# Difference between revisions of "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 . 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 .

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