Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
arithmetic progression; character sum; exponent pair method; square-full number
arithmetic progression; character sum; exponent pair method; square-full number
Summary:
Let $a$ and $b\in \mathbb {N}$. Denote by $R_{a,b}$ the set of all integers $n>1$ whose canonical prime representation $n=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_r^{\alpha _r}$ has all exponents $\alpha _i$ $(1\leq i\leq r)$ being a multiple of $a$ or belonging to the arithmetic progression $at+b$, $t\in \mathbb {N}_0:=\mathbb {N}\cup \{0\}$. All integers in $R_{a,b}$ are called generalized square-full integers. Using the exponent pair method, an upper bound for character sums over generalized square-full integers is derived. An application on the distribution of generalized square-full integers in an arithmetic progression is given.
References:
[1] Bateman, P. T., Grosswald, E.: On a theorem of Erdős and Szekeres. Ill. J. Math. 2 (1958), 88-98. DOI 10.1215/ijm/1255380836 | MR 0095804 | Zbl 0079.07104
[2] Chan, T. H.: Squarefull numbers in arithmetic progression. II. J. Number Theory 152 (2015), 90-104. DOI 10.1016/j.jnt.2014.12.019 | MR 3319056 | Zbl 1398.11124
[3] Chan, T. H., Tsang, K. M.: Squarefull numbers in arithmetic progressions. Int. J. Number Theory 9 (2013), 885-901. DOI 10.1142/S1793042113500048 | MR 3060865 | Zbl 1290.11130
[4] Cohen, E.: Arithmetical notes. II: An estimate of Erdős and Szekeres. Scripta Math. 26 (1963), 353-356. MR 0162770 | Zbl 0122.04901
[5] Erdős, P., Szekeres, S.: Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem. Acta Szeged 7 (1934), 95-102 German \99999JFM99999 60.0893.02.
[6] Liu, H., Zhang, T.: On the distribution of square-full numbers in arithmetic progressions. Arch. Math. 101 (2013), 53-64. DOI 10.1007/s00013-013-0525-0 | MR 3073665 | Zbl 1333.11094
[7] Munsch, M.: Character sums over squarefree and squarefull numbers. Arch. Math. 102 (2014), 555-563. DOI 10.1007/s00013-014-0658-9 | MR 3227477 | Zbl 1297.11097
[8] Richert, H.-E.: Über die Anzahl Abelscher Gruppen gegebener Ordnung. I. Math. Z. 56 (1952), 21-32 German. DOI 10.1007/BF01215034 | MR 0050577 | Zbl 0046.25002
[9] Richert, H.-E.: Über die Anzahl Abelscher Gruppen gegebener Ordnung. II. Math. Z. 58 (1953), 71-84 German. DOI 10.1007/BF01174132 | MR 0054594 | Zbl 0050.02302
[10] Srichan, T.: Square-full and cube-full numbers in arithmetic progressions. Šiauliai Math. Semin. 8 (2013), 223-248. MR 3265055 | Zbl 1318.11120
Search
Partner of
