Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 621-631, 2023
Sum of higher divisor function with prime summands
Yuchen Ding, Guang-Liang Zhou
Received May 14, 2022. Published online January 31, 2023.
Abstract: Let $l\geqslant2$ 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.
Keywords: higher divisor function; circle method; prime
Affiliations: Yuchen Ding, School of Mathematical Science, Yangzhou University, 180 Siwangting Rd, Wei Yang Qu, Yangzhou 225002, Jiangsu, P. R. China, e-mail: ycding@yzu.edu.cn; Guang-Liang Zhou (corresponding author), School of Mathematical Science, Tongji University, No.1239 Siping Road, Shanghai 200092, P. R. China, e-mail: guangliangzhou@126.com
