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The Ramanujan Journal

, Volume 34, Issue 1, pp 11–38 | Cite as

Representation of integers by a family of cubic forms

  • Manoj Verma
Article

Abstract

Under certain conditions on the coefficients, we derive asymptotic formulas for the number of representations of zero and of large positive integers by the cubic forms that can be written as \(L_{1}(x_{1},x_{2},x_{3}) Q_{1}(x_{1},x_{2},x_{3})+ L_{2}(x_{4},x_{5},x_{6}) Q_{2}(x_{4},x_{5}, x_{6}) + a_{7} x_{7}^{3}\), where L 1 and L 2 are linear forms, Q 1 and Q 2 are quadratic forms, and a 7 is a nonzero integer.

Keywords

Circle method Representation problems Cubic forms 

Mathematics Subject Classification

11D45 11D85 11P55 

Notes

Acknowledgements

I would like to thank my adviser Prof. Robert C. Vaughan for his supervision and help throughout this work and the referee for useful comments and suggestions.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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