Dirichlet's Class Number Formula Research Articles

A Handy Technique for Fundamental Unit in Specific Type of Real Quadratic Fields

Abstract Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ-function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h(d) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d > 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ-function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d > 0 and d ≡ 2,3 (mod4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Quadratic residues and related permutations and identities

Let p be an odd prime. In this paper we investigate quadratic residues modulo p and related permutations, congruences and identities. If a1<…<a(p−1)/2 are all the quadratic residues modulo p among 1,…,p−1, then the list {12}p,…,{((p−1)/2)2}p (with {k}p the least nonnegative residue of k modulo p) is a permutation of a1,…,a(p−1)/2, and we show that the sign of this permutation is 1 or (−1)(h(−p)+1)/2 according as p≡3(mod8) or p≡7(mod8), where h(−p) is the class number of the imaginary quadratic field Q(−p). To achieve this, we evaluate the product ∏1⩽j<k⩽(p−1)/2(cot⁡πj2/p−cot⁡πk2/p) via Dirichlet's class number formula and Galois theory. We also obtain some new congruences and identities in product forms; for example, we determine the exact value of∏1⩽j<k⩽p−1cos⁡πaj2+bjk+ck2p for any a,b,c∈Z with ac(a+b+c)≢0(modp).

Finite expressions for higher derivatives of the Dirichlet L-function and the Deninger R-function

We show the equivalence of the finite expression of Deninger's `R-function' at the rational arguments and the Kronecker limit formula on the line of our past study on the Gauss formula for the digamma function and the Dirichlet class number formula. Here the Gauss formula and the class number formula will be replaced by its analogue for the `R-function' and by the Kronecker limit formula or rather a closed form for the derivative of the Dirichlet L-function respectively. We also make a systematic study of the `$R_k$-function' by appealing to the Lipschitz-Lerch transcendent in which there is the vector space structure built in of these special functions.

The Class Number Formula of a Real Quadratic Field and an Estimate of the Value of a Unit

AbstractOur aim is to give an arithmetical expression of the class number formula of real quadratic fields. Starting from the classical Dirichlet class number formula, our proof goes along arithmetical lines not depending on any analytical method such as an estimate for

A note on a deformation of Dirichlet's class number formula

A note on a deformation of Dirichlet's class number formula

On Dirichlet's Class Number Formula

On Dirichlet's Class Number Formula

Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory.

Fermat Euler from Euler to Kummer Kummer's theort of ideal factors Fermat's last theorem for regular primes determination of the class number divisor theory for quadratic integers Gauss's theory of binary quadratic forms Dirichlet's class number formula the natural numbers.