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Growth of the Selberg zeta-function. (English) Zbl 1466.11061

The paper under review studies the growth of the Selberg zeta-function \(Z(s)\) of a compact Riemman surface \(F\) of genus \(g\geq 2\). The main theorem is proved in the paper states that, for \(\sigma_2<1/2\), there exists a constant \(c=c(\sigma_2)\) such that for \(T\geq 2\) it holds the upper bound \[ |Z(\sigma+iT)|\leq \left \{ \begin{array}{ll} \exp\left(\frac{cT}{\log T}\right) & \text{ if \ }\sigma\geq \frac{1}{2},\\ \exp\left(\mathrm{area }(F)(\frac{1}{2}-\sigma)T+\frac{cT}{\log T}\right) & \text{ if \ }\sigma_2\leq \sigma\leq \frac{1}{2}. \end{array} \right. \] As a corollary, the author counts the number of non-trivial zeros of \(Z'(s)\) with imaginary part \(0 < t \leq T\) and gives an asymptotic formula with an error term \(O\left(\frac{T}{\log^{1/3}T}\right)\) as \(T \to \infty\), which improves the formula given by W. Luo with an error term \(O(T)\) in [Am. J. Math. 127, No. 5, 1141–1151 (2005; Zbl 1109.11040)].

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)

Citations:

Zbl 1109.11040
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References:

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