On the cubic \(L\)-function (Q2849060)

Let \(\omega = \exp(2\pi i /3)\) denote the third root of unity. The Kubotta-Patterson theta function \(\Theta_{K-P}\) is a generalized theta function related to the law of cubic reciprocity in \(\mathbb{Q}[\omega]\), introduced by \textit{T. Kubota} [On automorphic functions and the reciprocity law in a number field. Tokyo: Kinokuniya (1969; Zbl 0231.10017)] and studied by \textit{S. J. Patterson} [J. Reine Angew. Math. 296, 125--161 (1977; Zbl 0358.10011); ibid. 217--220 (1977; Zbl 0358.10012)].NEWLINENEWLINEThe cubic \(L\)-function is a function related to Kubota-Patterson theta function via the Mellin transform. Namely, let NEWLINE\[NEWLINE \Theta_{K-P}(z,v)=v^{2/3} + (6\pi v)^{2/3} \sum_{\nu} \widetilde{\tau}(\nu)Ai((6\pi |\nu| v)^{2/3})\exp(2\pi i (\nu z + \overline{\nu z})) NEWLINE\]NEWLINE be the Fourier expansion of the theta function, where \(Ai\) denotes the Airy function, the sum runs over all \(\nu\) of the form \((\sqrt{-3})^{-3} l \), \(l \in \mathbb{Z}[\omega]\) and coefficients \(\widetilde{\tau}(\nu)\) can be expressed in terms of the cubic Gauss sums. The cubic \(L\)-function is defined as a Dirichlet series NEWLINE\[NEWLINE L(\tau; s):= \sum_{\nu} \frac{\tau(\nu)}{\|\nu\|^s} NEWLINE\]NEWLINE converging absolutely for \(\text{Re} (s)>1\), where \(\tau(\nu) := \widetilde{\tau}(\nu) \|\nu\| ^{1/6}\), the sum runs over the same set of \(\nu\) as above and \(\|\cdot\|\) denotes the norm mapping from \(\mathbb{Q}[\sqrt{-3}]\) to \(\mathbb{Q}\).NEWLINENEWLINEIn the paper under review it is proved that the cubic \(L\)-function extends meromorphically to the whole complex plane with its only singularity being a simple pole at \(s=5/6\) and such that the completed \(L\)-function \(\Lambda(s):= (\sqrt{3}/2) (2\pi)^{-2s} \Gamma(s-1/6) \Gamma(s+1/6) L(\tau; s)\) satisfies the functional equation \(\Lambda(s)= \Lambda (1-s)\). Furthermore, the author derives an approximate functional equation for \(L(\tau;s)\), determines the asymptotic behavior of the sum \(\sum_{\|\nu\| \leq x} \tau(\nu)\), as \(x \to \infty\) and shows how to represent \(L(\tau; s)\) for \(\text{Re}(s)>1\) as a classical Dirichlet series ( i.e. a sum over positive integers).NEWLINENEWLINESince the function \(\tau\) is not multiplicative, the cubic \(L\)-function does not possess an Euler product representation, hence it is not expected that all non-trivial zeros of \(L(\tau;s)\) lie on the line \(\text{Re}(s)=1/2\). In the paper under review, the author conducts an extensive study of zeros of the cubic \(L\)-function. He derives an asymptotic formula for the number \(N(T)\) of zeros \(\rho\) of \(L(\tau; s)\) such that \(0 < \Im (\rho) <T\), proves that \(L(\tau; s)\) is non-vanishing in the half-plane \(\text{Re}(s)>6/5\) and shows that if \(\sigma >3/4\), then \(N(\sigma, T) \ll T\), as \(T \to \infty\), where \(N(\sigma, T)\) denotes the number of zeros \(\rho\) such that \(\text{Re}(\rho) \geq \sigma\) and \(0 < \text{Im} (\rho) <T\).NEWLINENEWLINEFinally, the author presents a histogram of the distribution of real parts of the first 8724 zeros \(\rho\) of \(L(\tau; s)\) such that \(\text{Re}(\rho) >1/2\), \(\text{Im} (\rho)\geq 0\) and poses many conjectures about the zeros of \(L(\tau; s)\), based on numerical evidence. Numerical methods for calculating zeros of cubic \(L\)-function are presented by the author in [J. Math. Sci., New York 157, No. 4, 646--654; translation from Zap. Nauchn. Semin. POMI 357, 180-194 (2009; Zbl 1223.11109)] and other related papers.