Theorem 5.5, it is not hard to see that we are reduced to compute But this equals I 4 I cosπkθ cosπlθ + α cosπmθ + βdθ. cosπkθcosπl + mθ + lα + mβ + cosπl mθ + lα mβdθ cosπlα + mβ if l + m k, 4 cosπlα mβ if l m k, otherwise. Suggestion 8. a For any positive integer h, let f and g be functions such that f h dx πi x < and g h/h dx πi x <. x Then, by Hölder s inequality, we get that 8. πi x h fg dx x πi x x f h dx x πi In particular, taking fx log P x and gx, we get that m P h m h P. x g h/h dx x h. b On the other hand, by taking fx log P x and gx, and taking h instead of h in 8. we get that mp h m h P. Suggestion 9. These values can by computed with mahler.sage a b c d

31 Suggestion 3. We set a in Theorem 4.4: m + x + xy i π Li i Li i i i n π n in n n i k π k + ik k k k π k π k n χ 4 n n, where if n mod 4, χ 4 n if n mod 4, if n mod. Suggestion 3. a and c are obvious. For b, we may use the Taylor expansions in λ; s + λp s λ k P k k, log + λp λ k P k. k k In particular, we may write Zs, + λp k k m k + λp sk k! k ss s k + Zk, P λ. k! k In other words, the coefficients with respect to the monomial basis are the k-logarithmic Mahler measures m k + λp, while the coefficients with respect to the shifted monomial basis are the special values of zeta Mahler measures Zk, P λ k. Combining these observations, we obtain the three equalities. Suggestion 3. We have m + x + y + x + z k m + x + y m + x + y, + x + z + m + x + z 3 π π Ls 3 + π π π Ls 3 π 3 + 5π 54. Département de mathématiques et de statistique, Université de Montréal. CP 68, succ. Centre-ville. Montreal, QC H3C 3J7, Canada address: mlalin@dms.umontreal.ca, 3