## 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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