Zeros of the Estermann zeta function (Q2852273)

The Estermann zeta-function \(E(s; \lambda, \alpha)\), \(s=\sigma+it\), with parameters \(\alpha\in\mathbb{C}\) and \(\lambda\in \mathbb{R}\) is defined, for \(\sigma>\max (1+\text{Re}(\alpha), 1)\), by NEWLINE\[NEWLINE E(s; \lambda, \alpha)=\sum_{n=1}^\infty {\sigma_\alpha(n)\over n^s}\exp(2\pi i n\lambda), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \sigma_\alpha(n)=\sum_{d| n}d^\alpha NEWLINE\]NEWLINE is the generalized divisor function, and by analytic continuation elsewhere. For irrational \(\lambda\), analytic continuation of \(E(s; \lambda, \alpha)\) remains an open problem.NEWLINENEWLINEThe authors study the continuous dependency of zeros of the function \(E(s; \lambda, \alpha)\) with respect to the parameters \(\alpha\) and \(\lambda\). The notion of the continuous dependency of zeros is defined as follows. Let \(I\subset \mathbb{R}\) be an interval, and let \(S\) be a dense subset of \(I\). Suppose that a function \(f(s, \beta)\) is analytic in \(s\) for each \(\beta\in S\), and that \(s=\rho_0\) is a zero of multiplicity \(m\) of \(f(s, \beta_0)\), \(\beta_0\in S\). Then the \(\rho_0\) is called \(S\)-continuous at \(\beta_0\) if, for every sufficiently small open disk \(D\) with center at \(\rho_0\) in which the function \(f(s, \beta_0)\) has no other zeros, except for \(\rho_0\), there exists a positive number \(\delta=\delta(D)\) such that each function \(f(s, \beta)\), where \(\beta\in(\beta_0-\delta, \beta_0+\delta)\cap S\), has exactly \(m\) zeros (counted with multiplicity) in the disk \(D\).NEWLINENEWLINEFirst, the authors obtain that each zero of the function \(E(s; \lambda, \alpha)\) is \(S\)-continuous at every value of the parameter \(\alpha\in S=(-1, 0)\). Furthermore, they prove that each zero of \(E(s; \lambda, \alpha)\), \(\text{Re}( \alpha)\leq 0\), in the region \(\sigma>1\) is \(\mathbb{R}\)-continuous at every \(\lambda\in \mathbb{R}\).NEWLINENEWLINEThe main result of the paper involves the set NEWLINE\[NEWLINE \mathbb{Q}_{m, n}=\left\{{m^2q\over r}: \, (mnq)^2\equiv 1 ({\text{mod}}\, r), q\in \mathbb{Z}, r\in \mathbb{N}\right\}, NEWLINE\]NEWLINE with \(m, n \in \mathbb{N}\). It is showed that each set \(\mathbb{Q}_{m, n}\) is a dense subset of \(\mathbb{Q}\). Suppose that \(E(\rho; \lambda, \alpha)=0\), where \(\lambda \in \mathbb{Q}\), \(-1<\alpha\leq 0\) and \(\text{Re} (\rho)<\alpha\). Then the authors prove an interesting statement that the zero \(\rho\) is not \(\mathbb{Q}\)-continuous at \(\lambda\in \mathbb{Q}\), however, for \(\lambda\in \mathbb{Q}_{m, n}\), the zero \(\rho\) is \(\mathbb{Q}_{m, n}\)-continuous at \(\lambda\).NEWLINENEWLINEAlso, the authors consider a horizontal distribution of zeros of the function \(E(s; {k\over l}, \alpha)\) with relatively coprimes \(k, l\in \mathbb{N}\). Let \(N(T; {k\over l}, \alpha)\) be the number of nontrivial zeros \(\rho=\beta+i\gamma\) of \(E(s; {k\over l}, \alpha)\) satisfying \(|\gamma|\leq T\), and let \(N(\sigma, T; {k\over l}, \alpha)\) be the number of nontrivial zeros \(\rho\) of \(E(s; {k\over l}, \alpha)\) with \(\beta>\sigma\) and \(|\gamma|\leq T\). Then the authors show that if \(\text{Re} (\alpha)<0\), \(0<\varepsilon<{1\over 2}\) and \({1\over 2}+\varepsilon\text{Re}(\alpha)\leq \sigma<{1\over 2}\), then NEWLINE\[NEWLINE\lim\limits_{T\to\infty}{N(\sigma, T; {k\over l}, \alpha) \over N(T; {k\over l}, \alpha)}\leq {1\over 2}+\varepsilon. NEWLINE\]NEWLINE The latter result supports their conjecture that a positive proportion of nontrivial zeros of \(E(s; {k\over l}, \alpha)\) is clustered around the lines \(\sigma={1\over 2}\) and \(\sigma={1\over 2}+\text{Re} (\alpha)\).