On the number of representations of certain integers as sums of 11 or 13 squares

Cooper, S. Barry

doi:10.1016/j.jnt.2003.06.001

Cited by 13 publications

(14 citation statements)

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“…We wish to make a remark about a conjecture made by S. Cooper [5,Conjecture 10.14]. It is interesting to note that the q-series appearing in that conjecture are modular forms of weight 2k on 0 (4).…”

Section: Discussionmentioning

confidence: 99%

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On the representation of integers as sums of an odd number of squares

Gun

Balasubramaniam

2008

Ramanujan J

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In a recent work, S. Cooper (J. Number Theory 103:135-162, 1988) conjectured a formula for r 2k+1 (p 2 ), the number of ways p 2 can be expressed as a sum of 2k + 1 squares. Inspired by this conjecture, we obtain an explicit formula for r 2k+1 (n 2 ), n ≥ 1.A classical problem in number theory is to give an explicit formula for the number of ways one can represent a non-negative integer n as a sum of k squares, where k is a positive integer. The study of r k (n) has a long history. For k = 2, 4, 6, 8, elegant formulae for r k (n) were found by Jacobi. A general formula for r k (n), when k is even was stated by Ramanujan [12]. It was proved by Mordell [10] and S. Cooper [3] gave an elementary proof using Ramanujan's 1 ψ 1 summation formula. It is also known that r k (n) can be expressed in terms of coefficients of Eisenstein series and cusp forms and Rankin [13] showed that the cusp form part is non-trivial for k > 8. Recently, combining a variety of methods, S. Milne [9] obtained formulas for r 4s 2 (n) and r 4s 2 +4s (n) Dedicated to Srinivasa Ramanujan.S. Gun · B. Ramakrishnan ( )

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Section: Discussionmentioning

confidence: 99%

“…For odd values of k a general formula is not known, though formulas for certain values of k are known. For example, the formulas for r k (n) or r k (n 2 ) are known for k = 1, 3,5,7,9,11,13. See [2,4,5,7,8,[14][15][16] for details.…”

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confidence: 99%

On the representation of integers as sums of an odd number of squares

Gun

Balasubramaniam

2008

Ramanujan J

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“…The following recurrence relations have been proved by Cooper (see [1,2]) r 9 (4t) = 129r 9 (t), t ≡ 5 (mod 8), r 9 (16t) = 16513r 9 (t), t ≡ 5 (mod 8), r 11 (4t) = 513r 11 (t), t ≡ 7 (mod 8), r 11 (16t) = 262657r 11 (t), t ≡ 7 (mod 8).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…where a(n) is the n-th Fourier coefficient of 16 and U(2) is the Hecke operator acting on M k (2). (2), where M k (N ) denotes the complex vector space of modular forms of weight k on 0 (N ).…”

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confidence: 99%

On special values of certain Dirichlet L-functions

Gun

Balasubramaniam

2008

Ramanujan J

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Let r k (n) denote the number of ways n can be expressed as a sum of k squares. Recently, S. Cooper (Ramanujan J. 6:469-490, 2002), conjectured a formula for r 9 (t), t ≡ 5 (mod 8), r 11 (t), t ≡ 7 (mod 8), where t is a square-free positive integer. In this note we observe that these conjectures follow from the works of Lomadze (Akad. Nauk Gruz. Tr. Tbil. Mat. Inst. Razmadze 17: 281-314, 1949; Acta Arith. 68(3):245-253, 1994). Further we express r 9 (t), r 11 (t) in terms of certain special values of Dirichlet L-functions. Combining these two results we get expressions for these special values of Dirichlet L-functions involving Jacobi symbols.

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“…(3.2)-(3.6) were proved in [4] and (3.7) was proved in [5]. All of the proofs used the theory of modular forms of half integer weight.…”

Section: Case 2 N ≡ 0 (Mod 4)mentioning

confidence: 99%