A famous theorem of Zudilin states that at least one of the Riemann zeta values <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\zeta (5), \zeta (7), \zeta (9), \zeta (11)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>9</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ζ</mi> <mo stretchy="false">(</mo> <mn>11</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is irrational. In this paper, we establish a p-adic analogue of Zudilin’s theorem. As a weaker form of our result, it is proved that for any prime number <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p \geqslant 5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>⩾</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> there exists an odd integer i in the interval <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$[3,p+p/\log p+5]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>3</mn> <mo>,</mo> <mi>p</mi> <mo>+</mo> <mi>p</mi> <mo stretchy="false">/</mo> <mo>log</mo> <mi>p</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> such that the p-adic zeta value <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\zeta _p(i)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is irrational.

On the irrationality of certain p-adic zeta values

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2025年10月4日

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Li Lai,
Cezar Lupu,
Johannes Sprang

摘要

A famous theorem of Zudilin states that at least one of the Riemann zeta values $\zeta (5), \zeta (7), \zeta (9), \zeta (11)$ $ζ (5), ζ (7), ζ (9), ζ (11)$ is irrational. In this paper, we establish a p-adic analogue of Zudilin’s theorem. As a weaker form of our result, it is proved that for any prime number $p \geqslant 5$ $p ⩾ 5$ there exists an odd integer i in the interval $[3,p+p/\log p+5]$ $[3, p + p / log p + 5]$ such that the p-adic zeta value $\zeta _p(i)$ $ζ_{p} (i)$ is irrational.