Abstract: (1973). A History of the Prime Number Theorem. The American Mathematical Monthly: Vol. 80, No. 6, pp. 599-615.

Abstract: In this note we study quaternary even positive definite quadratic forms of prime discriminant. In § 1 we classify quaternary even positive definite quadratic forms of prime discriminant p = 1 mod 4 (called simply nice quaternary lattices in this note) which represent two. We note that the class number of such forms is closely related to the dimension of the space of certain automorphic forms. (Remark 4 in the text). By using the classification in § 1 and the theory of integral representations of cyclic groups we show that the orthogonal group of a nice quaternary lattice is generated by ±1 and symmetries (of the lattice). In § 3, we calculate the class number of nice quaternary lattices. Notations and terminologies will generally be those of O'Meara [5]. Any exceptions to this convention will be stated explicitly. Through this note Q(x) and B(x, y) denote quadratic forms and corresponding bilinear forms (i.e., 2B(x,y) = Q(x + y) — Q(x) — Q(y)), and p denotes a fixed prime number Ξlmod4. § 1. We say that a quadratic lattice N over the ring of rational integers Z is even if and only if Q(x) = 0 mod 2 for any element in N. For brevity, a quadratic lattice N is called nice in this note if and only if N is an even positive definite quadratic lattice over Z, its discriminant d(N) is p, 2p or 4p according as N is quarternary, ternary or binary respectively, and moreover the Hasse invariant S2(N) at the prime two of N is — (— i when N is binary.

Abstract: We consider a Beurling generalized prime system for which the distribution function N( x) of the integers satisfies f x -s I N(Cv)-Ayl) dx 0 We shall prove that the Chebyshev type estimates 0 0 and /3 < xo such that (1) liminf ( ) a, limsup ?)3 X '00 x * 00 x The prime number theorem (PNT) asserts that a = / = 1 Here we shall study Chebyshev type estimates for Beurling generalized primes Let 9 = { P, },=1, where 1 < Pi ? P2i ,p*,, be a set of Beurling generalized (henceforth g-) prime numbers and X= {n,i} 7 be the associated set of g-integers (see [1, 2]) Define N(x)= L 1, +(x)= E logp,

Abstract: It is proved that there exist systems of generalized primes in which the asymptotic distribution of integers is N(x)= Ax+O(x log7 x) with A>0 and y C [0, 1) and in which the Chebyshev inequalities lim inf "(x)log x 0 lim su 7r(x)log x< 00 X-cO x *CO x

Abstract: Some useful combinatorial selection lemmas are shown to be directly equivalent to the prime ideal theorem for boolean algebras. 1. The theorems we shall consider are intimately related to R. Rado's selection lemma (Theorem 2, below) which first appeared in [10] and subsequently has found wide application (see [1], [3], [4], [12]). Our main theorem is Theorem 1 which we use to derive other forms of Rado's lemma and to prove A. Robinson's valuation lemma which was shown by Robinson in [9] to be a fundamental result in model theory. We also show that Theorem 1 and some of the theorems we derive from it are equivalent to the prime ideal theorem for boolean algebras and thus we have some useful abstract versions of this theorem, versions divorced from any particular algebraic structure. Whether the original lemma of Rado is as strong as the prime ideal theorem appears to be an open question; note, however, that E. S. Wolk [12] has shown that Rado's lemma plus the axiom of choice for families of finite sets implies the Tychonoff theorem for finite spaces, which, in turn, implies the prime ideal theorem (see [8]). We conjecture that Rado's lemma is strictly weaker than the prime ideal theorem. 2. Preliminaries. In this paper we shall work, informally, within the framework of Zermelo-Fraenkel set theory (ZF). All uses of the axiom of choice will be explicitly noted. Given a set I, by a partial function on I, we mean a function whose domain is a subset of L A partial valuation on I is a partial function on I whose range is included in {0, 1}. As in ZF we consider functions to be special sets of ordered pairs and we even allow 0, the empty function (the empty set of ordered pairs). Iff is a partial function on I we write D(f) for its domain, and if Uc I we writef L U for the restriction off to UriD(f) Presented to the Society, August 30, 1972; received by the editors November 28,

Abstract: Submittal of an algorithm for consideration for publication in Communications of the ACM implies unrestricted use of the algorithm within a computer is permissible. General permission to republish, but not for profit, all or part of this material is granted provided that ACM's copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprinting privileges were granted by permission of the Association for Computing Machinery. The procedure Coulomb can be used very weU to generate the Coulomb wave functions FL and GL and their derivatives, needed in elastic scattering calculations in nuclear physics. When the procedure is used many times for many values of rho and eta, it is not only very useful but also necessary to have in each instance an indication about the accuracy of the results. It is obvious to use the Wronskian relations FL'GL-FLGL' ~ l for the purpose of checking the results, as Fr6berg [1] states after formula (3.4). However, one has to be very careful in using these relations. The most significant check is given later on, but first it is shown what can go wrong. K61big pointed out already in the certification that Lutz and Karvelis [2] failed to notice discrepancies exceeding 100 units in the sixth significant digit in their tables although they state \"when all the functions are generated we test to see how closely the Wron-skian relation FL'GL-FLGL' = 1 is obeyed.\" The way Lutz and Karvelis generate the functions goes as follows. First they calculate Go and Go'; then they use recurrence relations to get GL and GL' for L > 0; and lastly them use backward recurrence This investigation was part of the research program of the \"Stichting voor Fundamental Onderzoek der Materie (F.O.M.),\" which is financially supported by the \"Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)\". relations together with the relation FoG~-GoF1 = (n 2 + I)-~ to get FL and FL' for all L. This last relation is in fact a di:::rent form of the Wronskian relation, see e.g. Fr6berg [1] formula (3.5). The use of the Wronskian relations to check the results now gives information only about the stability in the use of the recurrence relations, not about the accuracy of the Coulomb wave functions. As an independent check on the function values, the following procedure can be used. It is easy to …

Abstract: This chapter presents the illustrated description of some important sequences. There are externally generated sequences, such as the sequences in which the nth term is the number of graphs with n nodes or the n th triangular number. The chapter presents method that seeks these externally generated sequences. The method looks for a systematic way of going from n to a n , for example, a n is the number of divisors of n , or the number of trees with n nodes, or the n th prime number.

Abstract: If X is an integer greater than one which has no factors other than itself and one, then it is a prime number. Being greater than one and having no factors other than itself and one are necessary conditions of an integer’s being a prime number: no integer that is not greater than one or has factors other than itself and one can be a prime number. And being an integer greater than one and having no factors other than itself and one are also sufficient conditions for being a prime number: we need know nothing else about X except that it is an integer greater than one and has no factors other than itself and one to conclude with logical certainty that X is a prime number. “X is an integer greater than one which has no factors other than itself and one” implies “X is a prime number.” And the term “prime number” is condition-governed in that there are logically necessary-and-sufficient conditions for any number being prime, namely, that it is an integer greater than one and that it has no factors other than itself and one.

Abstract: (1973) Prime Numbers and Brownian Motion The American Mathematical Monthly: Vol 80, No 10, pp 1099-1115

Abstract: (1973). Correction to “A History of the Prime Number Theorem”. The American Mathematical Monthly: Vol. 80, No. 10, pp. 1115-1115.