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<p>In the present paper, we construct theta functions with two parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(a, b \in \mathbb{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> that satisfy Jacobi's modular relation. Furthermore, we define zeta functions, also depending on two real parameters <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(a, b \in \mathbb{R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> and which are derived from these theta functions, which satisfy Riemann's functional equation. To the best of our knowledge, these zeta functions are the first known examples that simultaneously satisfy Riemann's functional equation and involve two independent parameters.</p>

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Bilateral Lerch theta and theta star function and Quadrilateral Lerch zeta and zeta star functions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月1日

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  • T. Nakamura

摘要

In the present paper, we construct theta functions with two parameters \(a, b \in \mathbb{R}\) a , b ∈ R that satisfy Jacobi's modular relation. Furthermore, we define zeta functions, also depending on two real parameters \(a, b \in \mathbb{R}\) a , b ∈ R and which are derived from these theta functions, which satisfy Riemann's functional equation. To the best of our knowledge, these zeta functions are the first known examples that simultaneously satisfy Riemann's functional equation and involve two independent parameters.