Additive divisor problem

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


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


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].


[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)


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


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098