An asymptotic formula related to the divisors of the quaternary quadratic form (Q2928541)

Let \(d(n)\) and \(\Lambda(n)\) stand for the divisor function and the Mangoldt function. This paper gives an asymptotic formula for the following sums: NEWLINE\[NEWLINE\begin{aligned}& \sum\limits_{m_{i}\leq x}d(m^{2}_{1}+m^{2}_{2}+m^{2}_{3}+m^{2}_{4}), \\ &\sum\limits_{m_{i}\leq x}\Lambda(m^{2}_{1}+m^{2}_{2}+m^{2}_{3}+m^{2}_{4}).\end{aligned}NEWLINE\]NEWLINE The proof uses the circle method.