No Access

THE CONVOLUTION SUM

KENNETH S. WILLIAMS

Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada

Search for more papers by this author

https://doi.org/10.1142/S1793042105000091Cited by:28 (Source: Crossref)

Abstract

The evaluation of the sum ∑_m<n/9σ(m)σ(n - 9m) is carried out for all positive integers n. This evaluation is used to detemine the number of solutions to

in integers x₁, x₂, x₃, x₄, x₅, x₆, x₇, x₈.

Keywords:

AMSC: 11A25, 11E20, 11E25

References

B. C. Berndt , Ramanujan's Notebooks, Part II ( Springer-Verlag , New York, Berlin , 1989 ) . Crossref, Google Scholar
B. C. Berndt , Ramanujan's Notebooks, Part V ( Springer-Verlag , New York, Berlin , 1998 ) . Crossref, Google Scholar
B. C. Berndt, S. Bhargava and F. G. Garvan, Trans. Amer. Math. Soc. 347, 4163 (1995). Web of Science, Google Scholar
M. Besge, J. Math. Pures Appl. 7, 256 (1862). Google Scholar
E. T. Copson , An Introduction to the Theory of Functions of a Complex Variable ( Oxford University Press , 1960 ) . Google Scholar
J. W. L. Glaisher, Mess. Math. 14, 156 (1885). Google Scholar
J. W. L. Glaisher , Mathematical Papers, 1883–1885 ( Cambridge University Press , 1885 ) . Google Scholar
J. G. Huardet al., Number Theory for the Millennium II, eds. M. A. Bennettet al. (A. K. Peters, Natick, Massachusetts, 2002) pp. 229–274. Google Scholar
D. B. Lahiri, Bull. Calcutta Math. Soc. 38, 193 (1946). Google Scholar
D. B. Lahiri, Bull. Calcutta Math. Soc. 39, 33 (1947). Google Scholar
J. Lützen , Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics ( Springer-Verlag , Berlin, Heidelberg , 1998 ) . Google Scholar
G. Melfi, Some Problems in Elementary Number Theory and Modular Forms, PhD thesis, University of Pisa (1998) . Google Scholar
G. Melfi, Number Theory, eds. K. Györy, A. Pethö and V. Sós (de Gruyter, Berlin, 1998) pp. 371–382. Google Scholar
L. J. Mordell, Proc. Cambridge Philos. Soc. 19, 117 (1917). Google Scholar
S. Ramanujan, Trans. Cambridge Phil. Soc. 22, 159 (1916). Google Scholar
S. Ramanujan , Collected Papers ( AMS Chelsea Publishing , Providence, Rhode Island , 2000 ) . Google Scholar
K. S. Williams, Math. Proc. Cambridge Philos. Soc. 137, 519 (2004). Crossref, Web of Science, Google Scholar