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<p>We consider the cubic theta functions <i>a</i>, <i>b</i>, and <i>c</i> of the Borwein brothers, from a Weierstrassian perspective. We determine their zeros and the precise ranges of their ratios, as functions on the punctured unit disc: in particular, the theta function <i>a</i> has <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_168_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">\(- \exp \{ - \pi / \sqrt{3} \}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mo>exp</mo> <mo stretchy="false">{</mo> <mo>-</mo> <mi>π</mi> <mo stretchy="false">/</mo> <msqrt> <mn>3</mn> </msqrt> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> as its unique zero, while the ratio <i>a</i>/<i>b</i> omits as values precisely the cube roots of unity. Our account also includes a new proof of the cubic identity <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40627_2025_168_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">\(a^3 = b^3 + c^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>a</mi> <mn>3</mn> </msup> <mo>=</mo> <msup> <mi>b</mi> <mn>3</mn> </msup> <mo>+</mo> <msup> <mi>c</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>, along with properties of the cubic theta functions that yield an independent proof of the transformation law for the Dedekind eta function.</p>

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Borwein theta functions: zeros and ratios

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • P. L. Robinson

摘要

We consider the cubic theta functions a, b, and c of the Borwein brothers, from a Weierstrassian perspective. We determine their zeros and the precise ranges of their ratios, as functions on the punctured unit disc: in particular, the theta function a has \(- \exp \{ - \pi / \sqrt{3} \}\) - exp { - π / 3 } as its unique zero, while the ratio a/b omits as values precisely the cube roots of unity. Our account also includes a new proof of the cubic identity \(a^3 = b^3 + c^3\) a 3 = b 3 + c 3 , along with properties of the cubic theta functions that yield an independent proof of the transformation law for the Dedekind eta function.