Additive divisor problem

The problem of finding asymptotic values for sums of the form:

$$\tag{1} \left . { {\sum _ {m \leq n} \tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( m + a ) ,} \atop {\sum _ {m < n}\tau _{ {k _ 1}} ( m ) \tau _{ {k _ 2}} ( n - m ) ,}} \right \}$$

where $\tau _{k} (m)$ is the number of different factorizations of an integer $m$ in $k$ factors, counted according to multiplicity. Here $k _{1}$ and $k _{2}$ are integers $\geq 2$, $a$ is a fixed integer different from zero and $n$ is a sufficiently large number. In particular $\tau _{2} (m) = \tau (m)$ is the number of divisors of the number $m$. Sums of the form (1) express the number of solutions of the equations

$$\tag{2} x _{1} \dots x _{ {k _ 2}} \ - \ y _{1} \dots y _{ {k _ 1}} \ = \ a ,$$

$$\tag{3} x _{1} \dots x _{ {k _ 1}} \ + \ y _{1} \dots y _{ {k _ 2}} \ = \ n ,$$

respectively. Particular cases of the additive divisor problem ( $k _{1} = k _{2} =2$, $k _{1} = 2$ and $k _{2} = 3$) are considered in [1]–[3]. The additive divisor problem with $k _{1} = 2$ and an arbitrary positive integer $k _{2}$ was solved using the dispersion method of Yu.V. Linnik [4].

Comments

The function $\tau_{2} (m) = \tau (m)$ is also denoted by $d (m)$ or $\sigma_{0} (m)$, cf. [a1], Sect. 16.7.

References

[1]	A.E. Ingham, "Some asymptotic formulae in the theory of numbers" J. London Math. Soc. (1) , 2 (1927) pp. 202–208
[2]	T. Esterman, "On the representations of a number as the sum of two products" Proc. London Math. Soc. (2) , 31 (1930) pp. 123–133
[3]	C. Hooly, "An asymptotic formula in the theory of numbers" Proc. London Math. Soc. (3) , 7 (1957) pp. 396–413
[4]	Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc. (1963) (Translated from Russian)
[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979)