First-hit analysis of algorithms for computing quadratic irregularity

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani’s formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [Minking Eie, The special values at negative integers of Dirichlet series associated with polynomials of several variables, Proceedings of A. M. S. 119 (1993), 51–61] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.

In this paper, by using q-deformed bosonic p-adic integral, we give <TEX>$\lambda$</TEX>-Bernoulli numbers and polynomials, we prove Witt's type formula of <TEX>$\lambda$</TEX>-Bernoulli polynomials and Gauss multiplicative formula for <TEX>$\lambda$</TEX>-Bernoulli polynomials. By using derivative operator to the generating functions of <TEX>$\lambda$</TEX>-Bernoulli polynomials and generalized <TEX>$\lambda$</TEX>-Bernoulli numbers, we give Hurwitz type <TEX>$\lambda$</TEX>-zeta functions and Dirichlet's type <TEX>$\lambda$</TEX>-L-functions; which are interpolated <TEX>$\lambda$</TEX>-Bernoulli polynomials and generalized <TEX>$\lambda$</TEX>-Bernoulli numbers, respectively. We give generating function of <TEX>$\lambda$</TEX>-Bernoulli numbers with order r. By using Mellin transforms to their function, we prove relations between multiply zeta function and <TEX>$\lambda$</TEX>-Bernoulli polynomials and ordinary Bernoulli numbers of order r and <TEX>$\lambda$</TEX>-Bernoulli numbers, respectively. We also study on <TEX>$\lambda$</TEX>-Bernoulli numbers and polynomials in the space of locally constant. Moreover, we define <TEX>$\lambda$</TEX>-partial zeta function and interpolation function.

Let k=Q(d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(d)={x+yd|x,y∈Q}. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either wd=d=[a0;a1,a2,⋯,al(d)−1,2a0¯] in the case of d ≡ 2,3(mod 4) or wd=1+d2=[a0;a1,a2,⋯,al(d)−1,2a0−1¯] in the case of d ≡ 1(mod 4) where l(d) is the period length of continued fraction expansion.The purpose of this paper is to obtain classification of some types of real quadratic fields Q(d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is l=l(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain...

Bernoulli numbers appear as special values of zeta functions at integers and identities relating the Bernoulli numbers follow as a consequence of properties of the corresponding zeta functions. The most famous example is that of the special values of the Riemann zeta function and the Bernoulli identities due to Euler. In this paper we introduce a general principle for producing Bernoulli identities and apply it to zeta functions considered by Shintani, Zagier and Eie. Our results include some of the classical results of Euler and Ramanujan. Kummers congruences play important roles in the investigation of p -adic interpolation of the classical Riemann zeta function.

Explicit computation of Gross–Stark units over real quadratic fields

A note on [formula omitted]-Bernoulli numbers and polynomials

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1–5, 7, and 11 partitions.

Uniform distribution and real fields

Well-rounded twists of ideal lattices from real quadratic fields

We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(√D). By extending Shanks' infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step–giant step method for evaluating the regulator of K in O(|D|¼) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|⅕) polynomial operations. By implementing both methods on a computer, we found that the O(|D|⅕) algorithm tends to be far better than the baby step–giant step algorithm in those cases where the regulator exceeds 108.

To date, the only non-group structure that has been suitably employed as the key space for Diffie-Hellman-type cryptographic key exchange is the infrastructure of a real quadratic (number or function) field. We present an implementation of a Diffie-Hellman-type protocol based on real quadratic number field arithmetic that provides a significant improvement in performance over previous versions of this scheme. This dramatic speed-up is achieved by replacing the ordinary multiplication and reduction procedures for reduced ideals by a new version of the NUCOMP algorithm due to Shanks.

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

Let $K = \\mathbf{Q}(\\sqrt{m})$ be a real quadratic field, $\\mathcal{O}_K$\nits ring of integers and $G = \\operatorname{Gal}(K/\\mathbf{Q})$.\nFor $\\gamma \\in H^1(G, \\mathcal{O}_K^{\\times})$, we associate\na module $M_c/P_c$ for $\\gamma = [c]$. It is known that\n$M_c/P_c \\approx \\mathbf{Z}/\\Delta_m \\mathbf{Z}$ where $\\Delta_m = 1$\nor 2 and we will determine $\\Delta_m$.

Sums of nonvanishing integral squares in real quadratic fields