Harold G. Diamond

Elementary methods in the study of the distribution of prime numbers

Table of

The prime number theorem for Beurling's generalized numbers

Abstract Let P = {pj}j=1∞, where 1 Beurling [Acta Math, 1937] , we call such a sequence a system of generalized (g−) primes. Let N denote the multiplicative semigroup generated by P . Set N(x) = N P (x) = {n ∈ N : n ≤ x}. Beurling proved that if N P satisfies the asymptotic relation (1) N P (x) = Ax + O(x log−γ x) with some numbers A > 0 and γ > 3 2 , then the conclusion of the prime number theorem (P.N.T.) is valid for the system P . He gave an example of a g-prime system which satisfies (1) with γ = 3 2 , but for which the P.N.T. does not hold. The following theorem lies in the narrow range between the abovementioned results of Beurling. Let N P (x) = Ax + O{x (log x)−3/2 exp (−[log log x]a)} for some numbers A > 0 and α > 1 3 . Then the P.N.T. holds for P .

A Hundred Years of Prime Numbers

The twiddle notation is shorthand for the statement limxO, 7r(x)/{x/log x} = 1. Here we shall survey early work on the distribution of primes, the proof of the PNT, and some later developments. Since the time of Euclid, the primes, 2, 3, 5, 7, 11, 13, . . ., have been known to be infinite in number. They appear to be distributed quite irregularly, and early attempts to find a closed formula for the nth prime were unsuccessful. By the end of the 18th century many mathematical tables had been computed, and examination of tables of prime numbers led C. F. Gauss and A. M. Legendre to change the question under investigation. Instead of seeking an exact formula for the nth prime, they considered the counting function 7r(X) and asked for approximations to this function, evidently a new kind of question in number theory. Each of the two men conjectured the PNT, though neither did so in the form we have given. In 1808 Legendre published the formula 7r(X) = x/(log x + A(x)), where A(x) tends to a constant as x oc. Gauss recorded his conjecture in one of his favorite books

An elementary proof of the prime number theorem with a remainder term

AbstractAn elementary proof of the prime number theorem in the form

NEARLY PARALLEL VECTORS

\psi (x) - x = O(x exp\{ - (\log x)^{\tfrac{1}{7}} (log log x)^{ - 2} \} )

Erdős and Multiplicative Number Theory

is given. The proof uses a generalization of Selbergs formula and a tauberian argument.

Primes in Arithmetic Progressions

We study the distribution of a family{ (P)}of generalized Euler constants arising from integers sieved byfinite sets of primes P. For P = Pr, the set of the firstr primes, (Pr) ! exp( ) asr ! 1. Calculations suggest that (Pr) is monotonic in r, but we prove it is not. Also, we show a connection between the distribution of (Pr) exp( ) and the Riemann hypothesis.

The Theory of Algebraic Numbers

There are four exceptional pairs of vectors making a small angle. The minimal angles achieved by pairs of vectors of small norm are listed in Table 1. Notice that for ^3 4 < r < ^3 7 we have two essentially different ways of realizing 6(r). We take up this phenomenon of ties in §5. Some computation suggests that the minimizing vectors belong to class 1 for about 3/4 the values of r, to class 2 for about 1/5 the values of r, and to class 3 and class 4 each for about 1/50 the values of r. We ask whether each of the four cases actually provides a minimal angle for a sequence of rs tending to infinity. We answer this question in the