Logarithmic derivatives of theta functions

Hershel M. Farkas; Yves Godin

doi:10.1007/BF02775438

Logarithmic derivatives of theta functions

Hershel M. Farkas^*
, Yves Godin

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We provide two new proofs of the identity ∑_n=0 ^∞δ(3n + 1)xⁿ = ∏_n=1∞ (1-x³ⁿ)³/(1-xⁿ) where δ(n) = d ₁(n) - d₂(n) and d_i(n) is the number of divisors of n congruent to i mod 3. Furthermore, we express the number of solutions of the Diophantine equation x² + 3y² = N in terms of δ(N).

Original language	English
Pages (from-to)	253-265
Number of pages	13
Journal	Israel Journal of Mathematics
Volume	148
DOIs	https://doi.org/10.1007/BF02775438
State	Published - 2005

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title = "Logarithmic derivatives of theta functions",

abstract = "We provide two new proofs of the identity ∑n=0 ∞δ(3n + 1)xn = ∏n=1∞ (1-x3n)3/(1-xn) where δ(n) = d 1(n) - d2(n) and di(n) is the number of divisors of n congruent to i mod 3. Furthermore, we express the number of solutions of the Diophantine equation x2 + 3y2 = N in terms of δ(N).",

author = "Farkas, \{Hershel M.\} and Yves Godin",

year = "2005",

doi = "10.1007/BF02775438",

language = "אנגלית",

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journal = "Israel Journal of Mathematics",

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AU - Farkas, Hershel M.

AU - Godin, Yves

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N2 - We provide two new proofs of the identity ∑n=0 ∞δ(3n + 1)xn = ∏n=1∞ (1-x3n)3/(1-xn) where δ(n) = d 1(n) - d2(n) and di(n) is the number of divisors of n congruent to i mod 3. Furthermore, we express the number of solutions of the Diophantine equation x2 + 3y2 = N in terms of δ(N).

AB - We provide two new proofs of the identity ∑n=0 ∞δ(3n + 1)xn = ∏n=1∞ (1-x3n)3/(1-xn) where δ(n) = d 1(n) - d2(n) and di(n) is the number of divisors of n congruent to i mod 3. Furthermore, we express the number of solutions of the Diophantine equation x2 + 3y2 = N in terms of δ(N).

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U2 - 10.1007/BF02775438

DO - 10.1007/BF02775438

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AN - SCOPUS:32544443621

SN - 0021-2172

VL - 148

SP - 253

EP - 265

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

ER -

Logarithmic derivatives of theta functions

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