scholarly journals The signs of the Stieltjes constants associated with the Dedekind zeta function

2018 ◽  
Vol 94 (10) ◽  
pp. 93-96
Author(s):  
Sumaia Saad Eddin
Keyword(s):  
Zeta Function ◽  
Dedekind Zeta Function ◽  
Stieltjes Constants

The (α, β)-ramification invariants of a number field

2021 ◽  
Vol 71 (1) ◽  
pp. 251-263
Author(s):  
Guillermo Mantilla-Soler
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Residue Class ◽  
Interesting Application ◽  
Dedekind Zeta Function ◽  
Arithmetic Properties

Abstract Let L be a number field. For a given prime p, we define integers α p L $ \alpha_{p}^{L} $ and β p L $ \beta_{p}^{L} $ with some interesting arithmetic properties. For instance, β p L $ \beta_{p}^{L} $ is equal to 1 whenever p does not ramify in L and α p L $ \alpha_{p}^{L} $ is divisible by p whenever p is wildly ramified in L. The aforementioned properties, although interesting, follow easily from definitions; however a more interesting application of these invariants is the fact that they completely characterize the Dedekind zeta function of L. Moreover, if the residue class mod p of α p L $ \alpha_{p}^{L} $ is not zero for all p then such residues determine the genus of the integral trace.

scholarly journals EXPLICIT ZERO-FREE REGIONS FOR DEDEKIND ZETA FUNCTIONS

2012 ◽  
Vol 08 (01) ◽  
pp. 125-147 ◽  
Author(s):  
HABIBA KADIRI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Zeta Functions ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
The Absolute

Let K be a number field, nK be its degree, and dK be the absolute value of its discriminant. We prove that, if dK is sufficiently large, then the Dedekind zeta function ζK(s) has no zeros in the region: [Formula: see text], [Formula: see text], where log M = 12.55 log dK + 9.69nK log |ℑ𝔪 s| + 3.03 nK + 58.63. Moreover, it has at most one zero in the region:[Formula: see text], [Formula: see text]. This zero if it exists is simple and is real. This argument also improves a result of Stark by a factor of 2: ζK(s) has at most one zero in the region [Formula: see text], [Formula: see text].

A note on Dedekind zeta values at 1/2

2021 ◽  
pp. 1-11
Author(s):  
Neelam Kandhil
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Number Field ◽  
Dedekind Zeta Function ◽  
Derivative Formula ◽  
Zeta Values

For a number field [Formula: see text], let [Formula: see text] be the Dedekind zeta function associated to [Formula: see text]. In this paper, we study non-vanishing and transcendence of [Formula: see text] as well as its derivative [Formula: see text] at [Formula: see text]. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of [Formula: see text] and non-vanishing of [Formula: see text]-series at [Formula: see text], J. Number Theory 161 (2016) 444–456].

scholarly journals ON THE MERTENS–CESÀRO THEOREM FOR NUMBER FIELDS

2015 ◽  
Vol 93 (2) ◽  
pp. 199-210 ◽  
Author(s):  
ANDREA FERRAGUTI ◽  
GIACOMO MICHELI
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Number Fields ◽  
Algebraic Integers ◽  
Ring Of Integers ◽  
Dedekind Zeta Function ◽  
Natural Density ◽  
Suitable Notion

Let $K$ be a number field with ring of integers ${\mathcal{O}}$. After introducing a suitable notion of density for subsets of ${\mathcal{O}}$, generalising the natural density for subsets of $\mathbb{Z}$, we show that the density of the set of coprime $m$-tuples of algebraic integers is $1/{\it\zeta}_{K}(m)$, where ${\it\zeta}_{K}$ is the Dedekind zeta function of $K$. This generalises a result found independently by Mertens [‘Ueber einige asymptotische Gesetze der Zahlentheorie’, J. reine angew. Math. 77 (1874), 289–338] and Cesàro [‘Question 75 (solution)’, Mathesis 3 (1883), 224–225] concerning the density of coprime pairs of integers in $\mathbb{Z}$.

New summation relations for the Stieltjes constants

2006 ◽  
Vol 462 (2073) ◽  
pp. 2563-2573 ◽  
Author(s):  
Mark W Coffey
Keyword(s):  
Zeta Function ◽  
Hypergeometric Function ◽  
Laurent Expansion ◽  
Hurwitz Zeta Function ◽  
Stieltjes Constants ◽  
Positive Integers

The Stieltjes constants have been of interest for over a century, yet their detailed behaviour remains under investigation. These constants appear in the Laurent expansion of the Hurwitz zeta function about . We obtain novel single and double summatory relations for , including single summation relations for and , where a and b are real and p and q are positive integers. In addition, we obtain new integration formulae for the Hurwitz zeta function and a new expression for the Stieltjes constants . Portions of the presentation show an intertwining of the theory of the hypergeometric function with that of the Hurwitz zeta function.

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