Extreme residues of Dedekind zeta functions

Abstract

Abstract. In a family of S d+1 -fields (d = 2, 3, 4), we obtain the true upper and lower bound of the residues of Dedekind zeta functions except for a density zero set. For S5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bound, resp.

Extreme Values of Euler-Kronecker Constants

“…In this paper, following [2], we show that if n ≤ 5, except for a density zero set, the above are true upper and lower bounds. More precisely, we prove Ì ÓÖ Ñ 1.1º Let L(X) be the set of S n -fields (n ≤ 5) with X/2 ≤ d K ≤ X.…”

“…As in Proposition 3.1, we can show that for 0 < a < 1 4 , if L(s, ρ) is entire and is zero free in the rectangle Ê Ñ Ö 4.2º We would like to make corrections on [2]. They do not affect the results of the paper:…”

“…By applying Kowalski-Michel zero density theorem [7] to the family L(X) (See [2] for the details), we may assume that every L-function in L(X) outside the exceptional set in Proposition 3.2 has the desired zero free region of the form in Proposition 3.1. Then outside the exceptional set, In order to prove Proposition 3.2, we imitate the proof of Proposition 4.2 in [2]. Namely, we prove ÈÖÓÔÓ× Ø ÓÒ 3.3º Let n ≤ 5, and x = (log X) β and y = c 1 log X.…”

“…. , r u ) of positive integers such that r 1 + • • • + r u = 2r, and (2) p 1 ,...,p u means the sum over the u-tuples (p 1 , . .…”

The error term in the truncated Perron formula for the logarithm of an L-function

“…In these cases, the error term in (1.1) with is on the order of Granville and Soundararajan used 1.2 with Dirichlet L -functions to study large character sums [6, equation (8.1)]. Cho and Kim applied it to Artin L -functions to obtain asymptotic bounds on Dedekind zeta residues [1, Proposition 3.1]. A bilinear relative of (1.2) appears in Selberg’s work on primes in short intervals [14, Lemma 4].…”