Explicit bounds for generators of the class group

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group SICK of a number field K may be generated using prime ideals whose norm is bounded by 121og(2)delta(K), and by (4 + o(l)) log(2) delta(K) asymptotically, where delta(K) is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates SICK and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that SICK is generated by prime ideals whose norm is bounded by the minimum of 4.01 log(2) delta(K), 4(l + (2 pi e(gamma))N-_(K))(2) log(2) delta(k) and 4( log delta(k) + log log delta(K) - (gamma + log 2 pi)N-K + 1 + (N-K + 1) log(7log delta(K)/log delta(K))(2). Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedma's algorithms, confirming that it has size SIC (log delta(K) log log delta(K))(2). In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log(2) delta(K) except for 7 out of the 31292 fields we tested

Explicit short intervals for primes in arithmetic progressions on GRH

We prove explicit versions of Cram\ue9r's theorem for primes in arithmetic progressions, on the assumption of the generalised Riemann hypothesis

Morfologia e taxonomia das espécies do gênero Discosoma Ruppell & Leuckart, 1828 (Cnidaria, Corallimorpharia) no Brasil

This work aims to describe the species of the genus Discosoma Rüppell & Leuckart, 1828 (Corallimorpharia, Discosomatidae), from the Brasilian coast. There is only one record of the species Discosoma carlgreni (Watzl, 1922) in the north of Espirito Santo, Brazil ( SCHLENZ & BELÉM, 1892). Since then no other discosomatid has been recorded from the Brazilian shore. Two species belonging to the genus Discosoma have been studied: D. carlgreni (Watzl, 1922) and D. sanctithomae (Duchassaing & Michelotti, 1860). Moreover, other samples exchanged from foreign institutions were examined. The systematic was based on external morphology, anatomy, microanatomy as well as a cnidom qualitativa study. The cnidae were classified according to SCHMIDT (1969, 1972, 1974) and HART0G (1980). The study of cnidom allowed to distinguish the two species on the basis of their discal tentacle, marginal tentacle, column and pharynx. Such features had not been utilized in previous works. Externally D. sanctithomae differs from D. carlgreni especially due to the presence of a smooth, transparent, peripherial zone, separating the discal tentacles from the marginal ones. D. car7greni was found in several places on the brazilian coast such as in the south of Espírito Santo, south of Bahia as well as in the Abrolhos Archipelago. The D. sanctithomae on the other hand was only be found in the Caribean islands and Abrolhos Archipelago.FAPERJCAPESEste trabalho consiste no estudo descritivo das espécies do gênero Discosoma Rüppell & Leuckart, 1828 (Corallimorpharia, Discosomatidae) da costa brasileira. Na literatura específica, existe apenas o registro da espécie Discosoma carlgreni (Watzl, 1922) no Brasil, ao norte do Espirito Santo (SCHLENZ & BELÉM, 1982) . Desde então, nenhum outro discosomatídeo havia sido registrado para o litoral brasileiro. Foram estudadas duas espécies pertencentes ao gênero Discosoma: D. carlgreni (Watzl, 1922) , e D. sanctithomae (Duchassaing & Michelotti, 1860), sendo esta um novo registro para o Brasil. Além disso, foram examinados exemplares adquiridos por permuta junto a Instituições estrangeiras. A sistemática foi baseada na morfol ogia externa, anatomia, microanatomia e no estudo qual itativo do cnidoma. Os cnidae foram classificados segundo as nomenclaturas de SCHMI DT ( 1 969, 1 972, 1974) e HARTOG (1980). O estudo do cnidoma permitiu distinguir as duas espécies com base nos nematocistos dos tentáculos discais e marginais, coluna e faringe. Tais características não foram utilizadas em trabalhos anteriores. Externamente, Discosoma sanctithomae difere de Discosoma carlgreni, sobretudo pela presença de uma zona periférica lisa, transparente, no disco oral, separando os tentáculos discais dos marginais. D. carlgreni foi encontrada em várias localidades da costa brasileira: sul do Espirito Santo, sul da Bahia e Arquipélago dos Abrolhos. Por outro lado, Discosoma sanctithomae foi registrada apenas para as ilhas do Caribe e no Arquipél ago dos Abrolhos

Conditional upper bound for the k-th prime ideal with given Artin symbol

We prove an explicit upper bound for the k-th prime ideal with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions

Inequalities for the beta function

Let g(x):= (e/x)x\u393(x+1) and F(x,y):= g(x)g(y)/g(x+y). Let Dx,y(k) be the k th differential in Taylor's expansion of logF(x,y) . We prove that when (x,y) 08 R+2 one has (-1)k-1Dx,y(k) (X,Y) > 0 for every X,Y 08 R+, and that when k is even the conclusion holds for every X,Y 08 R with (X,Y) = (0,0). This implies that Taylor's polynomials for logF provide upper and lower bounds for logF according to the parity of their degree. The formula connecting the Beta function to the Gamma function shows that the bounds for F are actually bounds for Beta

An explicit Chebotarev density theorem under GRH

We prove an explicit version of the Chebotarev theorem for the density of prime ideals with fixed Artin symbol, under the assumption of the validity of the Riemann hypothesis for the Dedekind zeta functions. In appendix we also give some explicit formulas counting non-trivial zeros of Hecke's L-functions, in that case without assuming the truth of the Riemann hypothesis