Second moment of the Beurling zeta-function
Let \(1 < p_{1} \leq p_{2} \leq \dots \rightarrow \infty\) be a generalized prime number (g-prime) system and let \(1 = \nu_{1} < \nu_{2} \leq \dots\) be the increasing sequence of power-products of g-primes with \(a_{n}\) the corresponding multiplicity of \(\nu_{n}\). The g-prime system has a natural zeta function defined through \(\zeta(s) = \sum_{n} a_{n} \nu_{n}^{-s}\).\par The main theorem of the paper states that if the g-integer counting function satisfies \(A(x) := \sum_{\nu_{n} \leq x} a_{n} = \rho x + O(x \exp(-c \sqrt{\log x \log \log x}))\) for certain \(c > 0\) and some \(\rho\), and the \(g\)-prime system satisfies a certain gap condition then the zeta function admits a continuous extension to \(\operatorname{Re} (s) \geq 1, s \neq 1\) and moreover \[ \lim_{T \rightarrow \infty} \int^{T}_{1} |\zeta(1 + it)|^{2} \mathrm{d}t = \sum_{n = 1}^{\infty} \frac{a_{n}^{2}}{\nu_{n}^{2}}. \] A similar statement is also provided for \(1/\zeta(1+it)\) under additional estimates on the natural generalization of the summatory function of the Mobius function. One also provides a similar formula for integrating \(\zeta\) on the line \(\operatorname{Re} (s) = \sigma\) for \(\sigma > 1/2\) which is valid if the error term in the hypothesis for \(A(x)\) is strengthened to \(O(x^{\eta})\) where \(\eta < 2\sigma - 1\).
scientific article; zbMATH DE number 7139398
scientific article