For a number field K, the associated Dedekind zeta function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\zeta _K(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ζ</mi> <mi>K</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a simple pole at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and we denote its residue by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. Ihara introduced the Euler–Kronecker constant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gamma _K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> be an odd prime. We establish lower and upper bounds for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$R_K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\gamma _K$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>γ</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> when K is a cyclic extension of degree <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>. These bounds are stronger than those known under the Generalized Riemann Hypothesis (GRH) and are shown to be sharp. However, the trade-off is that they hold only almost surely. Finally, we compute the average of the Euler–Kronecker constants for cyclic fields K of degree <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\ell $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>.

On the residues and Euler–Kronecker constants of cyclic number fields

期刊论文
出版模式：

开放获取
发表日期：

2026年2月13日

查看全文

Peter J. Cho,
Gyeongseok Kim

摘要

For a number field K, the associated Dedekind zeta function $\zeta _K(s)$ $ζ_{K} (s)$ has a simple pole at $s=1$ $s = 1$ , and we denote its residue by $R_K$ $R_{K}$ . Ihara introduced the Euler–Kronecker constant $\gamma _K$ $γ_{K}$ . Let $\ell $ $ℓ$ be an odd prime. We establish lower and upper bounds for $R_K$ $R_{K}$ and $\gamma _K$ $γ_{K}$ when K is a cyclic extension of degree $\ell $ $ℓ$ over $\mathbb {Q}$ $Q$ . These bounds are stronger than those known under the Generalized Riemann Hypothesis (GRH) and are shown to be sharp. However, the trade-off is that they hold only almost surely. Finally, we compute the average of the Euler–Kronecker constants for cyclic fields K of degree $\ell $ $ℓ$ .