Abstract
We study the zeta Mahler function (ZMF, also zeta Mahler measure), which is closely related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials \(k + (x_1 + x_1^{-1}) \cdots \left( x_r + x_r^{-1}\right) \), where k is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore connections with the Mahler measure.
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1 Introduction
For a Laurent polynomial \(P \in \mathbb {C}[x_1^{\pm 1}, \dots , x_r^{\pm 1}] \setminus \{ 0 \},\) define the zeta Mahler function (ZMF) as
where s is a complex parameter. This was first introduced by Gelfand and Bernstein in [1]. The function, though still unnamed, was later studied by Atiyah [2], Bernstein [3] and by Cassaigne and Maillot [4]. Finally, in 2009 these functions were re-introduced by Akatsuka [5]. The values Z(P; s) can be interpreted as the average value of \(\left| P(x_1, \dots , x_r) \right| ^s\) on the torus \(\mathbb {T}^r = \{ (x_1, \dots , x_r) \in \mathbb {C}^r :|x_1| = \dots = |x_r| = 1\}\). If we let \(X_1, \dots , X_r\) be uniformly distributed random variables on the complex unit circle we can also interpret Z(P; s) as the s-th moment of the random variable \(|P(X_1, \dots , X_r)|\). The logarithmic Mahler measure of P is defined as
see [6]. In this paper we discuss properties of the zeta Mahler function of the polynomials \(k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})\) for real k. We denote this function by \(W_r(k;s)\),
It is easy to see that for real k, the quantity \(k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})\) is real-valued on the torus \(\mathbb {T}^r\). Further, a substitution \(x_1 \mapsto -x_1\) shows that \(W_r(|k|,s) = W_r(k;s)\), so it suffices to consider only \(k \ge 0\). In the computation we make a clear distinction between the cases \(k \ge 2^r\) and \(0 \le k < 2^r\). In the case \(k \ge 2^r\) (the “light” case) the structure is much simpler as we can drop the absolute value in the integral (2). The ZMFs are originally defined in a half-plane \({\text {Re}}(s)>s_0\) for some \(s_0<0\), and hypergeometric expressions that we prove in this paper provide one with an efficient way for analytic continuation of the ZMF to a meromorphic function of s with an explicit location and structure of poles.
For \(r=1\), these values are already recorded in [5, Theorem 4]. We give a simpler expression for \(W_1(k;s)\).
Theorem 1.1
For \(s \in \mathbb {C}\), the following expressions are valid.
-
(i)
For \(|k|>2\), we have
$$\begin{aligned} W_1(k;s) = |k|^s \cdot {}_{2} F_{1} \left( \frac{-s}{2}, \frac{1-s}{2} ; 1; \frac{4}{k^2} \right) . \end{aligned}$$ -
(ii)
For \(|k| = 2\), we have
$$\begin{aligned} W_1(k;s) = \frac{2^s \Gamma \left( \frac{1}{2} + s\right) }{\Gamma \left( 1+\frac{s}{2}\right) \Gamma \left( \frac{1+s}{2}\right) }. \end{aligned}$$ -
(iii)
For \(|k| < 2\), we have
$$\begin{aligned} W_{1}(k;s) = \frac{4^s \Gamma \left( \frac{1 + s}{2}\right) ^2}{\pi \Gamma \left( 1 + s\right) } \cdot {}_{2} F_{1} \left( \frac{-s}{2}, \frac{-s}{2} ; \frac{1}{2}; \frac{k^2}{4} \right) . \end{aligned}$$(3)
Here and in what follows
is the hypergeometric function, where
denotes the Pochhammer symbol. A generalization of the hypergeometric function we also need is the Meijer G-function,
see [7, Sect. 16.17] for the definition.
The explicit formulae in Theorem 1.1 allow us to compute a functional equation in the variable s, using the symmetry of the \({}_{2} F_{1}\) hypergeometric function.
Theorem 1.2
-
(i)
For \(|k|>2\) and \(s \in \mathbb {C}\), we have
$$\begin{aligned} W_1(k;-s-1) = (k^2 - 4)^{-s-\frac{1}{2}} W_1(k;s). \end{aligned}$$(4) -
(ii)
For \(|k|<2\) and \(-1< {\text {Re}}(s) < 0\), we have
$$\begin{aligned} W_1(k;-s-1) = \cot \left( -\frac{\pi s}{2} \right) (4-k^2)^{-s-\frac{1}{2}} W_1(k;s). \end{aligned}$$(5)
The functional equation for \(|k|>2\) was already recorded in [5, Theorem 3], the \(|k|<2\) functional equation is new. Note that in both cases the line of symmetry of \(W_1(k;s)\) is \({\text {Re}}(s) = -\frac{1}{2}\). In fact, all the zeros of \(W_1(k;s)\) lie on this line. Furthermore, equation (5) defines an analytic continuation of \(W_1(k;s)\) to a meromorphic function on \(\mathbb {C}\).
Theorem 1.3
For any \(k \in \mathbb {R}\), all the non-trivial zeros of \(W_1(k;s)\) lie on the critical line \({\text {Re}}(s) = -\frac{1}{2}\).
Furthermore, we present a general formula for all r in the “light” case.
Theorem 1.4
Let \(r \ge 1\).
-
(i)
For \(|k| > 2^r\) and all \(s \in \mathbb {C}\),
$$\begin{aligned} W_r(k;s) = |k|^s \cdot {}_{r+1} F_{r} \left( \frac{-s}{2}, \frac{1-s}{2}, \frac{1}{2}, \dots , \frac{1}{2}; 1, \dots , 1; \frac{4^r}{k^2} \right) . \end{aligned}$$(6) -
(ii)
For \(|k| = 2^r\) and \({\text {Re}}(s) > -\frac{r}{2}\),
$$\begin{aligned} W_r(k;s) = 2^{rs} {}_{r+1} F_{r} \left( \frac{-s}{2}, \frac{1-s}{2}, \frac{1}{2}, \dots , \frac{1}{2}; 1, \dots , 1; 1 \right) . \end{aligned}$$(7)
The case \(r=2\) and \(|k|>4\) was already recorded in [5, Theorem 6].
For \(0< k < 2^r\) we show that, as a function of k, \(W_r(k;s)\) always satisfies the same differential equation as \(W_r(k;s)\) for \(|k|>2^r\). So that the knowledge of the solutions of this differential equation allows us to give an explicit formula for \(W_r(k;s)\) for \(|k| < 2^r\). For real s we have the following result.
Theorem 1.5
For real \(s>0\), s not an odd integer, and real k,
More generally, for \(r=2\) we find the following formula.
Theorem 1.6
For \(|k| < 4\) and \({\text {Re}}(s) > -1\) and s not an odd integer, we have
Furthermore for \(r=3\) we find the following extension.
Theorem 1.7
For \(|k|<8\) and \({\text {Re}}(s) > -1\), s not an odd positive integer, we have
where \(G_{p,q}^{m,n}\) denotes the Meijer G-function.
Apart from the hypergeometric expressions for \(W_r(k;s)\) above, we compute the probability density \(p_r(k;-)\) of the random variable \(|k + (X_1 + X_1^{-1}) \cdots (X_r + X_r^{-1})|\), where the \(X_i\) are independent uniformly distributed random variables on the complex unit circle. We can relate this \(p_r\) to \(W_r\) via
so that \(p_r(k;-)\) is the inverse Mellin transform of \(W_r(k;s)\) (see Sect. 4.1.2).
This paper is structured as follows. In Sect. 2 we prove Theorem 1.1, and in Sect. 3 we prove Theorems 1.2 and 1.3. In Sect. 4, we study properties of the probability densities \(p_r\) and prove Theorems 1.4 and 1.5, using methods different from Sect. 2 and generalizing Theorem 1.1. We finish Sect. 4 with the proofs of Theorems 1.6 and 1.7. Furthermore, we use our findings to give representations of Mahler measures not only in a hypergeometric form but also as multiple Euler type integrals.
2 Proof of Theorem 1.1
2.1 Proof of Theorems 1.1.(i) and 1.1.(ii)
Assume that \(k>2\). We compute the probability distribution \(p_1(k;-)\) of the random variable \(|k + X + X^{-1}|\), where X is uniformly distributed on the unit circle \(\{z \in \mathbb {C} :|z| = 1 \}\). It can be easily seen that the density of this random variable for \(k>2\) is given by
with the support on \((k-2 , k + 2)\), so that
Using the substitution \(y = (x-k+2)/4\) to normalize the integral, we find that
Recall now the Euler integral
valid for \({\text {Re}}(c)> {\text {Re}}(b) > 0\) and \(z \not \in [1, \infty )\), see [8, p. 4]. We may write
Applying the quadratic transformation
for \(|{\text {arg}}(1-z)|<\pi \) (see [9, Eq. (9.6.17)]), Eq. (11) simplifies to
For \(k = \pm 2\), we find
valid for \({\text {Re}}(s) > -\frac{1}{2}\). Letting \(z \rightarrow 1\) in (10) (in other words, using the Gauss summation formula) gives
2.2 Proof of Theorem 1.1.(iii)
We now consider \(0 \le k<2\). Then the probability density of the random variable \(|k + X_1 + X_1^{-1}|\) is given by
leading to
Substituting \(y = \frac{x}{k+2}\), we obtain
Using this and the symmetric representation for \(\int _{0}^{2-k}\), we deduce that
for \({\text {Re}}(s) > - 1\). Applying the transformation
valid for \(|z| \le \frac{1}{2}\), we find
The latter expression can be simplified to the form (3), using the quadratic transformation
for \(|z| \le 1\) (see [7, Eq. (15.8.27)]).
3 Proof of Theorems 1.2 and 1.3
In this section we discuss special properties of \(W_r(k;s)\) for \(r=1\). We give a functional equation for \(W_1(k;s)\) and we state and prove the “Riemann hypothesis” for \(s \mapsto W_1(k;s)\). It is not clear to what extend these results generalize for \(r > 1\).
3.1 Functional Equations for \(W_1(k;s)\)
3.1.1 Proof of Theorem 1.2.(i)
We start with the proof of Theorem 1.2.(i). We give two proofs of this fact. One proof uses hypergeometric transformations, and the other uses the probability distribution \(p_1\).
For the first proof, we use Euler’s transformation formula
which is just the double iteration of the transformation (). We find out that
For the second proof, we use the symmetry of formula (9):
for \(x \in (k-2,k+2)\). Now applying it to
gives the functional equation (4).
3.1.2 Proof of Theorem 1.2.(ii)
We start with the proof of Theorem 1.2.(ii). Using Euler’s transformation formula (13) we obtain
which is precisely the functional Eq. (5).
3.2 Zeros of \(W_1(k;s)\)
For \(\lambda \in \mathbb {C}\) and \((\alpha ,\beta ) \in \mathbb {C}^2\), consider the Jacobi function
see [10]. When \((\alpha ,\beta ) \in \mathbb {C}^2\) is fixed, the sequence \(\{ \varphi ^{(\alpha ,\beta )}_{\lambda } \}_{\lambda \ge 0}\) forms a continuous orthogonal system on \(\mathbb {R}_{\ge 0}\) with respect to the weight function
In this way \(\varphi _{\lambda }^{(\alpha , \beta )}(t)\) becomes the unique even \(C^{\infty }\)-function f on \(\mathbb {R}\) satisfying
where the dash denotes the derivative with respect to t. For brevity we write \(\varphi _{\lambda }\) for \(\varphi ^{(\alpha ,\beta )}_{\lambda }\), similarly for \(\Delta \).
Lemma 3.1
For any \(x > 0\), \(\lambda , \mu \in \mathbb {C}\) with \(\lambda \ne \pm \mu \) and \(\alpha , \beta \in \mathbb {R}\),
Proof
Note that we can write (15) as
Now performing integration by parts twice and using \(\varphi _\lambda (0) = 0\) gives
which is Eq. (15). \(\square \)
Lemma 3.2
For any \(x > 0\) and \(\alpha , \beta \in \mathbb {R}\), if \(\varphi _\lambda (x) = 0\) then \(\lambda \in \mathbb {R} \cup i \mathbb {R}\). Moreover, if \(\min (\alpha +\beta +1,\alpha -\beta +1) > 0\) then \(\varphi _{i \mu }(x) > 0\) for \(\mu \le 0\).
The idea of the proof of Lemma 3.2 is based on the proof of Lommel’s theorem on the zeros of Bessel function, see [11, p. 482].
Proof of Lemma 3.2
Fix \(x > 0\) and assume that there is a \(\lambda \not \in \mathbb {R} \cup i \mathbb {R}\) such that \(\varphi _\lambda (x) = 0\). Choose \(\mu = \overline{\lambda }\) and apply Lemma 3.1. Clearly, \(\overline{\varphi _\lambda } = \varphi _\mu \), hence
But this gives a contradiction as \(\int _0^x |\varphi _\lambda (t)|^2 \Delta (t) \, \mathrm {d}t\) is transparently positive. Thus \(\lambda \in \mathbb {R} \cup i \mathbb {R}\). If additionally \(\min (\alpha +\beta +1,\alpha -\beta +1) > 0\), the coefficients of the hypergeometric function in (14) are strictly positive provided that \(\lambda = i \mu \) for \(\mu \le 0\), showing that \(\varphi _{i\mu } >0\). \(\square \)
If \((\alpha , \beta ) = (-\frac{1}{2},0)\) or \((0, -\frac{1}{2})\), then a consequence of this result is Theorem 1.3.
Proof of Theorem 1.3
Using Lemma 3.2 it suffices to show that \(s \mapsto W_1(k;s)\) has no real zeros. For \(s \ge 0\) this is obvious, as \(W_1(k;s)\), according to its definition (2), coincides with an integral of a positive function. \(\square \)
4 The zeta Mahler function \(W_r(k;s)\) for general r
We start this section with noticing that for \(r = 2\), the value of \(W_2(k;s)\) coincides with \(Z(k + x + x^{-1} + y + y^{-1}; s)\) (see (1)). Indeed, the substitution \(x = x_1x_2\) and \(y = x_1x_2^{-1}\) in the latter leads to
For \(r \ge 3\), the value of (2) is different from \(Z(k + x_1 + x_1^{-1} + \dots + x_r + x_r^{-1};s)\).
In this section we discuss the densities \(p_r(k;-)\).
4.1 The probability densities \(p_r(k;-)\)
In this section we explicitly compute the probability distributions \(p_r(k;-)\) for \(r = 1,2,3\) and discuss how to obtain them for any r.
Define \(\hat{p}_r\) to be the density of the random variable \((X_1 + X_1^{-1}) \cdots (X_r + X_r^{-1})\) on \(\mathbb {C}^r\); note that the quantity assumes real values only. We can relate the density of \(\hat{p}_r\) to \(p_r(k;-)\) in the following way (compare with Sect. 2.2).
Lemma 4.1
For \(|k| < 2^r\), we have
For \(|k| \ge 2^r\), we have \(p_r(k;x) = \hat{p}_r(x - |k|)\).
Proof
We have
If \(|k| \ge 2^r\), then
so that \(p_r(k;x) = \hat{p}_r(x - |k|)\).
If \(|k| < 2^r\), then
where the symmetry \(\hat{p}_r(-z) = \hat{p}_r(z)\) was employed. \(\square \)
Using Lemma 4.1, it is clear that it suffices to compute the distributions \(\hat{p}_r\).
4.1.1 Computation of \(\hat{p}_r\)
We will now compute \(\hat{p}_r\) explicitly for \(r = 1,2\) and 3.
It follows from (9) that
for \(|x| < 2\) and \(\hat{p}_1(x) = 0\) otherwise.
For \(r \ge 1\), define \(G_r(y) = \hat{p}_r(2^r \sqrt{1-y})\) for \(0 < y \le 1\) and \(G_r(y) = 0\) otherwise, so that \(\hat{p}_r(y) = G_r(1-x^2/4^r)\). By the above,
for \(0 < y \le 1\) and \(G_1(y) = 0\) otherwise.
Further, using basic properties of the Mellin transform, it follows that the \(G_r\) satisfy a recurrence for \(r \ge 2\):
Then using the substitution \(u = t^2/4^{r-1}\) in (16), we find
Finally, let \(y = 1 - x^2/4^r\) and \(v = u/y\) to arrive at the following result.
Theorem 4.2
(Recursive formula for the \(G_r\)) For \(G_r\) with \(r \ge 2\) we have the following recursion:
Applying this recursion with \(r = 2\) we obtain for \(0< y < 1\),
In the last step we use the integral representation (10). This shows that
for \(0 < |x| \le 4\) and \(\hat{p}_2(x) = 0\) otherwise.
In the case \(r = 3\) we proceed similarly. For \(0< y < 1\),
The expression
satisfies a third-order linear differential equation
After solving this differential equation (for example, with Mathematica [12]) and checking the initial conditions we find out that
so that
for \(0< |x| \le 8\) and \(\hat{p}_3(x) = 0\) for \(|x| > 8\).
Already at this stage it is pretty suggestive that \(G_r(1-y)\) satisfies the same differential equation as
We discuss this in the next subsection.
4.1.2 Mellin Transform of \(\hat{p}_r\)
In this section we find an expression for
when \(v \in \mathbb {C}\). This is the v-th moment of the random variable \((X_1 + X_1^{-1}) \dots (X_r + X_r^{-1})\).
Lemma 4.3
For \(\text {Re}(v) > -1\), we have
and
Proof
Since
and for the expected value
we have
For \(r = 1\), we obtain
Therefore,
and, clearly,
\(\square \)
Let \(f :(0, \infty ) \rightarrow \mathbb {R}\) be continuously differentiable and \(s \in \mathbb {C}\). Denote by M(f; s) its Mellin transform
Note that our definition of the Mellin transform differs slightly from the standard one, this does not affect the properties discussed below. Further, let \(\theta \) be the differential operator \(x \frac{\mathrm {d}}{\mathrm {d}x}\).
Proposition 4.4
(General properties of Mellin transforms; see [13, Sect. 3.1.2])
-
(i)
\(M(\theta f(x);s) = -(s+1) M(f(x);s)\);
-
(ii)
\(M(xf(x);s) = M(f(x);s+1) \).
Let \(H_r(y) := G_r(1-y)\) and consider the Mellin transform
Substituting \(y = x^2/4^r\) and using Lemma 4.3, we obtain
It is clear that \(M(H_r;s)\) satisfies a recursion:
If \(H_r\) were continuously differentiable, it would follow that
and
In other words,
so that \(H_r(y)\) is annihilated by the differential operator \(\theta ^{r} - y(\theta + 1/2)^r\). This means that \(H_r\) satisfies the same differential equation as \({}_r F_{r-1} \left( \frac{1}{2}, \dots , \frac{1}{2};1, \dots , 1; y \right) \). Though \(H_r\) is not continuously differentiable, giving a similar argument as in the proof of [14, Theorem 2.4] it follows rigorously that \(H_r\) satisfies the same differential equation in a distributional sense.
4.2 \(W_r\) for \(|k| \ge 2^r\)
In this part, we compute the value of \(W_r(k;s)\) for \(|k| \ge 2^r\).
Define for \(s \in \mathbb {C}\) the following function of \(z \in \mathbb {C}\):
Proposition 4.5
(Properties of \(F_{r,s}\))
-
(i)
\(F_{r,s}\) is analytic on \(\mathbb {C} \setminus (-\infty , 2^r]\).
-
(ii)
\(F_{r,s}\) is continuous on \(\{ z \in \mathbb {C} \ :{\text {Im}}(z) \ge 0\}\); in other words for real \(z \ge 0\).
Proof
This follows immediately as we can write
and notice that \(z \mapsto (x + z)^s\) is holomorphic for \(z \not \in (-\infty , 2^r]\). \(\square \)
First of all, we can find an explicit expression for \(F_{r,s}(z)\) if \(|z|>2^r\). The value of \(F_{r,s}(z)\) when \(|z| \le 2^r\) will follow from the analytic continuation of \(F_{r,s}\), with the help of Proposition 4.5.
Lemma 4.6
For \(|z| > 2^r\), we have
Proof
Note that
as \({\text {Re}}\left( 1+ \frac{x}{z} \right) > 0\). Since \(|x/z|<1\), we can write
so that
Now using Lemma 4.3, we find
Thus,
\(\square \)
Now for \(|k| > 2^r,\) \(k \in \mathbb {R}\) we can deduce an explicit formula for \(W_r\).
Proof of Theorem 1.4.(i)
Clearly,
so that
where Lemma 4.6 was used. \(\square \)
We see that, for a fixed \(|k| > 2^r\), the mapping \(s \mapsto W_r(k;s)\) defines an entire function.
Remark 1
If s is a non-negative integer, the hypergeometric sum is terminating. Hence, \(W_r(k;s)\) becomes a polynomial in |k|.
The value of \(W_r(k;s)\) for \(k = \pm 2^r\) when \(r>1\) can be found by taking the corresponding limit.
Proof of Theorem 1.4.(ii)
This follows from the absolute convergence of
on the domain \(|z| \le 1\) if
(see [15, p. 156]).
Example 4.7
For \(r = 1\) and \(|k|>2\), Eq. (6) gives
and for \(|k| = 2\), Eq. (7) gives
for \({\text {Re}}(s) > -\frac{1}{2}\). This leads to a second proof of Theorem 1.1.
4.3 \(W_r\) for \(|k| < 2^r\)
For \(|k|<2^r\), we write
Note that
hence
We now define the following analytic continuations \(F_{r,s}^{+}\) and \(F_{r,s}^{-}\). Let \(F_{r,s}^{+}(z)\) be the analytic continuation of the function
on the complement of the upper half-disk \(D(0,2^r)^\mathsf {c} \cap \mathbb {H}^{+}\) to \(\mathbb {C} \setminus ((-\infty ,0] \cup [2^r, \infty ))\), while \(F_{r,s}^{-}(z)\) is the analytic continuation of the function
on the complement of the lower half-disk \(D(0,2^r)^\mathsf {c} \cap \mathbb {H}^{-}\) to \(\mathbb {C} \setminus ((-\infty ,0] \cup [2^r, \infty ))\). Here \(\mathbb {H}^+\) and \(\mathbb {H}^{-}\) denote the (strict) upper- and lower-half planes, respectively.
Using Proposition 4.5, it is clear that for \(|k| < 2^r\) we have
Define
which is analytic on \(\mathbb {C} \setminus ((-\infty ,0] \cup [2^r, \infty ))\), with the following motivation.
Theorem 4.8
For \(|k| < 2^r\),
In fact, in the case of real s, we have an additional structure.
Lemma 4.9
For real positive s, we have
Proof
For notational convenience assume \(0 \le k < 2^r\). Then
and taking the complex conjugate, we deduce that
which is the desired claim. \(\square \)
Now we establish Theorem 1.5.
Proof of Theorem 1.5
Observe that for real s Theorem 1.4 coincides with the statement of Theorem 1.5, as
is real-valued for \(|k| \ge 2^r\). Therefore, we can assume \(|k| < 2^r\). Then by Lemma 4.9 and Theorem 4.8 we obtain
\(\square \)
Remark 2
Note that \(F_{r,s}^{+}(z), F_{r,s}^{-}(z)\) and hence also \(H_{r,s}(z)\) satisfy the same differential equation as
It is clear from the definition that \(F_{r,s}^{+}(z) = F_{r,s}^{-}(-z)\) for \(z \in D(0,2^r)^\mathsf {c} \cap \mathbb {H}^{+}\); therefore, from the theory of analytic continuation we conclude with the following.
Proposition 4.10
For all \(z \in \mathbb {H}^{+},\)
Proposition 4.11
(Real differentiability at \(k = 2^r\)) Suppose \({\text {Re}}(s) > n - \frac{r}{2}\). Then \(k \mapsto W_r(k;s)\) is n times (real) differentiable at \(k = 2^r\).
Proof
We will prove that \(k \mapsto W_r(k;s)\) is differentiable at \(k = 2^r \) if \({\text {Re}}(s) > \frac{1}{2}\). The general case follows by induction. For \(k > 2^r\), we have \(W_r(k;s) = F_{r,s}(k)\), so that the right derivative at \(k = 2^r\) is given by
when \({\text {Re}}(s) > 1 - \frac{r}{2}\). For \(|k|< 2^r\), we have
Therefore, the left derivative at \(k = 2^r\) is given by
This means that at \(k = 2^r\) the left and right derivatives coincide. \(\square \)
Proposition 4.12
(Real derivatives at \(k = 0\)) The right real derivatives \((\mathrm {d}^j/\mathrm {d}k^j)^{+}\) for \(0 \le j \le \lfloor {{\text {Re}}(s)}\rfloor \) of \(k \mapsto W_r(k;s)\) at \(k=0\) are given by
Proof
For \(k \ge 0\), we have
By induction,
for even j. Finally, notice that
\(\square \)
We now return to Theorem 1.1.(iii) and give another proof of it.
Proof of Theorem 1.1.(iii)
For \(|z|>2\), we have that
implying that \(F_{1,s}\) is a solution to the differential equation
Equation (17) has a basis of solutions
so that \(F_{1,s}^{+}(z) = C_0 Y_0(z) + C_1 Y_1(z)\), \(F_{1,s}^{-}(z) = \tilde{C_0} Y_0(z) + \tilde{C_1} Y_1(z)\) and \(H_{1,s}(z) = D_0 Y_0(z) + D_1 Y_1(z)\) for constants \(C_0, C_1, \tilde{C_0}, \tilde{C_1}, D_0\) and \(D_1\) depending only on s. Using Proposition 4.10, it follows that \(C_1 = -\tilde{C_1}\), hence \(D_1 = 0\).
As \(\lim _{z \rightarrow 0} H_{1,s}(z) = W_1(0;s)\) and we have
it follows that
Thus,
for \(|k|<2^r\).
Remark 3
In the proof above we see that \(H_{1,s}(z)\) extends to an analytic function in a neighborhood of \(z = 0\). We expect that this will only happen in the case \(r = 1\).
Remark 4
We use the symmetry of \(H_{1,s}\) to show that \(D_1 = 0\). Clearly, using
would lead to the same conclusion.
For \(r=2\) we follow the above strategy of the case \(r = 1\).
Proof of Theorem 1.6
We have, for \(|z|>4\),
hence \(F_{2,s}\) is a solution to the differential equation
A basis of solutions for (18) is given by
It follows that \(H_{2,s}(z) = D_0 Y_0(z) + D_1 Y_1(z) + D_2 Y_2(z)\) for some constants \(D_0,D_1\) and \(D_2\) depending only on s. Using the same argument as in the second proof of Theorem 1.5, it follows that \(D_1 = 0\). Since
and
we find out that
We can further simplify the coefficient in front of \(Y_2\). We have the following hypergeometric identity for the special value at \(z=1\) (see [8]):
Applying it with \(a_1 = 1/2, a_2 = 1/2-s/2, a_3 = -s/2, b_1 = 1\) and \( b_2 = 1\) gives
Hence \(H_{2,s}\) can be written as
It remains to apply Theorem 4.8 to arrive at the formula for \(W_2(k;s)\). \(\square \)
Remark 5
Notice that \(H_{2,s}(z)\) is not anymore analytic at \(z = 0\).
For odd positive values of s we need to compute the corresponding limits. We present an explicit formula, which can no longer be written in terms of hypergeometric functions.
Theorem 4.13
For odd positive integers n and \(|k|<4\),
Proof
For \({\text {Re}}(s)>-1\), not an odd integer, write
where
and
Using Mathematica, we have
where
Thus,
Taking the limit \(s \rightarrow n\), for n odd and positive, gives the result. \(\square \)
Using the explicit expression for \(W_2(k;s)\) in Theorem 1.6, the Mahler measure of the Laurent polynomial \(k + (x+x^{-1})(y+y^{-1})\) can be computed for \(|k|<4\).
Corollary 4.14
[[16, Theorem 3.1]] For \(|k| < 4\),
Proof
The Mahler measure of \(k + (x+x^{-1})(y+y^{-1})\) can be recovered as
Note that in the neighborhood of \(s=0\),
and
Thus,
\(\square \)
Remark 6
The Mahler measure \({\text {m}} (k + (X+X^{-1})(Y+Y^{-1}))\) can also be written as the double integral
using Corollary 4.14 and [7, Eq. (16.5.2)].
For the case \(r \ge 3\) and \(|k|<2^r\) we expect that \(W_r(k;s)\) cannot be written anymore as a linear combination of hypergeometric functions (not their real and imaginary parts as in Theorem 1.5). We now give the proof of Theorem 1.7.
Proof of Theorem 1.7
For \(|k| < 8\) and real \(s>1\), we have
Let \(\epsilon > 0\). Then for \(|k|<8\) we have
where
and
We expand \(\alpha _3(s)\) and \(\alpha _4(s)\) in powers of \(\epsilon \). Note that \(\Gamma (\epsilon ) = \Gamma (\epsilon +1)/ \epsilon \), so that for s fixed
and
hence
By L’Hôpital’s rule,
where \(\psi \) is the digamma function. Furthermore, notice that
where
Thus we can write
Finally, using (22) we arrive at the result. \(\square \)
We can alternatively represent the Meijer G-function in (8) as the following triple integral.
Proposition 4.15
For \({\text {Re}}(s)>-1\) and \(0< |k| < 8\),
Proof
First, for all \(z \in \mathbb {C}\) we have
Applying Nesterenko’s theorem [17, Proposition 1] with \(a_0 = \dots = a_3 = \frac{1}{2}\), \(b_1 = 1\), \(b_2 =1 + \frac{s}{2}\) and \(b_3 = \frac{3+s}{2}\) to the right hand side gives
\(\square \)
As a consequence of Theorem 1.7 we can compute the Mahler measure of the polynomial \(k + (x+x^{-1})(y+y^{-1})(z+z^{-1})\) for \(|k|<8\).
Corollary 4.16
The Mahler measure of \(k + (x+x^{-1})(y+y^{-1})(z+z^{-1})\) for \(|k|<8\) is given by
Proof
We expand Eq. (8) in s. Note that
and
Hence,
\(\square \)
We can further write this Mahler measure for \(0<|k|<8\) as the triple integral (compare with equation (21)):
5 Concluding remarks
Using Theorem 1.5, explicit formulas for \(W_r\) in terms of hypergeometric functions and Meijer G-functions can be found for general r; see Theorems 1.6 and 1.7. In this way, the Mahler measure of the polynomials \(k + (x_1+x_1^{-1}) \cdots (x_r + x_r^{-1})\) could be written in terms of certain Meijer G-functions, although it is not expected that this Mahler measure can be written as a single Meijer G-function, like in formulae (20) and (23).
The result in Theorem 1.3 about the location of the zeros of \(W_1\) does not seem to be generalizable to \(W_r\). Furthermore, the numerics suggests there is no functional equation in the general case. For the zeros, there seems to be a pattern, though hardly recognizable.
From an arithmetic point of view, it could be interesting to simplify the expression (19) of \(W_2(k;n)\) for specific odd integers n and integers k. This can be compared to the case \(r=1\), where it can be shown, for example, that \(W_1(1;n) \in \mathbb {Q} + \frac{\sqrt{3}}{\pi } \mathbb {Q}\).
For integers s, the expression (6) of \(W_r(k;s)\) for \(k > 2^r\) is a polynomial in k. For the case \(r=1\), the induced polynomials are, up to an appropriate transformation of the variable, equal to the Legendre polynomials. This implies that the zeros of the induced polynomials have a very specific structure. This structure seems to generalize to the induced polynomials \(W_r(k;s)\) for general r. More specifically, we expect all the zeros of these polynomials to lie on the imaginary axis. Discussion of this theme is outside the scope of this paper.
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Acknowledgements
The author would like to thank Tom Koornwinder for his help with the proof of Theorem 1.3, and Frits Beukers for his help with the proof of Theorem 1.4. Thanks to Riccardo Pengo for his helpful comments. Thanks to Wadim Zudilin for his support and useful comments. I would finally like to thank the referee for the careful reading and valuable comments.
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Ringeling, B. Special zeta Mahler functions. Res. number theory 8, 29 (2022). https://doi.org/10.1007/s40993-022-00326-9
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DOI: https://doi.org/10.1007/s40993-022-00326-9