A variance for k-free numbers in arithmetic progressions of given modulus
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 317-360.

An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes, see [9]. We improve one of the error terms as far as one would expect to be able to go.

Une formule asymptotique pour la variance des nombres entiers sans facteur carré dans une progression arithmétique de raison donnée a été trouvée par Nunes dans [9]. Pour l’un des termes d’erreur, nous donnons la meilleure amélioration que l’on puisse espérer d’avoir.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1163
Classification: 11N25, 11N37, 11N56, 11N60, 11N64, 11N69, 11B25, 11B83
Keywords: $k$-free number, variance, arithmetic progression
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JTNB_2021__33_2_317_0,
     author = {Tomos Parry},
     title = {A variance for $k$-free numbers in arithmetic progressions of given modulus},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {317--360},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {2},
     year = {2021},
     doi = {10.5802/jtnb.1163},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1163/}
}
TY  - JOUR
AU  - Tomos Parry
TI  - A variance for $k$-free numbers in arithmetic progressions of given modulus
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 317
EP  - 360
VL  - 33
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1163/
DO  - 10.5802/jtnb.1163
LA  - en
ID  - JTNB_2021__33_2_317_0
ER  - 
%0 Journal Article
%A Tomos Parry
%T A variance for $k$-free numbers in arithmetic progressions of given modulus
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 317-360
%V 33
%N 2
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1163/
%R 10.5802/jtnb.1163
%G en
%F JTNB_2021__33_2_317_0
Tomos Parry. A variance for $k$-free numbers in arithmetic progressions of given modulus. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 2, pp. 317-360. doi : 10.5802/jtnb.1163. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1163/

[1] Valentin Blomer The average value of divisor sums in arithmetic progressions, Q. J. Math, Volume 59 (2008) no. 3, pp. 275-286 | DOI | MR | Zbl

[2] M. J. Croft Squarefree numbers in arithmetic progressions, Proc. Lond. Math. Soc., Volume 30 (1975), pp. 143-159 | DOI | MR | Zbl

[3] Ofir Gorodetsky; Kaisa Matomäki; Maksym Radziwiłł; Brad Rodgers On the variance of squarefree integers in short intervals and arithmetic progressions (2020) (https://arxiv.org/abs/2006.04060) | Zbl

[4] Christopher Hooley A note on square-free numbers in arithmetic progressions, Bull. Lond. Math. Soc., Volume 7 (1975), pp. 133-138 | DOI | MR | Zbl

[5] Yuk-Kam Lau; Lilu Zhao On a variance of Hecke eigenvalues in arithmetic progressions, J. Number Theory, Volume 132 (2012) no. 5, pp. 869-887 | MR | Zbl

[6] Pierre Le Boudec On the distribution of squarefree integers in arithmetic progressions, Math. Z., Volume 290 (2018) no. 1-2, pp. 421-429 | DOI | MR | Zbl

[7] Hugh L. Montgomery Topics in Multiplicative Number Theory, Lecture Notes in Mathematics, 227, Springer, 1971 | MR | Zbl

[8] Hugh L. Montgomery; Robert C. Vaughan Multiplicative Number Theory I. Classical Theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, 2007 | MR | Zbl

[9] Ramon M. Nunes Squarefree numbers in arithmetic progressions, J. Number Theory, Volume 153 (2015), pp. 1-36 | DOI | MR | Zbl

[10] Francesco Pappalardi A survey on k-freeness, Number theory. Proceedings of the international conference on analytic number theory with special emphasis on L-functions held at the Institute of Mathematical Sciences (Chennai, India, 2002) (Ramanujan Mathematical Society Lecture Notes Series), Volume 1, Ramanujan Mathematical Society, 2005 | MR | Zbl

[11] Edward C. Titchmarsh The Theory of the Riemann Zeta-function, Oxford Science Publications, Clarendon Press, 1986

[12] Robert C. Vaughan A variance for k-free numbers in arithmetic progressions, Proc. Lond. Math. Soc., Volume 91 (2005) no. 3, pp. 573-597 | DOI | MR | Zbl

Cited by Sources: