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A lower bound for the Lindelöf function associated to generalized integers. (English) Zbl 1159.11036

Author’s summary: We study generalized prime systems for which the integer counting function \(N_{\mathcal P}(x)\) is asymptotically well-behaved, in the sense that \(N_{\mathcal P}(x)=\rho x+O(x^\beta)\), where \(\rho\) is a positive constant and \(\beta<{1\over 2}\). For such systems, the associated zeta function \(\zeta_{\mathcal P}(s)\) has finite order for \(\sigma= \text{Re}\,s> \beta\), and the Lindelöf function \(\mu_{\mathcal P}(\sigma)\) may be defined. We prove that for all such systems, \(\mu_{\mathcal P}(\sigma)\geq \mu_0(\sigma)\) for \(\sigma>\beta\), where \[ \mu_0(\sigma)= \begin{cases} {1\over 2}-\sigma\quad &\text{if}\;\sigma< {1\over 2},\\ 0\quad &\text{if}\;\sigma\geq {1\over 2}.\end{cases} \]

MSC:

11N80 Generalized primes and integers
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References:

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