Sum of higher divisor function with prime summands

@article{Ding2023,
abstract = {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.},
author = {Ding, Yuchen, Zhou, Guang-Liang},
journal = {Czechoslovak Mathematical Journal},
keywords = {higher divisor function; circle method; prime},
language = {eng},
number = {2},
pages = {621-631},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Sum of higher divisor function with prime summands},
url = {http://eudml.org/doc/299504},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Ding, Yuchen
AU - Zhou, Guang-Liang
TI - Sum of higher divisor function with prime summands
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 621
EP - 631
AB - 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.
LA - eng
KW - higher divisor function; circle method; prime
UR - http://eudml.org/doc/299504
ER -