The Mahler measure of $(x+1/x)(y+1/y)(z+1/z)+\sqrt{k}$

1.   Introduction

Let $f$ be a nonzero Laurent polynomial in $\mathbb{C}[T_1,\dots,\ T_n,T_1^{-1},\dots,T_n^{-1} ]$. The logarithmic Mahler measure of $f$ is defined as

There have been a lot of numerical evidence and theoretical results implying that there exist non-trivial relations between Mahler measure of certain kind of Laurent polynomials and special values of $L$-functions. The first example is provided by Smyth[17]:

where $\chi_D$ denotes the Jacobi symbol $\left(\frac{D}{\cdot}\right)$.

Suppose $P$ is a two-variable Laurent polynomial. If $P(x,y) = 0$ happens to define an elliptic curve $E$ over $\mathbb{Q}$, then $m(P)$ can sometimes be related to $L(E,2)$ by Beilinson's regulator maps. It is Deninger who first noticed this and conjectured in [5] that

where $E_{15}$ is an elliptic curve with conductor 15. This conjecture is proved by Rogers and Zudilin in [11] and [12]. After Deninger's discovery, Boyd proposed hundreds of conjectures in this direction [4].

In 2008, Bertin [1,2] investigated the Mahler measure of three-variable Laurent polynomials:

Later, Rogers [10] considered the following functions:

where $k\in \mathbb{C}$ is a constant. Samart's work shows that there are some appealing connections between the special values of the above functions and the special value of $L$-functions of certain modular forms [15]. Actually Samart made many more conjectures of this type in [15] and [16]. The following five identities conjectured in [15] have been proved by Guo, Peng and Qin in [7]:

where $g_{_N}$ denotes some newform with rational coefficients in the space of cusp forms of weight 3 and level $N$.

In this article, we prove six identities conjectured in 2015 in [16]:

Here, $d_k = L'(\chi_{-k},-1)$, $M_N = L'(g_N,0)$, $M_{N\otimes D} = L'(g_N\otimes\chi_D,0)$ and $g_N\otimes\chi_D$ is the quadratic twist of $g_N$ by $\chi_D$.

In addition to the complicated computation of the lattice sums, which is mainly due to the distinct forms of the integral basis of the number fields $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-7})$, some techniques that work well in [7] fail in our cases. For example, the method of finding the identities of the corresponding modular forms fails, as the $L$-functions are no longer equal to each other in one of our cases. In order to prove the identities of the special values of $L$-functions, we have to develop another method, see Theorem 4.2 in this paper. On the other hand, it seems that the identities like $f_2(1) = 8M_7$ in [16] cannot be proved by any of our previous techniques, although they appear very similar to the ones we have verified. In fact we are still not able to prove them.

2.   Modular functions

As usual we denote the Dedekind eta function by $\eta(\tau)$. Then we have $\Delta(\tau) = \eta(\tau)^{24}$, where $\Delta$ is the modular discriminant. Define

Write $\mathfrak{f}(\tau)$ for the classical Weber modular function $e^{-\frac{\pi i}{24}}\frac{\eta((\tau+1)/2)}{\eta(\tau)}$. The following relation holds (cf. [15], Lemma 2.2)

Let $\mathfrak{f}_1(\tau) = \frac{\eta(\tau/2)}{\eta(\tau)}$, $\mathfrak{f}_2(\tau) = \sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}$. We have $\mathfrak{f}(\tau)\mathfrak{f}_1(\tau)\mathfrak{f}_2(\tau) = \sqrt{2}$.

Moreover, $\mathfrak{f}(\tau)^{24}$, $-\mathfrak{f}_1(\tau)^{24}$, $-\mathfrak{f}_2(\tau)^{24}$ are exactly the three roots of the cubic equation

where $j(\tau)$ is the classical $j$-invariant. One can see more details on page 288 of [3].

Lemma 2.1. We have the following special values of $\mathfrak{f}(\tau)^{24}$:

(1)$\mathfrak{f}((1+\sqrt{-3})/2)^{24} = s_2(q((1+\sqrt{-3})/4)) = 16$.

(2) $\mathfrak{f}(1+\sqrt{-3})^{24} = s_2(q((1+\sqrt{-3})/2)) = -104-60\sqrt{3}$.

(3) $\mathfrak{f}((3+\sqrt{-3})/3)^{24} = s_2(q((3+\sqrt{-3})/6)) = -104+60\sqrt{3}$.

(4) $\mathfrak{f}(\sqrt{-7})^{24} = s_2(q(\sqrt{-7}/2)) = 4096$.

(5) $\mathfrak{f}(1+\sqrt{-7})^{24} = s_2(q((1+\sqrt{-7})/2)) = -2024-765\sqrt{7}$.

(6) $\mathfrak{f}((7+\sqrt{-7})/7)^{24} = s_2(q((7+\sqrt{-7})/14)) = -2024+765\sqrt{7}$.

Proof. (1)We have

where $\zeta = e^{\frac{\pi i}{24}}$(cf. page 1647 of [19]). Setting $x = 16$ in equation $(x-16)^3-j(\tau)x = 0$, we get $j(\tau) = 0$, which implies $\tau$ can be taken as $\frac{-1+\sqrt{-3}}{2}$.

It follows from $(x-16)^3 = 0$ that

and we are done.

Setting $x = -104\pm60\sqrt{3}$ in equation $(x-16)^3-j(\tau)x = 0$, one gets $j(\tau) = 54000$. The three roots of the equation $(x-16)^3-54000x = 0$ are

By Weber's table VI [18], $\mathfrak{f}(\sqrt{-3}) = \sqrt[3]{2}$, i.e., $\mathfrak{f}(\sqrt{-3})^{24} = 2^8 = 256$, hence

By numerical computations, one can deduce that $\mathfrak{f}_1(\sqrt{-3})^{24} = 104+60\sqrt{3}$ and $\mathfrak{f}_2(\sqrt{-3})^{24} = 104-60\sqrt{3}$.

Using $(2.1)$ we obtain

(3) Note that the transformation $\tau\mapsto-\frac{1}{\tau}$ fixes $\mathfrak{f}$, and exchanges $\mathfrak{f}_1$, $\mathfrak{f}_2$. Hence,

$\mathfrak{f}_1(-1/\sqrt{-3})^{24} = \mathfrak{f}_2(\sqrt{-3})^{24} = 104-60\sqrt{3}$, which implies

(4) By Weber's table VI [18], we have $\mathfrak{f}(\sqrt{-7}) = \sqrt{2}$, which implies

(5) & (6): Setting $x = -2024\pm765\sqrt{7}$ in equation $(x-16)^3-j(\tau)x = 0$, we get $j(\tau) = 16581375$. Solving the equation $(x-16)^3-16581375x = 0$ one obtains three roots:

From above, we have known that $\mathfrak{f}(\sqrt{-7})^{24} = 4096$. Thus

By numerical computation, one can decide that

and

Therefore

Moreover, we have

which implies

Lemma 2.2.([15], Proposition 2.1(i)). Assume that $q\in(0,1)$. If $\mathrm{Im}(\tau)\ge \frac{1}{2}$, then

where

The restriction on $q$ can be removed and the lower bound of $\mathrm{Im}(\tau)$ is unnecessary. Applying a similar argument as in [8], one can show that the identity in this lemma holds everywhere. Indeed, both $f_2(k)$ and $\frac{2\mathrm{Im}(\tau)}{\pi^3}(-A+16B)$ are the real parts of holomorphic functions at least when $k\in\mathbb{C}\setminus[-64,64]$ and they coincide on a non-discrete set of points, see [10] and [15].

The $L$-function with respect to a quadratic Dirichlet character $\chi$ is denoted by $L(s,\chi)$. Let $\chi_{-4}$ be the character $\left(\frac{-1}{\cdot}\right)$, $\chi_{-8}(\cdot) = \left(\frac{-2}{\cdot}\right)$, $\chi_{8}(\cdot) = \left(\frac{2}{\cdot}\right)$ and $\chi_{-3}(\cdot) = \left(\frac{-3}{\cdot}\right)$, etc.

We will use the notations as in [6]. For any integer $n$, write $L_n(s) = L(\chi_n,s)$, in particular, $L_1(s) = \zeta(s)$ the Riemann zeta function. The following results prove to be crucial to our computation.

Lemma 2.3. (Glasser & Zucker [6], Table Ⅵ). For any complex $s$ with $Re(s)>1$,

(1) $\underset{x,y\in\mathbb{Z}}{\sum\nolimits^{'}}\frac{1}{\left(x^2+3y^2\right)^s} = 2(1+2^{1-2s})L_1(s)L_{-3}(s)$

(2)$\underset{x,y\in\mathbb{Z}}{\sum\nolimits^{'}}\frac{1}{\left(x^2+7y^2\right)^s} = 2(1-2^{1-s}+2^{1-2s})L_1(s)L_{-7}(s)$

(3) $\underset{x,y\in\mathbb{Z}}{\sum\nolimits^{'}}\frac{1}{\left(x^2+12y^2\right)^s} = (1+2^{-2s}+2^{2-4s})L_{1}(s)L_{-3}(s)+L_{-4}(s)L_{12}(s)$

(4) $\underset{x,y\in \mathbb{Z}}{\sum\nolimits^{'}}\frac{1}{(x^2+28y^2)^2} = \\ (1-2^{1-s}+3\cdot2^{-2s}-2^{2-3s}+2^{2-4s})L_{1}(s)L_{-7}(s)+L_{-4}(s)L_{28}(s)$

Remark 1. The following results are useful for our computation: $L_1(2) = \frac{\pi^2}{6}$, $L_{12}(2) = \frac{\sqrt{3}}{18}\pi^2$, $L_{28}(2) = \frac{2\pi^2}{7\sqrt{7}}$.

Remark 2. For the sake of brevity, let us do the following convention of notations in this paper. Let $R(l,m)$ be any function of $l$ and $m$. We denote

where $a,b\in \mathbb{Z}$. As usual, the symbol $\sideset{}{^{'}}\sum_{\substack{(a,b)}}R(l,m)$ means the sum does not include $(a,b) = (0,0)$. Similarly, the notation $\Sigma'_{x\equiv y(d)}$ means $\Sigma'_{x\equiv y\pmod d}$, where the primed summation sign means to sum over all integer pairs except $(x,y) = (0,0)$.

3.   Computation of $s_2(k)$ for $k=16,-104\pm60\sqrt{3}$

Following Samart's method, we will calculate $A$ and $B$ in the statement of Lemma 2.2 for our case, where $\tau = \frac{1+\sqrt{-3}}{4}$ by Lemma 2.1(1).

and

Thus,

Let

By lemma 2.7 in [15], $f(\tau)\in S_3(\Gamma_0(12),\chi_{-3})$. Thus,

Therefore, we obtain:

as desired.

3.1.   $k=-104-60\sqrt{3}$

In this subsection, $K = \mathbb{Q}(\sqrt{-3})$. By Lemma 2.1, we know that

So in this case

To ease the notation, we set

By Samart's formulas in Lemma 2.2, we have

and

3.1.1.   The part of the $L$-Series of a modular form

We continue to set

From the proof of Lemma 2.4 in [15], we know that $h = \sum_{x\equiv y\pmod 2}\left(\frac{x^2-3y^2}{2}\right) q^{x^2+3y^2} = 0$. As a multiple of the $L$-function associate to $h$, $A_M = 0$. Thus

On the other hand, we let

It follows from [15] (cf. page 248, last line) that $f\otimes\chi_{-4}(\tau)\in S_3(\Gamma_0(48),\chi_{-3})$. We see that

and

Therefore,

Applying the functional equation (the sign can be settled by finite precision numerical computation, it is '$+$' for our case)

we obtain

3.1.2.   The part of the Dirichlet $L$-Series

Now let us deal with the Dirichlet $L$-function part. It is clear that

One can easily check that

where $\sqcup$ means disjoint union.

Thus we have

which implies

Here we used the fact that $|U(K)| = 6$.

By Lemma 2.3, we know that the following results hold

Hence,

Therefore,

Here, we used the functional equation

In conclusion,

3.2.   $k=-104+60\sqrt{3}$

In this subsection, we still denote $\mathbb{Q}(\sqrt{-3})$ by $K$. By Lemma 2.1, we have

So in this case,

To ease the notation, we set

We need to calculate the values of $A$ and $B$ in Samart's formula, where

and

3.2.1.   The part of the $L$-Series of a modular form

From the computation in the last section, we know that $A_M = 0$. Define $f(\tau)$ just in the same way as in the last section. We obtain

Thus,

3.2.2.   The part of the Dirichlet $L$-Series

We now treat the part involving the Dirichlet $L$-Series. By Lemma 2.3, we have

where we used Remark 2.4 that $L_{12}(2) = \frac{\sqrt{3}}{18}\pi^2$. It is clear that $Y = C-X-D$. By identities (2)-(5), we do the following calculation

Thus

4.   Computation of $s_2(k)$ for $k=4096,-2024\pm765\sqrt{7}$

In this section, let $K = \mathbb{Q}(\sqrt{-7})$. Then $Cl(O_K) = 1\Rightarrow O_K$ is PID, $U(K) = \{\pm 1\}$ and 2 split completely in $K$. Let

and

Hence,

On the other hand, let us do the following settings

where $I$ denotes the ideals in $O_K$. In addition we set

4.1.   $k=4096$

By Lemma 2.1, we know that

So in this case,

To ease the notation, we set

We have

and

4.1.1.   The part of the $L$-Series of a modular form

Let $g_7 = \eta(\tau)^3\eta(7\tau)^3 = \sum_{m,n\in \mathbb{N}} \chi_{-4}(mn)mn q^{\frac{m^2+7n^2}{8}}$. We have following result.

Theorem 4.1. Let

Then,

Proof. Let $\Lambda' = (2)$ and define $\phi(\mathfrak{a}) = \alpha^2$ for a generator of an integral ideal $\mathfrak{a}$ which satisfies $\alpha\equiv 1(\Lambda')$.

In fact, if $(\alpha)$ is coprime to $\Lambda'$, then $\alpha\equiv 1(\Lambda')$. To see this, consider the norm of $\alpha$, it is odd (since $(\alpha)+\Lambda' = O_K$) and moreover,

Hence,

Thus if $(\alpha)$ is coprime to $\Lambda'$, then $\alpha\equiv 1(\Lambda')$. It is clear that $\phi$ is multiplicative and satisfies $\phi(\alpha O_K) = \alpha^2$. Therefore, $\phi$ is a Hecke charactor which satisfies the condition of Theorem 1.31 in [9]. So

is a newform in $S_3(\Gamma_0(28),\chi_{-7})$.

As $y$ is even, we can rewrite the norm as

Here, $x'',y''$ satisfy that $(x'',y'')\in \mathbb{Z}$ and $x''-y''\equiv 1\pmod 2$ since $x = x''-y''$ should be odd from above.

Note that $\alpha = x+y\sqrt{-7}$ and $\beta = x-y\sqrt{-7}$ share the same norm value, so we have:

So one can deduce that $g\otimes\chi_{-4} = g_7\otimes\chi_{-4}$ whose level divides 112 by considering the Sturm Bound (with the help of SageMath[13]). Indeed, the Sturm Bound for our case is at most 48. In addition,

and

Since

one can get

Here we applied the functional equation (1) to $L(g_7\otimes \chi_{-4},3)$.

4.1.2.   The part of the Dirichlet $L$-Series

Now let us deal with the Dirichlet $L$-Series part, or more exactly, the computation of the term

Firstly, by

we obtain

Thus,

By Lemma 2.3,

On the other hand, we have

In addition, one can easily check that

which implies

By Lemma 2.3,

Therefore,

Since $L_{28}(2) = 2\pi^2/(7\sqrt{7})$ and by the functional equation (5) we have

Hence,

Finally, we obtain

as desired.

4.2.   $k=-2024-765\sqrt{7}$

To ease the notation, in this subsection we set

Following Samart's method, we will calculate $A$ and $B$ in the statement of Lemma 2.2 for our case, where $\tau = \frac{1+\sqrt{-7}}{2}$ by Lemma 2.1(2).

4.2.1.   The part of the $L$-Series of a modular form

To prove this identity, we need to find another expression for $g_7$ defined in last subsection.

Theorem 4.2. We have

Proof. Take modulus $\Lambda = (1)$ and define $\phi(\mathfrak{a}) = \alpha^2$ for a generator $\alpha$ of $\mathfrak{a}$ which satisfies $\alpha\equiv 1(\Lambda)$. It is clear that $\phi$ is multiplicative and satisfies $\phi(\alpha O_L) = \alpha^2$. Therefore, $\phi$ is a Hecke character which satisfies the conditions of Theorem 1.31 in [9]. So

is a newform in $S_3(\Gamma_0(7),\chi_{-7})$. Here, the sum in the definition of $G_7(\tau)$ is over all ideals in $O_K$.

Rewrite $\alpha = x+y\frac{1+\sqrt{-7}}{2}$ as $\alpha = \frac{2x+y+y\sqrt{-7}}{2} = \frac{x'+y'\sqrt{-7}}{2}$, $x,y\in \mathbb{Z}$ and $x' = 2x+y$, $y' = y$. Then we have $x'\equiv y'(\text{mod }2)$.

Note that $\alpha = x+y\sqrt{-7}$ and $\beta = x-y\sqrt{-7}$ share the same norm value, so we have

Finally one can show that $g_7 = G_7$ with the help of SageMath [13] and Sturm bound.

Theorem 4.3. With the notation above, the following identity holds:

Proof. Firstly, by the fact that,

we have

Secondly, we have

where

And it is clear that:

where

Write $g_7$ as following form

i.e.,

We can find that

In addition,

where

Let $h(\tau) = \sum^{\infty}_{j = 1}(w^{2j}+\bar{w}^{2j})g(2^j\tau)$. Then

Note that

Therefore,

Combining the relation (6), we have

Thus

as desired.

By functional equations, one can show that

In addition

On the other hand,

Comparing (7) and (8) and by Theorem 4.2, 4.3, we are done.

4.2.2.   The part of the Dirichlet $L$-Series

From last subsection, one can get

as desired. Here we used the following facts:

4.3.   $k=-2024+765\sqrt{7}$

To ease the notation, we set:

in this subsection.

Following Samart's method, we will calculate $A$ and $B$ in the statement of Lemma 2.2 for our case, where $\tau = \frac{7+\sqrt{-7}}{14}$ by Lemma 2.1(3).

and

4.3.1.   The part of the $L$-Series of a modular form

By functional equations, we have

And,

On the other hand,

By Theorem 4.2 and Theorem 4.3, comparing (9) and (10), we are done.

4.3.2.   The part of the Dirichlet $L$-Series

From previous computation, we have

Hence,

as desired. Here we used the following facts:

Acknowledgments

The first author would like to thank Dr.Wei Lu for deducing the very useful form of $g_7$ in theorem 4.2 in an elegant way, which inspires him to come up with the proof given in this paper. The authors would like to thank the referee for many valuable comments and suggestions which have been incorporated herein.