# Additive divisor problem

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

(1) |

where is the number of different factorizations of an integer in factors, counted according to multiplicity. Here and are integers , is a fixed integer different from zero and is a sufficiently large number. In particular is the number of divisors of the number . Sums of the form (1) express the number of solutions of the equations

(2) |

(3) |

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

#### 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) |

#### Comments

The function is also denoted by or , cf. [a1], Sect. 16.7.

#### References

[a1] | G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1979) |

**How to Cite This Entry:**

Additive divisor problem. B.M. Bredikhin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Additive_divisor_problem&oldid=17105