Geodesic
On generalized square-full numbers in an arithmetic progression
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 149-163
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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.
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.
DOI : 10.21136/CMJ.2021.0362-20
Classification : 11B50, 11N25, 11N69
Keywords: arithmetic progression; character sum; exponent pair method; square-full number 
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Sripayap, Angkana; Ruengsinsub, Pattira; Srichan, Teerapat. On generalized square-full numbers in an arithmetic progression. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 1, pp. 149-163. doi: 10.21136/CMJ.2021.0362-20

[1] Bateman, P. T., Grosswald, E.: On a theorem of Erdős and Szekeres. Ill. J. Math. 2 (1958), 88-98. | DOI | MR | JFM

[2] Chan, T. H.: Squarefull numbers in arithmetic progression. II. J. Number Theory 152 (2015), 90-104. | DOI | MR | JFM

[3] Chan, T. H., Tsang, K. M.: Squarefull numbers in arithmetic progressions. Int. J. Number Theory 9 (2013), 885-901. | DOI | MR | JFM

[4] Cohen, E.: Arithmetical notes. II: An estimate of Erdős and Szekeres. Scripta Math. 26 (1963), 353-356. | MR | JFM

[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 | MR | JFM

[7] Munsch, M.: Character sums over squarefree and squarefull numbers. Arch. Math. 102 (2014), 555-563. | DOI | MR | JFM

[8] Richert, H.-E.: Über die Anzahl Abelscher Gruppen gegebener Ordnung. I. Math. Z. 56 (1952), 21-32 German. | DOI | MR | JFM

[9] Richert, H.-E.: Über die Anzahl Abelscher Gruppen gegebener Ordnung. II. Math. Z. 58 (1953), 71-84 German. | DOI | MR | JFM

[10] Srichan, T.: Square-full and cube-full numbers in arithmetic progressions. Šiauliai Math. Semin. 8 (2013), 223-248. | MR | JFM

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