EXTREME VALUES OF DERIVATIVES OF THE DEDEKIND ZETA FUNCTION OF A CYCLOTOMIC FIELD

Abstract In this paper, we establish a lower bound for the maxima of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double-version convolution formula and combined with special GCD sums, our result generalizes the work of Bondarenko et al . [‘A dichotomy for extreme values of zeta and Dirichlet L-functions’, Bull. Lond. Math. Soc. 55 (2023), 2963–2975]. We also establish a lower bound via the resonance method when the real part is near the critical line; both of the above results refine parts of Yang’s [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510] work.