Estimating Mahler measures using periodic points for the doubling map

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Estimating Mahler measures using periodic points for the doubling map M. Pollicott ∗ , P. Felton

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Department of Mathematics, Warwick University, Coventry, CV4 7AL, UK

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Abstract

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The Mahler measure m(P) of a polynomial P is a numerical value which is useful in number theory, dynamical systems and geometry. In this article we show how this can be written in terms of periodic points for the doubling map on the unit interval. This leads to an interesting algorithm for approximating m(P) which we illustrate with several examples. c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). ⃝

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1. Introduction The Mahler measure m(P), or height, of a polynomial P(z) ∈ C[z] is a numerical value which plays an important role in number theory, dynamical systems and geometry [10]. It is named after the mathematician Kurt Mahler, by whom it was introduced to provide a simple proof of Gelfond’s inequality for the product of polynomials in many variables. However, it has been used with great success in a variety of settings. For example, it was also used previously in Lehmer’s investigation of certain cyclotomic functions and led him to ask his famous question (now known as Lehmer’s conjecture) about which polynomial has the least Mahler measure. It is conjectured to be x 10 +x 9 −x 7 −x 6 −x 5 −x 4 +x 3 +x +1 [8], but in spite of considerable work by many authors on this conjecture, it has not yet been proved. In another direction, connections have been found between the Mahler measure of certain two variable polynomials and invariants of hyperbolic 3manifolds such as the volume, via the dilogarithm and work of Milnor, Zagier and others [11,16]. Last, but not least, the Mahler measure of an integer polynomial in k variables gives the ∗ Corresponding author.

E-mail address: [email protected] (M. Pollicott). http://dx.doi.org/10.1016/j.indag.2014.04.002 c 2014 Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG). 0019-3577/⃝

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topological entropy of certain Zk -dynamical system (for k ≥ 2) canonically associated to the polynomial in the work of Lind, Schmidt and Ward [9]. These can be viewed as generalizations of Yuzvinskii’s formula for automorphisms of solenoid’s in the case k = 1. However, unlike the case of Z-actions these entropies need not be the logarithms of algebraic integers. There exist more classical approaches to estimating the Mahler measure, including relating it to L-functions, as in the work of Smyth [14]. However, in this note we want to give a new explicit formula for the Mahler measure. This is in terms of a rapidly convergent series, the terms of which are explicitly given in terms of the values of the polynomial at a finite set of points. However, perhaps the most satisfying aspect of this approach is that we use periodic points of a simple dynamical system (the doubling map on the interval) to describe the series, particularly given that one of the recent applications described above is to computing the entropy of Z2 -dynamical systems.

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Theorem 1.1. We can write the logarithmic Mahler measure log m(P) as an infinite series

1 2 3 4 5 6 7 8 9 10 11

14

log m(P) =

∞ 

an

n=1 15 16 17 18

where 1. the terms an are explicitly given in terms of the values of the polynomial P at the points { 2nk−1 : k = 0, 1, . . . , 2n − 1}; and 2 2. there exist 0 < θ < 1 and C > 0 such that |an | ≤ Cθ n .

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The use of infinite series as explicit expressions for the Mahler measure of polynomials is not new, occurring as it does (for example) in the work of Smyth and others. In that case, it is related to expressions in terms of L-functions, i.e., generalizations of zeta functions in number theory. Our approach is based on another class of complex functions d(z, s), which are closely related to dynamical zeta functions.

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1.1. Overview of the proof of Theorem 1.1

19 20 21 22

25 26 27 28 29 30 31

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One can write log m(P) in terms of the integral of log |P(·)| around the unit circle. In the special case that the polynomial P(z) is non-zero on the unit circle then the function θ → log |P(eiθ )| is real analytic on the interval [0, 2π ]. In particular, Theorem 1.1 follows by a direct application of the method in [6] (recalled in Section 4). In the general case we simply need to replace the integral in the expression for m(P) by sum of two integrals, i.e., writing  P(z) = P0 (z)P1 (z) where P0 (z) has no zeros on the unit circle and P1 (z) = mj=1 |z − e2πiθ j |, where e2πiθ1 , . . . , e2πiθm are the zeros of P(z) on the unit circle, we can then trivially write  2π  2π  2π 2πiθ 2πiθ log |P(e )|dθ = log |P0 (e )|dθ + log |P1 (e2πiθ )|dθ. (1.1) 0

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The method in [6] now applies to estimate the first integral makes no contribution, i.e.,  2π  2π log |P1 (eiθ )|dθ = m log |1 − eiθ |dθ = 0 0

36 37

0

0

 2π 0

P0 (e2πiθ )dθ . The second integral

0

(cf. [4, pp. 8–9]). In particular, we obtain the series expansion for log m(P) by carrying out the expansion for the first term on the right hand side of (1.1).

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1.2. Plan of the article

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In Section 2 we recall the definition of the Mahler measure and some examples. We also give the definition of the Mahler measure for polynomials of two variables, the general case being similar, and give more examples. Although Theorem 1.1 is formulated for polynomials of one variable, a variant of our approach works for many examples in several variables. In Section 3 we illustrate this principle via some specific examples. As explained above, our approach to proving Theorem 1.1 is based on the dynamical method of evaluating integrals of analytic functions developed in [7] (cf. also [6]). We describe the basic ideas in this approach in Section 4. Since the series converges quite rapidly it makes accurate numerical evaluation quite easy, as we will illustrate in Section 5 with explicit examples. The speed of the convergence is faster than in several other more familiar approaches. For example, if one uses the Ulam method of approximating the integrals then one would expect that the error in approximating the Mahler measure using the same data to be at most polynomial. However, if one used the well known equidistribution results of periodic points of period n then one would still only expect an exponential error term. In Section 6 we conclude with a discussion of some related topics. 2. Definition and examples

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We now proceed to describing in detail the Mahler measure. We first define the Mahler measure for a polynomial in one variable, and then extend this to the case of polynomials of several variables. 2.1. Polynomials in one variable

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We begin with the definition of the Mahler measure for a complex valued polynomial P(z). This is defined using the zeros and the leading coefficient of the polynomial. Definition 2.1. If P(z) = a(z − α1 ) · · · (z − αn ) then  m(P) = |a| |αi |

2

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25

(2.1)

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|αi |≥1

where a ∈ R and the product is over those zeros αi ∈ C of P(z) of modulus at least 1. From the definition (2.1) it is easy to see that a product of cyclotomic polynomials has Mahler measure precisely 1, and that for the linear polynomial P(z) = az + b the Mahler measure is m(az + b) = max{|a|, |b|}. There is another equivalent definition which is useful in making explicit numerical estimates. Using Jensen’s theorem from complex analysis we can rewrite the definition in the following form incorporating a definite integral:    2π 1 iθ log |P(e )|dθ . (2.2) m(P) = exp 2π 0 The value of the Mahler measure can be explicitly computed using (2.2) in other simple examples.

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Example 2.2. For the famous example of a polynomial featuring in Lehmer’s conjecture

1

P(x) = x 10 + x 9 − x 7 − x 6 − x 5 − x 4 + x 3 + x + 1

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3

Q3

4

Q4

5

6

7 8

9

10

11 12 13

14

15

16

we can compute log m(P) = 1.176280818 . . . , which is conjectured to be the smallest value for non-cyclotomic polynomials [15]. 2.2. Polynomials in two variables One can similarly define the Mahler measure for polynomials of more than one variable. Definition 2.3. For a polynomial of two variables P(z 1 , z 2 ) we can define the Mahler measure by    2π  2π 1 iθ1 iθ2 log |P(e , e )|dθ1 dθ2 . m(P(z 1 , z 2 )) = exp (2π )2 0 0 The definition for n-variables is similar. There are explicit series for the Mahler measures of particular polynomials, originally due to Smyth in 1981, which allow the Mahler measure to be written in terms of zeta functions and L-functions. We recall a couple of examples from [14]. Example 2.4. When P(z 1 , z 2 ) = 1 + z 1 + z 2 we can write √ 3 3 log m(1 + z 1 + z 2 ) = L(2, χ3 ) 4π ∞ where L(s, f ) = n=1 f (n)n −s is the L-function and thus L(2, χ3 ) = 1 −

17

18

1 1 1 + − + ··· 4 16 25

where  if n = 1 mod 3 1 χ3 (n) = −1 if n = 2 mod 3  0 if n = 0 mod 3.

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The characterization of Mahler measure in terms of zeta-functions and L-functions extends to other examples.

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Example 2.5. Following Smyth we can also write

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log m(1 + z 1 + z 2 + z 3 ) =

23

24

25

26

27 28

7 ζ (3) 2π 2

and log m(1 + z 1−1 + z 2 + (1 + z 1 + z 2 )z 3 ) =

14 ζ (3) 3π 2

where ζ (s) is the Riemann zeta function. These identities come from integrating series expansions term by term. However, the very slow convergence of these series makes then difficult to compute numerically.

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3. Modifying the integrals

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We would like to rewrite the integrals giving the Mahler measures in specific cases so that they are integrals of analytic functions defined in a neighbourhood of bounded intervals in the real line. This is an essential requirement of our approach. 3.1. Rewriting the integrals

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Traditionally, the formulae for the logarithmic Mahler measure come from expanding the integrands as a power series prior to integration. In particular, one can take the Taylor series of sin(·) and cos(·). However we instead want to rewrite the definition in terms of an integral of an analytic function, perhaps over a different interval. This is illustrated by the following simple example. Lemma 3.1. We can write

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log m(1 + z 1 + z 2 ) =

5/6



log(2 sin(π y))dy.

11

1/6

Proof. Using Jenson’s formula  1 log |α + e2πt |dt = log(max{|α|, 1})

12

13

0

we can write

14

log m(1 + z 1 + z 2 ) =



2π

0

 =

2π



log |1 + eiθ1 + eiθ2 |dθ1 dθ2

15

0 2π

log(max{|1 + eiθ1 |, 1})dθ1

16

0



2π/3

=

log |1 + eiθ1 |dθ1

17

−2π/3

since |1 + eiθ1 | ≤ 1 for θ1 ∈ [2π/3, 4π/3].



It is at this stage that the traditional approach would be to expand   ∞  iθ n−1 inθ log |1 + e | = Re (−1) /ne

18 19

20

n=1

and integrate term by term. The resulting series can then be interpreted as a suitable zeta function or L-function, but would likely have slow convergence. However, since sin(π y) is nonzero and analytic on [1/6, 5/6] we see that log(2 sin(π y)) is analytic in a neighbourhood U of the interval [1/6, 5/6] which is a necessary ingredient in our approach. Thus, as we will see later, we can express the integrand instead in terms of a summation involving periodic points for the doubling map. 3.2. Removing the singularity from the integrand We now recall a second useful approach for replacing the integrand by an analytic function. In Section 1.1 we described the reduction for polynomials of one variable. In some cases with more variables the integral can be rewritten so that the singularity in the integrand can be removed

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and dealt with as a separate term, rather than eliminated completely. This is illustrated in the following example.

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Lemma 3.2. We can write

1

m(1 + z 1 + z 2 + z 3 ) =

4

5

Q5

6

7

2 π2

9

=

10

=

11

=

12

=

13

=

15

16 17

18

19 20

23

24 25

26

θ log(2 sin(θ/2))dθ.

0

+ eiθ3 (1 + ei(θ2 −θ3 ) )|dθ1 dθ2 dθ3  π π 1 log(max{|1 + eiu |, |1 + eit |})dudt π2 0 0  π π 2 log |1 + eit |dudt π2 0 t  π 2 (π − t) log |1 + eit |dt π2 t  π 2 − 2 t log |1 + eit |dt π 0    π 2 θ θ log 2 sin dθ.  2 π2 0

However, we see that we can write         2 θ θ = θ log sin + log (θ ) θ log 2 sin 2 θ 2

14

21

π

Proof. Using Jensen’s formula again we can write  2π  2π  2π 1 log |1 + eiθ1 + eiθ2 + eiθ3 |dθ1 dθ2 dθ3 log m(1 + z 1 + z 2 + z 3 ) = 2 π 0 0 0  2π  2π  2π 1 = 2 log |(1 + eiθ1 ) π 0 0 0

8

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

Q6

and the first term on the right hand side is analytic since we can expand it as a series about θ = 0 to write:    n ∞  2 θ (−1)n θ sin =1+ . θ 2 (n + 1)! 2 n=1   In particular, the integrand F : [0, π] → R defined by F(θ ) = θ log θ2 sin θ2 is analytic in a neighbourhood on [0, 1] allowing it to be efficiently integrated. On the other hand we can write π 1 2 θ log θ dθ = 0 4 π (2 log π − 1). Thus the value of the Mahler measure can be written as the sum of an explicit special value (coming from the expression involving the singularity) and an explicit series (for F) which we can effectively evaluate. Example 3.3. In the book of Everest and Ward [4, p. 55] there is a specific integral which comes from computation of the Mahler measure m(1 + x1 + x2 ):  1/3 2 log |1 + e2πiθ |dθ. 0

M. Pollicott, P. Felton / Indagationes Mathematicae xx (xxxx) xxx–xxx

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This has an integrand which is a function analytic on the region of integration and the integral reduces to  1/3 2 log(1 + cos(2π θ ))dθ + log 2.

1 2

3

0

After a change of variables this becomes

1 0

log(1 + cos(2π θ/3))dθ .

4. Linear operators and determinants In this section we recall the method used to compute the integrals of analytic functions. In particular, this leads to the series expansion in Theorem 1.1. We refer the reader to [7,6] for more details of the following standard constructions. Consider the contractions on the unit interval T0 , T1 : [0, 1] → [0.1] defined by x +1 x and T1 (x) = . 2 2 There is no particular reason to take these contractions, except for the simplicity. We could as easily take b-branches for any b ≥ 2 and consider the base b-expansion. (We might even take non-linear contractions, but since this complicates computation and has no clear advantage then we prefer not.) The fixed points of Ti1 ◦ · · · ◦ Tin are of the form   k where 0 ≤ k ≤ 2n − 2 (2n − 1) T0 (x) =

where {·} denoted the fractional part (i.e., the periodic points for the doubling map). In principle we could use any number of contractions and consider the corresponding expansion to that base. Let g : [0, 1] → R be any analytic function (i.e., one which extends analytically to a neighbourhood U of [0, 1]). We can define expressions for s ∈ C, n ≥ 1 by  n −2  2 n−1   2m k Z (s, n) := exp s g . (2n − 1) k=0 m=0 We can then define

4

5

6 7 8 9

10

11 12 13 14 15

16

17 18 19 20 21

22

23

 d(z, s) = exp −

∞ 



zn

n=1

Z (s, n) . 2n − 1

Finally, we can expand this as a power series in z using the standard expansion exp (−w) = 1 +

∞ 

(−1)n

n=1

wn n!

and collect together the terms. Example 4.1. We can write   Z (s, 2) Z (s, 3) − z3 + ··· d(z, s) = exp −z Z (s, 1) − z 2 3 7

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  Z (s, 2) Z (s, 3) = 1 − −z Z (s, 1) − z 2 − z3 + ··· 3 7  2 Z (s, 2) Z (s, 3) 1 −z Z (s, 1) − z 2 − z3 + ··· + ··· + 2 3 7   1 2 2 = 1 − z Z (s, 1) +z Z (s, 2) + Z (s, 1) + · · · .    2    a1 (s)

1

2

3

=a2 (s)

4 5

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Theorem 4.2. The function is analytic for all z, s ∈ C. In particular, there exists 0 < θ < 1 such that we can write N  2  d(z, s) = 1 + an (s)z n + O θ N . n=1

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10

11

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Proof. This follows from the results in [6]. It is a consequence of the work of Ruelle and Grothendieck on determinants.  We can also identify the integral of g using the following. Lemma 4.3. We have that   −1   ∂d(z, 0)  ∂d(1, s)  . g(x)d x = ∂z z=1 ∂s s=0 Proof. Let z(s) =: e P(s) be the implicit solution to d(z(s), s) = 0. We recall that P ′ (0) =  g(x)d x [6] and use the implicit function theorem.  This allows us to get an explicit solution for the integral and a method of approximation. More precisely, we can write ∞ 

 16

g(x)d x =

nan (0)

n=1 ∞ 

an′ (0)

n=1 N  17

=

nan (0)

n=1 N 

 2 + O θN

(4.1)

an′ (0)

n=1 18 19 20 21 22 23

for N ≥ 1. Finally, as explained in Section 1.1 we can write P(z) = P0 (z)P1 (z) where P0 (z) has no zeros on the unit circle and then let g(x) = log |P0 (x)|. Thus (4.1) gives an expansion for the first term on the right hand side of (1.1). The second term on the right hand side of (1.1) is zero. This completes the proof of Theorem 1.1. We can adapt the previous methods in this note to generalize the theorem in the introduction to the following:

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M. Pollicott, P. Felton / Indagationes Mathematicae xx (xxxx) xxx–xxx Table 1 Estimates of the Mahler measure in Example 5.1. Number of periodic points

Estimates of the integral in Example 5.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14

−0.73549747504528931000667846082397379751639080360 −1.84202654991039430194174776526265912284109415264 −0.41020092553869771648063831402510910580500970512 0.35078322259934296680006253145588524287430068804 0.24116255252055803299059913083170710785113957869 0.24542857346024114670842280178219590825871238613 0.24534857031232659117819606317355596494688961054 0.24534930689656192540530394777826583002603529892 0.24534929854268029753863865962331685140064221906 0.24534929849516570221896433191494480661519380576 0.24534929849570290031194886408119307347555279953 0.24534929849570876956895611422129258292207677225 0.24534929849570876971918507874358597891822930992 0.24534929849570876968022777221437569625646981881

Theorem 4.4. Let P be a polynomial in d ≥ 2 variables. We can write the logarithmic Mahler measure m(P) as an infinite series log m(P) =

∞ 

1 2

an

3

n=1

where 1. (an ) is explicitly given in terms of the values of the polynomial P at the points    k1 kd n ,..., n : 0 ≤ k1 , . . . , kd ≤ 2 − 1 . 2n − 1 2 −1 2. There exist 0 < θ < 1 and C > 0 such that |an | ≤ Cθ n

  1+ d1

4 5

6

.

The proof of this theorem is based on periodic points for the expanding map T : the d-dimensional torus given by T (x1 , . . . , xd ) = (nx1 , . . . , nxd ).

7

Td

→

Td

on

8 9

5. Numerical evaluation

10

Whereas the interest in the formulae for Mahler measures we present is intrinsic, it is interesting to use these series to estimate the Mahler measure in the examples we have presented. This is achieved by truncating the series at the nth term, say,

11 12

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Example 5.1 (Lemma 3.1 Revisited). We can evaluate  5/6 log m(1 + z 1 + z 2 ) = log(2 sin(π y))dy

13

14

15

1/6

by first changing variables by x = 32 (y − 16 ) to get     2 1 2π 1 log 2 sin x+ d x. 3 0 3 6 This can then be evaluated using the periodic point expansion and (4.1). This is illustrated in Table 1.

16

17

18

Q8 Q9

19

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M. Pollicott, P. Felton / Indagationes Mathematicae xx (xxxx) xxx–xxx Table 2 Estimates of the integral in Example 5.2.

1

2

3

4

Number of periodic points

Estimates of integral in Example 5.2

1 2 3 4 5 6 7 8 9 10

0.0 0.0710058106659420626139608915475710754171537460 −0.0674079942679211711371277070046100911840936746 −0.1137418084423140487912693319297018815707912345 −0.1091292853392956782029628035362785456971448144 −0.1092261716068447267310745709643933696618902089 −0.1092257521118177154674761773445224803026556852 −0.1092257434238678523468850000209060091707199826 −0.1092257435162616483165814424580769513185926387 −0.1092257435159467354031492882515473442102548640

Example 5.2 (Lemma 3.2 Revisited). We can evaluate     π 2 2 θ θ log sin dθ 2 θ 2 π 0 by first changing variables by x = θ/π to get    1  1   1  π x  π2x3 π4x5 2 sin x log xd x +2 x − − − · · · d x. x log =3 2 πx 2 24 2880 0 0  0   =−1/4

In particular, the first integral in the final term can be explicitly evaluated. The second integral in the final term can be evaluated using the periodic point expansion and Eq. (4.1). This is illustrated 7 Q10 in Table 2. 5 6

8

6. Generalizations and related values

10

The evaluation of similar integrals plays an important role in a number of other settings. In this section we propose a few.

11

6.1. Generalizations

9

12 13

14

15

Borwein and Straub [1] considered multiple (logarithmic) Mahler measures based on several polynomials, of the following form. Definition 6.1. We define multiple (logarithmic) Mahler measures  1  1 k µ(P1 , . . . , Pk ) = ··· log |P j (e2πiθ1 ) · · · P j (e2πiθn )|dt1 · · · dtn . 0

16 17 18

19

20

0

j=1

In the particular case P = P1 = · · · Pk we denote this as µk (P). When k = 1 this reduces to the original Mahler measure µ1 (P). Central to the evaluation of such values are the log – sin integrals defined by    σ  θ Lsn (σ ) := − logn−1 2 sin  dθ 2 0 for n ≥ 1.

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Example 6.2. In particular, Sasaki [12] showed that µ(1 + x + y1 , 1 + x + y2 , . . . , 1 + x + yk ) =

1



5/6

logk |1 − 22πit |dt

2

1/6

=

1 (Lsk+1 (π/3) − Lsk+1 (π )) . π

6.2. Related integrals: Catalan constants and Clausen’s integrals The problem of evaluating Mahler measures has parallels to evaluating other explicit integrals. A particularly interesting class of such integrals are the following:  2π  2π 1 cn := ··· (2 − 2 cos(θ1 )) · · · (2 − 2 cos(θn ))dθ1 · · · dθn (2π )n 0 0 called Catalan constants. These particular integrals occur as limits of determinants of a discrete Laplacian associated to a graph called a discrete torus, given with a suitable normalization, to the product of the non-zero eigenvalues of the Laplacian It is also related to Kirchhoff’s 1847 paper on spanning trees. Example 6.3 (n = 1). The constant is named in honour of E. C. Catalan (1814–1894), who first gave an equivalent series and expressions in terms of integrals. Based on methods developed in collaboration with M. Leclert, Catalan (1865) computed to 9 decimals. In 1867, M. Bresse subsequently computed to 24 decimal places using a technique from Kummer. Glaisher evaluated to 20 (Glaisher 1877) and subsequently 32 decimal digits (Glaisher 1913). Catalan’s constant may be computed to 108 digits in approximately 1 CPU-hour using modern hardware and a developmental version of Mathematica, and is now known to 3.1010 decimal places. From our point of view, the most useful form of the constant is as an integral  π/2 log(2 sin(t/2))dt = 0.915965 . . . . c1 : −

3

4

5 6

7

8 9 10 11

12 13

Q11 14 15 16 17 18 19

20

0

This is a special value of the Clausen function.  θ C(θ ) = − log(2 sin(t/2))dt 0  2      x  π x2 =i − + x log(1 − ei x ) − log 2 − iLi2 (ei x ) − x log sin . 6 4 2 We can change variables by t = π2 x then we can write  π/2 c1 = − log (sin(t/2)) dt − log(π/4) log 2

21

22

23

24

25

0

=−

π 2



1

log (sin(π x/4)) d x − log(π/4) log 2.

26

0

We can expand sin(π x/4) = (π x/4) −

27

(π x/4)3 3!

+

(π x/4)5 5!

−

(π x/4)7 7!

+ ···

28

12

M. Pollicott, P. Felton / Indagationes Mathematicae xx (xxxx) xxx–xxx Table 3 Estimates of the integral in Example 6.3. Number of periodic points

Estimates of integral

2 3 4 5 6 7 8 9 10

0.028783481132522005195827640528436831592870971980761 −0.017597596938717325443892198755114160250026672339315 −0.037035295847928556939358308020139133203154818774092 −0.034628688321030893474631521272671044247185943206331 −0.034705707340429574152204007920956903583981142752151 −0.034704504761600623768703825770410214250300363705901 −0.034704513378801624771335583022987871957717300165660 −0.034704513351055271758860768558861591782522325683191 −0.034704513351092439495683762246807986869029078814615

 (π x/4)4 (π x/4)6 (π x/4)2 + − + ··· = (π x/4) 1 − 3! 5! 7! =: (π x/4) f (x) 

1

2

3

4

5

6

where f (x) = 1 −

(π x/4)2 (π x/4)4 (π x/4)6 + − + ··· 3! 5! 7!

which is analytic and non-zero a neighbourhood of the interval [0, 1]. We can then write   c1 =

π 2

 1  1   π    log f (x)d x  − log log xd x − log 2. − log(π/4) −   4 0 0    =1

7 8

1

To estimate c1 remains to estimate 0 log f (x)d x = −0.0347045 . . . . In particular, we can apply the methods described in this paper to write N  1

 9

0

log f (x)d x =

n=1 N 

nan (0) 2

+ O(θ N ) nan′ (0)

n=1

where the an are expressed in terms of the periodic points for the doubling map, up to period n. In practice, we can truncate the series of f (x) at the 20th term, which is a very good approximation 12 Q12 due the factorials in the individual terms (see Table 3). 13 Whereas this may not be the best numerical approach to computation it does give another 14 explicit formula.

10

11

15

16

17

Remark 6.4 (n ≥ 2). More generally, one can consider the integrals  2π  2π 1 cn := · · · (2n − 2 (cos(θ1 ) + · · · + cos(θn ))) dθ1 · · · dθn (2π)n 0 0    2π  2π n 1 = log(2n) + ··· log 1 − cos(2π xi ) d x1 · · · d xn . n i=1 0 0

M. Pollicott, P. Felton / Indagationes Mathematicae xx (xxxx) xxx–xxx

13

These were studied by Chinta, Jorgenson and Karlsson [3] and Felker and Lyons [5] in the article. In particular, cd are related to spanning trees, since the value c2 has an explicit expression in terms of the Catalan constant. Moreover, c4 is close to 2 but numerically shown to be 1.999707644517 · · · (accurate to 7 decimal places).

4

Remark 6.5. There are many other interesting applications of this approach. For example:

5

1. there is a close connection between volumes of hyperbolic manifolds and special values of [16]. Moreover, these volumes can expressed in terms of the integrals Π (x) = L-functions x log(2 sin u)du, which can be estimated as before (cf. [11]). 0 2. The Mahler measure also has interesting uses in knot theory. For example, in the work of Silver and Williams [13] on the Alexander polynomial and the work of Champanerkar and Kofman on the Jones polynomials [2]. 3. We can write l moments of the n-step Random Walk by l  1  1  n    e2πitk  dt1 · · · dtk Wn (l) =    0 0 k=1 which can be similarly estimated. References [1] J. Borwein, A. Straub, Mahler measures, short walks and log-sine integrals, Theoret. Comput. Sci. 479 (2013) 4–21. [2] A. Champanerkar, I. Kofman, On the tail of Jones polynomials of closed braids with a full twist, Proc. Amer. Math. Soc. 141 (7) (2013) 2557–2567. [3] G. Chinta, J. Jorgenson, A. Karlsson, Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori, Nagoya Math. J. 198 (2010) 121–172. [4] G. Everest, T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, in: Universitext, Springer-Verlag London, Ltd., London, 1999. [5] J. Felker, R. Lyons, High-precision entropy values for spanning trees in lattices, J. Phys. A 36 (31) (2003) 8361–8365. (English summary). [6] O. Jenkinson, M. Pollicott, Calculating Hausdorff dimension of Julia sets and Kleinian limit sets, Amer. J. Math. 124 (2002) 495–545. [7] O. Jenkinson, M. Pollicott, A dynamical approach to accelerating numerical integration with equidistributed points, (with O. Jenkinson), Tr. Mat. Inst. Steklova 256 (2007) 290–304. Din. Sist. i Optim.. [8] D.H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933) 461–479. [9] D. Lind, K. Schmidt, T. Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (3) (1990) 593–629. [10] K. Mahler, On some inequalities for polynomials in several variables, J. Lond. Math. Soc. 37 (1962) 341–344. [11] J. Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. 6 (1982) 9–24. [12] Y. Sasaki, On multiple higher Mahler measures and multiple L values, Acta Arith. 144 (2) (2010) 159–165. [13] D. Silver, S. Williams, Mahler measure of Alexander polynomials, J. Lond. Math. Soc. 69 (2004) 767–782. [14] C. Smyth, On measures of polynomials in several variables, Bull. Aust. Math. Soc. 23 (1) (1981) 49–63. [15] C. Smyth, The Mahler measure of algebraic numbers: a survey, in: Number Theory and Polynomials, in: London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 322–349. [16] D. Zagier, Hyperbolic manifolds and special values of Dedekind zeta-functions, Invent. Math. 83 (1986) 285–301.

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