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<p>We investigate the positivity of the real part of the log-derivative of the Riemann <i>ξ</i>-function in the region <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1/2+1/\sqrt{\text{log}t}&lt;\sigma &lt;1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <msqrt> <mrow> <mtext>log</mtext> <mi>t</mi> </mrow> </msqrt> <mo>&lt;</mo> <mi>σ</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> with <i>t</i> sufficiently large. We provide an explicit lower bound for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathfrak{R}{\sum }_{\rho }1/\left(s-\rho \right),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">R</mi> <msub> <mo>∑</mo> <mi>ρ</mi> </msub> <mn>1</mn> <mo stretchy="false">/</mo> <mfenced close=")" open="("> <mi>s</mi> <mo>-</mo> <mi>ρ</mi> </mfenced> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the logderivative of the Riemann <i>ξ</i>-function in the provided region, assuming that there are nontrivial zeta-zeros off the critical line.</p>

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Note on the positivity of the real part of the log-derivative of the Riemann ξ-function near the critical line

  • 期刊论文

  • 发表日期：

    2026年1月28日

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  • Andrius Grigutis,
  • Lukas Turčinskas

摘要

We investigate the positivity of the real part of the log-derivative of the Riemann ξ-function in the region \(1/2+1/\sqrt{\text{log}t}<\sigma <1\) 1 / 2 + 1 / log t < σ < 1 with t sufficiently large. We provide an explicit lower bound for \(\mathfrak{R}{\sum }_{\rho }1/\left(s-\rho \right),\) R ∑ ρ 1 / s - ρ , where the summation runs over the zeta-zeros on the critical line. We also consider hypothetical cases of positivity of the logderivative of the Riemann ξ-function in the provided region, assuming that there are nontrivial zeta-zeros off the critical line.