TY - JOUR

T1 - Algebraic independence results for the values of the theta-constants and some identities

AU - Elsner, Carsten

AU - Kaneko, Masanobu

AU - Tachiya, Yohei

N1 - Funding Information:
The authors wish to express their sincere gratitude to the referee for his/her careful reading of our manuscript and for valuable comments. This work was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 18K03201.

PY - 2020/3

Y1 - 2020/3

N2 - In the present work, we give algebraic independence results for the values of the classical theta-constants ϑ2(τ), ϑ3(τ), and ϑ4(τ). For example, the two values ϑα(mτ) and ϑβ(nτ) are algebraically independent over Q for any τ in the upper half-plane when eπiτ is an algebraic number, where m, n ≥ 1 are integers and α, β ∈ {2, 3, 4} with (m, α) ≠ (n, β). This algebraic independence result provides new examples of transcendental numbers through some identities found by S. Ramanujan. We additionally give some explicit identities among the three theta-constants in particular cases.

AB - In the present work, we give algebraic independence results for the values of the classical theta-constants ϑ2(τ), ϑ3(τ), and ϑ4(τ). For example, the two values ϑα(mτ) and ϑβ(nτ) are algebraically independent over Q for any τ in the upper half-plane when eπiτ is an algebraic number, where m, n ≥ 1 are integers and α, β ∈ {2, 3, 4} with (m, α) ≠ (n, β). This algebraic independence result provides new examples of transcendental numbers through some identities found by S. Ramanujan. We additionally give some explicit identities among the three theta-constants in particular cases.

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M3 - Article

AN - SCOPUS:85088987233

VL - 35

SP - 71

EP - 80

JO - Journal of the Ramanujan Mathematical Society

JF - Journal of the Ramanujan Mathematical Society

SN - 0970-1249

IS - 1

ER -