Abstract
In this paper, we consider pairs of a prime and a prime power with a fixed difference. We prove an average result on the distribution of such pairs. This is a partial improvement of the result of Bauer (Acta Arith. 85:99–118, 1998).
Similar content being viewed by others
References
Baier, S., Zhao, L.: Primes in quadratic progressions on average. Math. Ann. 338(4), 963–982 (2007)
Baier, S., Zhao, L.: On primes in quadratic progressions. Int. J. Number Theory 5(6), 1017–1035 (2009)
Bateman, P.T., Horn, R.A.: A heuristic asymptotic formula concerning the distribution of prime numbers. Math. Comp. 16(79), 363–367 (1962)
Bauer, C.: On the sum of a prime and the \(k\)-th power of a prime. Acta Arith. 85(2), 99–118 (1998)
Bauer, C.: On the exceptional set for the sum of a prime and the \(k\)-th power of a prime. Stud. Sci. Math. Hung. 35, 291–330 (1999)
Foo, T., Zhao, L.: On primes represented by cubic polynomials. Math. Z. 274(1–2), 323–340 (2013)
Gallagher, P.X.: A large sieve density estimate near \(\sigma =1\). Invent. Math. 11(4), 329–339 (1970)
Iwaniec, H., Kowalski, E.: Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)
Kawada, K.: A zero density estimate for Dedekind zeta functions of pure extension fields. Tsukuba J. Math. 22(2), 357–369 (1998)
Landau, E.: Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Rie- mannschen Zetafunktion. In: 5th International Congress of Mathematicians (ICM) (1913). https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1912.1/ICM1912.1.ocr.pdf (Reprinted from Jahresbericht der Deutschen Mathematiker-Vereinigung 21, 208–228 (1912). https://eudml.org/doc/urn:eudml:doc:145337)
Liu, J.Y., Zhan, T.: On a theorem of Hua. Arch. Math. 69(5), 375–390 (1997)
Matomäki, K., Radziwiłł, M., Tao, T.: Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. Proc. Lond. Math. Soc 118(2), 284–350 (2019)
Mikawa, H.: On prime twins. Tsukuba J. Math. 15(1), 19–29 (1991)
Mikawa, H.: On the sum of a prime and a square. Tsukuba J. Math. 17(2), 299–310 (1993)
Mikawa, H.: On the sum of three squares of primes. In: Motohashi, Y. (ed.) Analytic number theory, pp. 253–264. Cambridge University Press, London (1997)
Mikawa, H., Peneva, T.: Sums of five cubes of primes. Stud. Sci. Math. Hungar. 46(3), 345–354 (2009)
Montgomery, H.L.: Topics in multiplicative number theory. Lecture notes in mathematics, vol. 227. Springer, Berlin (1971)
Montgomery, H.L., Vaughan, R.C.: Multiplicative number theory I. Classical theory, Cambridge studies in advanced mathematics, vol. 97. Cambridge University Press, Cambridge (2006)
Perelli, A., Pintz, J.: On the exceptional set for Goldbach’s problem in short intervals. J. Lond. Math. Soc. 47(1), 41–49 (1993)
Perelli, A., Pintz, J.: Hardy–Littlewood numbers in short intervals. J. Number Theory 54(2), 297–308 (1995)
Perelli, A., Zaccagnini, A.: On the sum of a prime and a \(k\)-th power. Izv. Ross. Akad. Nauk Ser. Math. 59(1), 185–200 (1995)
Tatuzawa, T.: On the number of the primes in an arithmetic progression. Jpn. J. Math. 21, 93–111 (1951)
Acknowledgements
The author would like to thank Kohji Matsumoto, Hiroshi Mikawa, Koichi Kawada and Alberto Perelli for their helpful comments and suggestions. This work was supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP16J00906).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Suzuki, Y. On prime vs. prime power pairs. Math. Z. 295, 681–710 (2020). https://doi.org/10.1007/s00209-019-02384-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02384-9