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Abstract
Let
ϑ 3 ( τ )
= 1
+ 2 ∑
ν = 1 ∞ e π i ν 2 τ
denote the classical Jacobi theta-constant. We prove that the two values
ϑ 3 ( m τ ) and
ϑ 3 ( n τ ) are algebraically
independent over
ℚ
for any
τ in the upper
half-plane such that
q
= e π i τ is an
algebraic number, where
m , n
≥ 2
are distinct integers.
Keywords
algebraic independence, Jacobi theta-constants, modular
functions
Mathematical Subject Classification 2010
Primary: 11J85
Secondary: 11J91, 11F27
Milestones
Received: 10 January 2018
Revised: 18 June 2018
Accepted: 3 July 2018
Published: 11 August 2018