On the representations of a positive integer by certain classes of quadratic forms in eight variables

2016/07/16 by B. Ramakrishnan, Ramakrishnan, B., Brundaban Sahu +3
Mathematics · #11F11 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11F25 #Secondary 11A25

paper · pdf · doi:10.48550/arxiv.1607.04764

openalex publication_date 2016/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form a₁x₁² + a₂ x₂² + a₃ x₃² + a₄ x₄² + b₁(x₅²+x₅x₆ + x₆²) + b₂(x₇²+x₇x₈ + x₈²), where a₁≤ a₂≤ a₃≤ a₄, b₁≤ b₂ and a_i's ∈ \1,2,3\, b_i's ∈ \1,2,4\. We also determine formulas for the number of representations of a positive integer by the quadratic forms (x₁²+x₁x₂+x₂²) + c₁(x₃²+x₃x₄+x₄²) + c₂(x₅²+x₅x₆+x₆²) + c₃(x₇²+x₇x₈+x₈²), where c₁,c₂,c₃∈ \1,2,4,8\, c₁≤ c₂≤ c₃.

Citations

• On the number of representations by certain octonary quadratic forms with coefficients 1, 2, 3, 4 and 6
• On Modular Forms of Half Integral Weight

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• Representations of an integer by some quaternary and octonary quadratic forms
• Representations by octonary quadratic forms with coefficients 1, 2, 3 or 6
• A Study on the Number of Representations of an Integer into Certain Quadratic Forms
• Representations by Quaternary Quadratic Forms with Coefficients 1,\n 2, 5 or 10
• Representation of integers by a family of cubic forms in seven variables II