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 \[ \mu_0(\sigma)= \begin{cases} {1\over 2}-\sigma\quad &\text{if}\;\sigma< {1\over 2},\\ 0\quad &\text{if}\;\sigma\geq {1\over 2}.\end{cases} \]


11N80 Generalized primes and integers
Full Text: DOI


[1] Apostol, T.M., Introduction to analytic number theory, (1976), Springer · Zbl 0335.10001
[2] Bateman, P.T.; Diamond, H.G., Asymptotic distribution of Beurling’s generalized prime numbers, (), 152-212 · Zbl 0216.31403
[3] Beurling, A., Analyse de la loi asymptotique de la distribution des nombres premiers généralisés I, Acta math., 68, 255-291, (1937) · JFM 63.0138.01
[4] Hilberdink, T.W., Well-behaved Beurling primes and integers, J. number theory, 112, 332-344, (2005) · Zbl 1154.11335
[5] Titchmarsh, E.C., The theory of functions, (1986), Oxford Univ. Press · Zbl 0601.10026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.