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<p>The paper contains a mean square estimate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\underset{T-H}{\overset{T+H}{\int }}{\left|L\left(\lambda ,\alpha ,\sigma +\text{i}t\right)\right|}^{2}\text{d}t{\ll }_{\lambda ,\alpha ,\sigma }H\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mover> <mo>∫</mo> <mrow> <mi>T</mi> <mo>+</mo> <mi>H</mi> </mrow> </mover> <mrow> <mi>T</mi> <mo>-</mo> <mi>H</mi> </mrow> </munder> <msup> <mrow> <mfenced close="|" open="|"> <mi>L</mi> <mfenced close=")" open="("> <mi>λ</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>σ</mi> <mo>+</mo> <mtext>i</mtext> <mi>t</mi> </mfenced> </mfenced> </mrow> <mn>2</mn> </msup> <mtext>d</mtext> <mi>t</mi> <msub> <mo>≪</mo> <mrow> <mi>λ</mi> <mo>,</mo> <mi>α</mi> <mo>,</mo> <mi>σ</mi> </mrow> </msub> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> for the Lerch zeta-function <i>L</i>(<i>λ, α, s</i>) with fixed parameters <i>λ, α</i> ∈ (0<i>,</i> 1], 1<i>/</i>2 <i>&lt; σ</i> ≤ 7<b>/</b>12, and <i>T</i><sup>27/82</sup> ≤ <i>H</i> ≤ <i>T</i><sup><i>σ</i></sup>. The estimate is uniform in <i>H</i>. The result extends the mean square estimates for the Hurwitz and Riemann zeta-functions. The obtained bound is applied for universality theorems in short intervals for the function <i>L</i>(<i>λ, α, s</i>).</p>

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On the mean square estimate for the Lerch zeta-function in short intervals

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • Birutė Gutauskienė,
  • Antanas Laurinčikas,
  • Darius Šiaučiūnas

摘要

The paper contains a mean square estimate \(\underset{T-H}{\overset{T+H}{\int }}{\left|L\left(\lambda ,\alpha ,\sigma +\text{i}t\right)\right|}^{2}\text{d}t{\ll }_{\lambda ,\alpha ,\sigma }H\) ∫ T + H T - H L λ , α , σ + i t 2 d t ≪ λ , α , σ H for the Lerch zeta-function L(λ, α, s) with fixed parameters λ, α ∈ (0, 1], 1/2 < σ ≤ 7/12, and T27/82 ≤ H ≤ Tσ. The estimate is uniform in H. The result extends the mean square estimates for the Hurwitz and Riemann zeta-functions. The obtained bound is applied for universality theorems in short intervals for the function L(λ, α, s).