Special zeta Mahler functions
The author derives several explicit expressions for the function \[ W_r(k,s)=\frac{1}{(2\pi i)^r} \int_{{\mathbb T}^r} \big|k+(x_1+x_1^{-1})\dots(x_r+x_r^{-1})\big|^s \>\> \frac{dx_1}{x_1} \dots \frac{dx_r}{x_r}, \] where \(k\) is a real number, \(r \in {\mathbb N}\) and \({\mathbb T}^r\) is the torus \(|x_1|=\dots=|x_r|=1\). For example, for \(r=1\) he gives several expressions for this function in terms of the gamma function and the hypergeometric function. The formulas are different for \(|k|>2\), \(|k|<2\) and \(|k|=2\). In the latter case it is proved that \[ W_1(k,s)=\frac{2^s \Gamma(1/2+s)}{\Gamma(1+s/2)\Gamma(1/2+s/2)}. \] Using the symmetry of the hypergeometric function the author derives some functional equations for the function \(W\). For example, for \(|k|>2\) and \(s \in {\mathbb C}\) he proves that \[ W_1(r,-s-1)=(k^2-4)^{-s-1/2} W_1(k,s). \] It is also proved that for any real \(k\) the nontrivial zeros of \(W_1(k,s)\) all lie on the critical line \(\Re(z)=-1/2\). For \(r \geq 2\) the formulas are more complicated.
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