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A closed-form expression for the Euler–Kronecker constant of a quadratic field

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Abstract

Given a number field, the Euler–Kronecker constant is defined as the constant term in the Laurent series expansion of the logarithmic derivative of the Dedekind zeta function at the point \(s=1\). In the case of real and imaginary quadratic fields, a closed-form expression for the Euler–Kronecker constants can be obtained with the help of suitable Kronecker limit formulas. In this article, we avoid the use of Kronecker limit formulas and derive an explicit series representation of these constants with the help of an asymptotic series representation of the coefficients appearing in the Laurent series expansion of the Dedekind zeta function at \(s=1\). As a result, the expressions obtained do not require evaluation of the special functions appearing in the Kronecker limit formulas.

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Notes

  1. The expression for \(\gamma _{0}(K)\) given in [28, Theorem 1] has a typo as there should not be extra \(\gamma _{-1}(K)\).

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  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, Uttar Pradesh, 208016, India

    Suraj Singh Khurana

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The author was supported by the National Board for Higher Mathematics (NBHM), Department of Atomic Energy (DAE), Government of India (DAE Ref no:0204/37/2021/R& D-II/16168).

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Khurana, S.S. A closed-form expression for the Euler–Kronecker constant of a quadratic field. Ramanujan J 63, 507–526 (2024). https://doi.org/10.1007/s11139-023-00772-8

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