Article
| Title: | Sum of higher divisor function with prime summands (English) |
| Author: | Ding, Yuchen |
| Author: | Zhou, Guang-Liang |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 621-631 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | Let $l\geqslant 2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $$ \sum _{1\leqslant n_{1},n_{2},\ldots ,n_{l}\leqslant x^{1/2}}\tau _{k}(n_{1}^{2}+n_{2}^{2}+\cdots +n_{l}^{2}), $$ where $\tau _{k}(n)$ represents the $k$th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $$ \sum _{1\leqslant p_{1},p_{2},\ldots ,p_{l}\leqslant x}\tau _{k}(p_{1}+p_{2}+\cdots +p_{l}), $$ where $p_1,p_2,\dots ,p_l$ are prime variables. (English) |
| Keyword: | higher divisor function |
| Keyword: | circle method |
| Keyword: | prime |
| MSC: | 11A41 |
| MSC: | 11N37 |
| MSC: | 11P55 |
| idZBL: | Zbl 07729528 |
| idMR: | MR4586915 |
| DOI: | 10.21136/CMJ.2023.0206-22 |
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| Date available: | 2023-05-04T17:52:16Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151678 |
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| Reference: | [1] Bellman, R.: Ramanujan sums and the average value of arithmetic functions.Duke Math. J. 17 (1950), 159-168. Zbl 0037.31202, MR 0035312, 10.1215/S0012-7094-50-01717-0 |
| Reference: | [2] Calderón, C., Velasco, M. J. de: On divisors of a quadratic form.Bol. Soc. Bras. Mat., Nova Sér. 31 (2000), 81-91. Zbl 1031.11057, MR 1754956, 10.1007/BF01377596 |
| Reference: | [3] Chace, C. E.: The divisor problem for arithmetic progressions with small modulus.Acta Arith. 61 (1992), 35-50. Zbl 0726.11056, MR 1153920, 10.4064/aa-61-1-35-50 |
| Reference: | [4] Chace, C. E.: Writing integers as sums of products.Trans. Am. Math. Soc. 345 (1994), 367-379. Zbl 0811.11061, MR 1257641, 10.1090/S0002-9947-1994-1257641-3 |
| Reference: | [5] Gafurov, N.: On the sum of the number of divisors of a quadratic form.Dokl. Akad. Nauk Tadzh. SSR 28 (1985), 371-375 Russian. Zbl 0586.10023, MR 0819343 |
| Reference: | [6] Gafurov, N.: On the number of divisors of a quadratic form.Proc. Steklov Inst. Math. 200 (1993), 137-148. Zbl 0790.11073, MR 1143362 |
| Reference: | [7] Guo, R., Zhai, W.: Some problems about the ternary quadratic form $m_1^2+m_2^2+m_3^2$.Acta Arith. 156 (2012), 101-121. Zbl 1270.11099, MR 2997561, 10.4064/aa156-2-1 |
| Reference: | [8] Hooley, C.: On the representation of a number as the sum of a square and a product.Math. Z. 69 (1958), 211-227. Zbl 0081.03904, MR 0096624, 10.1007/BF01187402 |
| Reference: | [9] Hooley, C.: On the number of divisors of a quadratic polynomials.Acta Math. 110 (1963), 97-114. Zbl 0116.03802, MR 0153648, 10.1007/BF02391856 |
| Reference: | [10] Hu, G., Lü, G.: Sums of higher divisor functions.J. Number Theory 220 (2021), 61-74. Zbl 1466.11065, MR 4177535, 10.1016/j.jnt.2020.08.009 |
| Reference: | [11] Ingham, A. E.: Some asymptotic formulae in the theory of numbers.J. Lond. Math. Soc. 2 (1927), 202-208 \99999JFM99999 53.0157.01. MR 1574426, 10.1112/jlms/s1-2.3.202 |
| Reference: | [12] Montgomery, H. L., Vaughan, R. C.: The exceptional set in Goldbach's problem.Acta Arith. 27 (1975), 353-370. Zbl 0301.10043, MR 0374063, 10.4064/aa-27-1-353-370 |
| Reference: | [13] Nathanson, M. B.: Additive Number Theory: The Classical Bases.Graduate Texts in Mathematics 164. Springer, New York (1996). Zbl 0859.11002, MR 1395371, 10.1007/978-1-4757-3845-2 |
| Reference: | [14] Shiu, P.: A Brun-Titchmarsh theorem for multiplicative functions.J. Reine Angew. Math. 313 (1980), 161-170. Zbl 0412.10030, MR 0552470, 10.1515/crll.1980.313.161 |
| Reference: | [15] Sun, Q., Zhang, D.: Sums of the triple divisor function over values of a ternary quadratic form.J. Number Theory 168 (2016), 215-246. Zbl 1396.11117, MR 3515816, 10.1016/j.jnt.2016.04.010 |
| Reference: | [16] Zhao, L.: The sum of divisors of a quadratic form.Acta Arith. 163 (2014), 161-177. Zbl 1346.11056, MR 3200169, 10.4064/aa163-2-6 |
| Reference: | [17] Zhou, G.-L., Ding, Y.: Sums of the higher divisor function of diagonal homogeneous forms.Ramanujan J. 59 (2022), 933-945. Zbl 1498.11200, MR 4496536, 10.1007/s11139-022-00579-z |
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