Daniel Shanks

The simplest cubic fields

Abstract. The cyclic cubic fields generated by x3 = ax2 + (a + 3)x + 1 are studied in detail. The regulators are relatively small and are known at once. The class numbers are al2 2 ways of the form A + 3B , are relatively large and easy to compute. The class groups are usually easy to determine since one has the theorem that if m is divisible only by primes -2 (mod 3), then the m-rank of the class group is even. Fields with different 3-ranks are treated separately.

On the Distribution of Parity in the Partition Function

Kolbcrg [1] proved, but by contradiction and without identifying the arguments n, that i nitely many p(n) are even, and infinitely many are odd. His proof is almost as simple as Euclids proof that there are infinitely many primes, but like that proof it offers only very little more in the way of exact information concerning questions of distribution. From Guptas tables [2], [3] we find the following cumulative distribution into odds and evens for 0 ^ n :£ 499.

On maximal gaps between successive primes

exists and perhaps it might be possible to determine its value, but it will probably not be possible to express ft(n) by a simple function of n and t (even for t = 3). If t is large compared to n our method used in the proof of our theorem no longer gives a good estimation, but it is not difficult to prove by a different method the following result. Let 1 ? a, 0 there is an no (a) so that if n > no (a) and

Integer sequences having prescribed quadratic character

For the odd primes p, = 3, p2 = 5, ■ • -, we determine integer sequences Np such that the Legendre symbol (N/p,) = +1 for all pi S P for a prescribed array of signs ± 1; (i.e., for a prescribed quadratic character). We examine six quadratic characters having special interest and applications. We present tables of these Np and examine some applications, particularly to questions concerning extreme values for the smallest primitive root (of a prime N), the class number of the quadratic field R(J — N), the real Dirichlet L functions, and quadratic character sums.

On the conjecture of Hardy & Littlewood concerning the number of primes of the form ²+

taken over all odd primes, w, which do not divide a, and for which (-a/w) is the Legendre symbol. In the trivial cases, a = -k2, since (k2/w) = +1 for every w, we have ha = 0 on the one hand, and on the other there can be at most one prime of the form n2 _ k2 = (n k) (n + k). For any other a, ha > 0, and the conjecture indicates that there are infinitely many primes. But for no a has this been proven. In particular, for a = 1, since (-1 /w) equals +1 or -1 according as w 4m + 1 or 4m1, we have

On Gauss's Class Number Problems

Let h be the class number of binary quadratic forms (in Gausss formulation). All negative determinants having some h = 6n i: 1 can be deter- mined constructively: for h = 5 there are four such determinants; for h = 7, six; for h = 11, four; and for h = 13, six. The distinction between class numbers for determinants and for discriminants is discussed and some data are given. The question of one class/genus for negative determinants is imbedded in the larger question of the existence of a determinant having a specific Abelian group as its composition group. All Abelian groups of order <25 so exist, but the noncyclic groups of order 25, 49, and 121 do not occur. Positive determinants are treated by the same composition method. Although most positive primes of the form n2 - 8 have h = 1, an interesting subset does not. A positive determinant of an odd exponent of irregularity also appears in the investigation. Gauss indicated that he could not find one. U

New types of quadratic fields having three invariants divisible by 3

Abstract Following a review and new results concerning the discriminants Δ ( A , B ) = A 6 + 4 B 6 and −3 Δ ( A , B ), a new construction develops four quartic formulas including D 6 (Z) = 108z 4 − 148z 3 + 84z 2 − 24z + 3 . The imaginary fields Q((−D 6 (z)) 1 2 ) have three invariants divisible by 3 for z = … −44, −41, −29, 28, 34, 46, …. Theorems are proven concerning the class groups generated by these quartics such as the result that all Q((−D 6 (z)) 1 2 ) have at least 2 such invariants if z = 3 k + 1 ≠ 1. Tables are given and open questions are stated. The paper concludes with a brief introduction to the following question: If f ( y ) is an irreducible polynomial, what algebraic substitutions y = g ( z ) cause f ( z ) to be factorable?

Gauss’s ternary form reduction and the 2-Sylow subgroup

An algorithm is developed for determining the 2-Sylow subgroup of the class group of a qtadratic field provided the complete factorization of the discriminant dis known. It uses Gausss ternary form reduction with some new improvements and is applicable even if d is so large that the class number h(d) is inaccessible. Examples are given for various d that illustrate a number of special problems.

Calculation and applications of Epstein zeta functions

Rapidly convergent series are given for computing Epstein zeta functions at integer arguments. From these one may rapidly and accurately compute Dirichlet L functions and Dedekind zeta functions for quadratic and cubic fields of any negative discriminant. Tables of such functions computed in this way are described and numerous applications are given, including the evaluation of very slowly convergent products such as those that give constants of Landau and of Hardy-Littlewood.

The calculation of certain Dirichlet series

These, and some closely related series, arise in several number-theoretic investigations, including the distribution of primes into arithmetic progressions, the class number of binary quadratic forms, and the distribution of Legendre and Jacobi symbols. Our own immediate interest in them stems from their utility in the calcution of certain other number-theoretic constants. These latter include the ha of references [1] and [2], the Si of reference [3], the constant 0.48762 of reference [4], and the constant \C of reference [5]. The last of these illustrates our point, for when it was first presented by Bateman and Stemmler [6], it was given as 0.76; subsequently, in [5], the improved value 0.761 was presented; but with the aid of a short table of L3(s), and utilizing a formula analogous to that in [1, eq. (18), p. 323], it is fairly easy to compute