AN ELEMENTARY PROOF OF THE CONDUCTOR-DISCRIMINANT FORMULA

2010 ◽  
Vol 06 (05) ◽  
pp. 1191-1197
Author(s):  
GABRIEL VILLA-SALVADOR
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Prime Number ◽  
Elementary Proof ◽  
Abelian Extension ◽  
Roots Of Unity ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Dirichlet Characters ◽  
Ramification Index

For a finite abelian extension K/ℚ, the conductor-discriminant formula establishes that the absolute value of the discriminant of K is equal to the product of the conductors of the elements of the group of Dirichlet characters associated to K. The simplest proof uses the functional equation for the Dedekind zeta function of K and its expression as the product of the L-series attached to the various Dirichlet characters associated to K. In this paper, we present an elementary proof of this formula considering first K contained in a cyclotomic number field of pn-roots of unity, where p is a prime number, and in the general case, using the ramification index of p given by the group of Dirichlet characters.

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].

scholarly journals A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

2010 ◽  
Vol 81 (2) ◽  
pp. 343-349 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Elementary Proof ◽  
Prime Number Theorem ◽  
Möbius Function ◽  
Partial Sums ◽  
Natural Numbers ◽  
Nontrivial Zeros ◽  
Mobius Function ◽  
Zeros And Poles

AbstractLet 〈𝒫〉⊂N be a multiplicative subsemigroup of the natural numbers N={1,2,3,…} generated by an arbitrary set 𝒫 of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈𝒫〉:n≤x(μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈𝒫(1−(1/p)) (the case where 𝒫 is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζ𝒫(s)≔∏ p∈𝒫(1−(1/ps))−1 on the line {Re(s)=1}. As equivalent forms of the first inequality, we have ∣∑ n≤x:(n,P)=1(μ(n))/n∣≤1, ∣∑ n∣N:n≤x(μ(n))/n∣≤1, and ∣∑ n≤x(μ(mn))/n∣≤1 for all m,x,N,P≥1.

scholarly journals Complex Roots of Unity and Normal Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Jean Marie De Koninck ◽  
Imre Kátai
Keyword(s):  
Prime Number ◽  
Roots Of Unity ◽  
Normal Numbers ◽  
Famous Result ◽  
Dirichlet Characters ◽  
Complex Roots ◽  
Selection Of

Given an arbitrary prime number q, set ξ=e2πi/q. We use a clever selection of the values of ξα, α=1,2,…, in order to create normal numbers. We also use a famous result of André Weil concerning Dirichlet characters to construct a family of normal numbers.

scholarly journals Class-number problems for cubic number fields

1995 ◽  
Vol 138 ◽  
pp. 199-208 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Zeta Function ◽  
Number Field ◽  
Class Number ◽  
Number Fields ◽  
Algebraic Integers ◽  
Absolute Value ◽  
Dedekind Zeta Function ◽  
Cubic Number Fields ◽  
Ring Of Algebraic Integers ◽  
The Absolute

Let M be any number field. We let DM, dM, hu, , AM and RegM be the discriminant, the absolute value of the discriminant, the class-number, the Dedekind zeta-function, the ring of algebraic integers and the regulator of M, respectively.we set If q is any odd prime we let (⋅/q) denote the Legendre’s symbol.

scholarly journals Leading terms of Artin L-functions at s=0 and s=1

2007 ◽  
Vol 143 (6) ◽  
pp. 1427-1464 ◽  
Author(s):  
Manuel Breuning ◽  
David Burns
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Galois Extension ◽  
Number Fields ◽  
Dedekind Zeta Function ◽  
Leading Term

AbstractWe formulate an explicit conjecture for the leading term at s=1 of the equivariant Dedekind zeta-function that is associated to a Galois extension of number fields. We show that this conjecture refines well-known conjectures of Stark and Chinburg, and we use the functional equation of the zeta-function to compare it to a natural conjecture for the leading term at s=0.

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 Minimal theta functions

1988 ◽  
Vol 30 (1) ◽  
pp. 75-85 ◽  
Author(s):  
Hugh L. Montgomery
Keyword(s):  
Functional Equation ◽  
Quadratic Form ◽  
Zeta Function ◽  
Theta Functions ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Epstein Zeta Function ◽  
Image Position

Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.Among such forms, let . The Epstein zeta function of f is denned to beRankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,We prove a corresponding result for theta functions. For real α > 0, letThis function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

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