On general prime number theorems with remainder
- Author
- Gregory Debruyne (UGent) and Jasson Vindas Diaz (UGent)
- Organization
- Abstract
- We show that for Beurling generalized numbers the prime number theorem in remainder form $$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right),$$ for all $n\in\mathbb{N}, is equivalent to (for some $a>0$) $$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right),$$ for all $n\in\mathbb{N}, where $N$ and $\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Cesaro sense.
- Keywords
- prime number theorem, zeta functions, Tauberian theorems for Laplace transforms, Beurling generalized primes, Beurling generalized integers
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU-8520069
- MLA
- Debruyne, Gregory, and Jasson Vindas Diaz. “On General Prime Number Theorems with Remainder.” Generalized Functions and Fourier Analysis, edited by Michael Oberguggenberger et al., vol. 260, Springer, 2017, pp. 79–94, doi:10.1007/978-3-319-51911-1_6.
- APA
- Debruyne, G., & Vindas Diaz, J. (2017). On general prime number theorems with remainder. In M. Oberguggenberger, J. Toft, J. Vindas Diaz, & P. Wahlberg (Eds.), Generalized functions and Fourier analysis (Vol. 260, pp. 79–94). https://doi.org/10.1007/978-3-319-51911-1_6
- Chicago author-date
- Debruyne, Gregory, and Jasson Vindas Diaz. 2017. “On General Prime Number Theorems with Remainder.” In Generalized Functions and Fourier Analysis, edited by Michael Oberguggenberger, Joachim Toft, Jasson Vindas Diaz, and Patrik Wahlberg, 260:79–94. Cham, Switzerland: Springer. https://doi.org/10.1007/978-3-319-51911-1_6.
- Chicago author-date (all authors)
- Debruyne, Gregory, and Jasson Vindas Diaz. 2017. “On General Prime Number Theorems with Remainder.” In Generalized Functions and Fourier Analysis, ed by. Michael Oberguggenberger, Joachim Toft, Jasson Vindas Diaz, and Patrik Wahlberg, 260:79–94. Cham, Switzerland: Springer. doi:10.1007/978-3-319-51911-1_6.
- Vancouver
- 1.Debruyne G, Vindas Diaz J. On general prime number theorems with remainder. In: Oberguggenberger M, Toft J, Vindas Diaz J, Wahlberg P, editors. Generalized functions and Fourier analysis. Cham, Switzerland: Springer; 2017. p. 79–94.
- IEEE
- [1]G. Debruyne and J. Vindas Diaz, “On general prime number theorems with remainder,” in Generalized functions and Fourier analysis, vol. 260, M. Oberguggenberger, J. Toft, J. Vindas Diaz, and P. Wahlberg, Eds. Cham, Switzerland: Springer, 2017, pp. 79–94.
@incollection{8520069,
abstract = {{We show that for Beurling generalized numbers the prime number theorem in remainder form
$$\pi(x) = \operatorname*{Li}(x) + O\left(\frac{x}{\log^{n}x}\right),$$
for all $n\in\mathbb{N}, is equivalent to (for some $a>0$)
$$N(x) = ax + O\left(\frac{x}{\log^{n}x}\right),$$
for all $n\in\mathbb{N}, where $N$ and $\pi$ are the counting functions of the generalized integers and primes, respectively. This was already considered by Nyman (Acta Math. 81 (1949), 299-307), but his article on the subject contains some mistakes. We also obtain an average version of this prime number theorem with remainders in the Cesaro sense.}},
author = {{Debruyne, Gregory and Vindas Diaz, Jasson}},
booktitle = {{Generalized functions and Fourier analysis}},
editor = {{Oberguggenberger, Michael and Toft, Joachim and Vindas Diaz, Jasson and Wahlberg, Patrik}},
isbn = {{9783319519104}},
issn = {{0255-0156}},
keywords = {{prime number theorem,zeta functions,Tauberian theorems for Laplace transforms,Beurling generalized primes,Beurling generalized integers}},
language = {{eng}},
pages = {{79--94}},
publisher = {{Springer}},
series = {{Operator Theory : Advances and Applications}},
title = {{On general prime number theorems with remainder}},
url = {{http://doi.org/10.1007/978-3-319-51911-1_6}},
volume = {{260}},
year = {{2017}},
}
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