Writing integers as sums of products
Author:
Charles E. Chace
Journal:
Trans. Amer. Math. Soc. 345 (1994), 367379
MSC:
Primary 11P55; Secondary 11D85, 11N37
MathSciNet review:
1257641
Fulltext 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 HardyLittlewood 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 (92k:11096)
 [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. HeathBrown and D. E. Freeman; Edited and prepared for
publication by T. D. Browning. MR 2152164
(2006a:11129)
 [E1]
T. Estermann, On the representations of a number as the sums of three products, Proc. London Math. Soc. (2) 29 (1929), 453478.
 [E2]
, On the representations of a number as the sum of two products, Proc. London Math. Soc. (2) 31 (1930), 123133.
 [HW]
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford, 1984.
 [H]
D.
R. HeathBrown, The divisor function 𝑑₃(𝑛)
in arithmetic progressions, Acta Arith. 47 (1986),
no. 1, 29–56. MR 866901
(88a:11088)
 [L]
Ju.
V. Linnik, The dispersion method in binary additive problems,
Translated by S. Schuur, American Mathematical Society, Providence, R.I.,
1963. MR
0168543 (29 #5804)
 [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 (87a:11095)
 [T]
E.
C. Titchmarsh, The theory of the Riemann zetafunction, 2nd
ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited
and with a preface by D. R. HeathBrown. MR 882550
(88c:11049)
 [V]
R.
C. Vaughan, The HardyLittlewood method, Cambridge Tracts in
Mathematics, vol. 80, Cambridge University Press, CambridgeNew York,
1981. MR
628618 (84b:10002)
 [C]
 C. E. Chace, The divisor problem for arithmetic progressions with small modulus, Acta Arith. 61 (1992), 3550. MR 1153920 (92k:11096)
 [D]
 H. Davenport, Analytic methods for Diophantine equations and Diophantine inequalities, Campus Publishers, Univ. of Michigan, Ann Arbor, Michigan, 1962. MR 2152164 (2006a:11129)
 [E1]
 T. Estermann, On the representations of a number as the sums of three products, Proc. London Math. Soc. (2) 29 (1929), 453478.
 [E2]
 , On the representations of a number as the sum of two products, Proc. London Math. Soc. (2) 31 (1930), 123133.
 [HW]
 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Clarendon Press, Oxford, 1984.
 [H]
 D. R. HeathBrown, The divisor function in arithmetic progressions, Acta Arith. 47 (1987), 2956. MR 866901 (88a:11088)
 [L]
 Ju. V. Linnik, The dispersion method in binary additive problems, Transl. Math. Monos., vol. 4, Amer. Math. Soc., Providence, R.I., 1983. MR 0168543 (29:5804)
 [M]
 K. Matsumoto, A remark on Smith's results on a divisor, Nagoya Math. J. 98 (1985), 3742. MR 792769 (87a:11095)
 [T]
 E. C. Titchmarsh, The theory of the Riemann zetafunction, 2nd ed. revised by D. R. HeathBrown, Clarendon Press, Oxford, 1986. MR 882550 (88c:11049)
 [V]
 R. C. Vaughan, The HardyLittlewood method, Cambridge Tracts in Math., vol. 80, Cambridge Univ. Press, 1981. MR 628618 (84b:10002)
Similar Articles
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:
http://dx.doi.org/10.1090/S00029947199412576413
PII:
S 00029947(1994)12576413
Keywords:
Additive divisor problem,
HardyLittlewood method
Article copyright:
© Copyright 1994
American Mathematical Society
