Sur la lacunarité des puissances de η

Serre, Jean-Pierre

doi:10.1017/s0017089500006194

Cited by 97 publications

(79 citation statements)

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“…For lacunarity, the reader should consult [9], [24], [25] The case k = 12: the arithmetic density of their non-zero Fourier coefficients is 0) this theorem implies that 6lo(n)= i ~-6a4,x(4n + 5) almost always.…”

Section: Rll2( 8"c) = Q3inl~)i ( ] -Qsn)212mentioning

confidence: 99%

“…lo(4z) , = oUnfortunately, qS°dl°(q4) is not an Eisenstein series; it is a linear combination of an Eisenstein series and a cusp with complex multiplication[24].Let F(0 be the modular form with complex multiplication by Q(i) defined as F(r) = qa(r)r12(2z)q4(4z) = ~ a(n)q" = qx(n) = ~ z(d)d 4.a[nThe character ;~ is the same Dirichlet character mod 4 which occurred when k = 6. lo(4z) , = oUnfortunately, qS°dl°(q4) is not an Eisenstein series; it is a linear combination of an Eisenstein series and a cusp with complex multiplication[24].Let F(0 be the modular form with complex multiplication by Q(i) defined as F(r) = qa(r)r12(2z)q4(4z) = ~ a(n)q" = qx(n) = ~ z(d)d 4.a[nThe character ;~ is the same Dirichlet character mod 4 which occurred when k = 6.…”

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confidence: 99%

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On the representation of integers as sums of triangular numbers

1995

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In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. If n > 0 is a non-negative integer, then the nth triangular number is 7", = n(n + 1)/2. Let k be a positive integer. We denote by 6k(n ) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate 6 k (n). The case where k = 24 is particularly interesting. It turns out that, if n > 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 212(212 -1)624(n -3). Furthermore the formula for (~24(r/) involves the Ramanujan z(n)-function. As a consequence, we get elementary congruences for z(n). In a similar vein, when p is a prime, we demonstrate 624(P k -3) as a Dirichlet convolution of alt (n) and r(n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.Consequently, the values of rk(n) are the formula coefficients of the power series Ok(q) = ~ rk(n)q". n>~O

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Section: Rll2( 8"c) = Q3inl~)i ( ] -Qsn)212mentioning

confidence: 99%

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confidence: 99%

On the representation of integers as sums of triangular numbers

1995

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“…The coefficients are called D'Arcais numbers [Co74]. Independently from D'Arcais, Newman and Serre [Ne55,Se85] studied the polynomials in the context of modular forms. Serre proved a famous theorem on lacunary modular forms, utilizing the factorization of P n (x) for 1 ≤ n ≤ 10 over Q.…”

Section: Introductionmentioning

confidence: 99%

On conjectures regarding the Nekrasov–Okounkov hook length formula

Heim

Neuhäuser

2019

Arch. Math.

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The Nekrasov-Okounkov hook length formula provides a fundamental link between the theory of partitions and the coefficients of powers of the Dedekind eta function. In this paper we examine three conjectures presented by Amdeberhan. The first conjecture is a refined Nekrasov-Okounkov formula involving hooks with trivial legs. We prove the conjecture. The second conjecture is on properties of the roots of the underlying D'Arcais polynomials. We give a counterexample and present a new conjecture. The third conjecture is on the unimodality of the coefficients of the involved polynomials. We confirm the conjecture up to the polynomial degree 1000. 1 24 ∞ m=1 (1 − q m ) (see [On03]).

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“…These coefficients were later considered by many authors, including Gupta, Atkin, Costello, Gordon, and finally Serre. Serre [16] showed that if r is an even integer, then {n : p r (n) = 0} has density zero if and only if r = 2, 4, 6, 8, 10, 14, or 26. Another natural question is about how large (as a function of r and n) the coefficients p r (n) are. In this regard, Newman's approach of expressing the coefficients p r (n) as polynomials in r is very ineffective.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%