View source for Additive divisor problem

{{TEX|done}}

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|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 [[#References|[1]]]–[[#References|[3]]]. The additive divisor problem with  $  k _{1} = 2 $
and an arbitrary positive integer  $  k _{2} $
was solved using the [[Dispersion method|dispersion method]] of Yu.V. Linnik [[#References|[4]]].

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

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A.E. Ingham, "Some asymptotic formulae in the theory of numbers"  ''J. London Math. Soc. (1)'' , '''2'''  (1927)  pp. 202–208</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> T. Esterman, "On the representations of a number as the sum of two products"  ''Proc. London Math. Soc. (2)'' , '''31'''  (1930)  pp. 123–133</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> C. Hooly, "An asymptotic formula in the theory of numbers"  ''Proc. London Math. Soc. (3)'' , '''7'''  (1957)  pp. 396–413</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> Yu.V. Linnik, "The dispersion method in binary additive problems" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press  (1979)</TD></TR>
</table>