Writing integers as sums of products

Author:
Charles E. Chace

Journal:
Trans. Amer. Math. Soc. **345** (1994), 367-379

MSC:
Primary 11P55; Secondary 11D85, 11N37

DOI:
https://doi.org/10.1090/S0002-9947-1994-1257641-3

MathSciNet review:
1257641

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we obtain an asymptotic expression for the number of ways of writing an integer *N* as a sum of *k* products of *l* factors, valid for and . The proof is an application of the Hardy-Littlewood method, and uses recent results from the divisor problem for arithmetic progressions.

**[C]**Charles E. Chace,*The divisor problem for arithmetic progressions with small modulus*, Acta Arith.**61**(1992), no. 1, 35–50. MR**1153920****[D]**H. Davenport,*Analytic methods for Diophantine equations and Diophantine inequalities*, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2005. With a foreword by R. C. Vaughan, D. R. Heath-Brown and D. E. Freeman; Edited and prepared for publication by T. D. Browning. MR**2152164****[E1]**T. Estermann,*On the representations of a number as the sums of three products*, Proc. London Math. Soc. (2)**29**(1929), 453-478.**[E2]**-,*On the representations of a number as the sum of two products*, Proc. London Math. Soc. (2)**31**(1930), 123-133.**[HW]**G. H. Hardy and E. M. Wright,*An introduction to the theory of numbers*, 5th ed., Clarendon Press, Oxford, 1984.**[H]**D. R. Heath-Brown,*The divisor function 𝑑₃(𝑛) in arithmetic progressions*, Acta Arith.**47**(1986), no. 1, 29–56. MR**866901****[L]**Ju. V. Linnik,*The dispersion method in binary additive problems*, Translated by S. Schuur, American Mathematical Society, Providence, R.I., 1963. MR**0168543****[M]**Kohji Matsumoto,*A remark on Smith’s result on a divisor problem in arithmetic progressions*, Nagoya Math. J.**98**(1985), 37–42. MR**792769****[T]**E. C. Titchmarsh,*The theory of the Riemann zeta-function*, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR**882550****[V]**R. C. Vaughan,*The Hardy-Littlewood method*, Cambridge Tracts in Mathematics, vol. 80, Cambridge University Press, Cambridge-New York, 1981. MR**628618**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
11P55,
11D85,
11N37

Retrieve articles in all journals with MSC: 11P55, 11D85, 11N37

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1257641-3

Keywords:
Additive divisor problem,
Hardy-Littlewood method

Article copyright:
© Copyright 1994
American Mathematical Society