The sum of divisors function σ(m) is defined by σ(m) = {∑d d∈ℕ d|m if m ∈ ℕ, 0 if m ∈ ℚ, m ∉ ℕ. Let H denote the upper half of the complex plane. Let η(z)(z ∈ H) be the Dedekind eta function. A class C of eta quotients is given for which the Fourier series of each member of C can be given explicitly. One example is η 2(2z)η 4(4z)η 6(6z)/ η 2(z)η 2(3z)η 4(12z)= 1+∑ ∞ n=1c(n)e 2πinz, z ∈ H, where c(n)=2σ(n)-3 σ(n/2) +4σ(n/4)+9σ(n/6)-36σ(n/ 12), n ∈ ℕ.

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Keywords Dedekind eta function, eta quotients, Fourier series
Persistent URL dx.doi.org/10.1142/S1793042112500595
Journal International Journal of Number Theory
Citation
Williams, K.S. (2012). Fourier series of a class of eta quotients. International Journal of Number Theory, 8(4), 993–1004. doi:10.1142/S1793042112500595