(PHD, 1982)

Abstract:

The notion of a reduced real quadratic number goes back to Gauss, who defined such a number to be reduced if it is greater than one, and its conjugate between negative one and zero. An equivalent characterization is that the continued fraction of a reduced quadratic number is purely periodic. Zassenhaus generalized this by defining a real algebraic number α to be reduced if α > 1 and -1 < Reα’ < 0 for the conjugates α’ of α distinct from α. In this thesis, several properties of these reduced numbers are developed. In particular it is shown that there exist reduced numbers α with the property that α has no reduced immediate predecessor, that is, u + 1/α is not reduced for any choice of the rational integer u. We call such a number α an ancestor. These ancestors have the property that every real algebraic number of degree at least three is equivalent to exactly one of them. Here, equivalence is in the sense of continued fractions; α ~ β means that there exist integers a, b, c, and d such that ad - bc = ±1 and α = aβ+b/cβ+d. This is equivalent to α and β having identical continued fractions after a certain point. This property of ancestors gives rise to an application to the problem of determining whether or not two given integral binary homogeneous forms are equivalent, assuming that each form has a real root. If the forms are equivalent, so are the roots of the forms; this can be checked by comparing the ancestors. This method is computationally effective.

In another direction, there is a connection between the reduced numbers defined above and the Pisot-Vijayaraghavan (PV) numbers (a PV number is a real algebraic integer greater than one all of whose other conjugates have absolute value less than one). It turns out that any reduced algebraic integer which is not an ancestor is a PV number; integral ancestors may or may not be. Part of the thesis is devoted to a more detailed comparison of PV numbers and integral ancestors. On one side, there is the theorem of Salem that the PV numbers are closed. On the other, it is proved here that if K is a field of degree at least three over the rationals, real but not totally real, then no integral ancestor in K is isolated (that is, there are other integral ancestors arbitrarily close). Much more is true; one can show in many cases that the integral ancestors in such a field lie in a set of non-trivial intervals in which they are dense. This decomposition is studied in more detail. For example, in Q(α), where α^{3} = α + 1, the integral ancestors are actually dense in [1,∞]. In contrast, in Q(∛2), the integral ancestors are dense in [1,2] U [3,5] U [6,8] U [9,11] U . . . and none of them occur in the gaps. It is proved that all cubic fields which are not totally real are like one of these two fields in the way the integral ancestors are distributed. Similar results hold for fields of higher degree, although the situation is somewhat more complicated.

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(BS, 1971)

Abstract: Preface: The main goal of this work is to give an introductory account of sieve methods that would be understandable with only a slight knowledge of analytic number theory. These notes are based to a large extent on lectures on sieve methods given by Professor Van Lint and the author in a number theory seminar during the 1970-71 academic year, but rather extensive changes have been made in both the content and the presentation.

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(PHD, 1968)

Abstract:

An explicit formula is obtained for the coefficients of the cyclotomic polynomial F_{n}(x), where n is the product of two distinct odd primes. A recursion formula and a lower bound and an improvement of Bang’s upper bound for the coefficients of F_{n}(x) are also obtained, where n is the product of three distinct primes. The cyclotomic coefficients are also studied when n is the product of four distinct odd primes. A recursion formula and upper bounds for its coefficients are obtained. The last chapter includes a different approach to the cyclotomic coefficients. A connection is obtained between a certain partition function and the cyclotomic coefficients when n is the product of an arbitrary number of distinct odd primes. Finally, an upper bound for the coefficients is derived when n is the product of an arbitrary number of distinct and odd primes.

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(PHD, 1966)

Abstract:

Let P_{K, L}(N) be the number of __unordered__ partitions of a positive integer N into K or fewer positive integer parts, each part not exceeding L. A distribution of the form

Ʃ/N≤x P_{K,L}(N)

is considered first. For any fixed K, this distribution approaches a piecewise polynomial function as L increases to infinity. As both K and L approach infinity, this distribution is asymptotically normal. These results are proved by studying the convergence of the characteristic function.

The main result is the asymptotic behavior of P_{K,K}(N) itself, for certain large K and N. This is obtained by studying a contour integral of the generating function taken along the unit circle. The bulk of the estimate comes from integrating along a small arc near the point 1. Diophantine approximation is used to show that the integral along the rest of the circle is much smaller.

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(PHD, 1961)

Abstract:

We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them. If Q is a subset of the n-dimensional integer lattice L^n, we write VQ for the set of points which can see every point of Q, and we call a set S a set of visible points if S = VQ for some set Q. In the first section we study the elementary properties of the operator V and of certain associated operators. A typical result is that Q is a set of visible points if and only if Q = V(VQ). In the second and third sections we study sets of visible points in greater detail. In particular we show that if Q is a finite subset of L^n, then VQ has a “density” which is given by the Euler product ^π_p (1 – r_p(Q)/p_n) where the numbers r_p (Q) are certain integers determined by the set Q and the primes p. And if Q is an infinite subset of L^ n, we give necessary and sufficient conditions on the set Q such that VQ has a density which is given by this or other related products. In the final section we compute the average values of a certain class of functions defined on L^n, and we show that the resulting formula may be used to compute the density of a set of visible points VQ generated by a finite set Q.

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(PHD, 1960)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document. We say that one lattice point is visible from another if no third lattice point lies on the line joining them. A lattice point visible from the origin is called a visible point. We study the manner in which the visible points are distributed throughout the lattice and show that, in a k-dimensional lattice, the fraction of such points in an expanding region “usually” tends to […]. On the other hand there exist arbitrarily large “gaps” containing no visible points. The following is a typical theorem: The maximum number of lattice points mutually visible in pairs is […], and if […], the “density” of points visible from each of a fixed set of n points, themselves mutually visible in pairs, is […]. The last section is devoted to a study of the function […], which is defined to be the number of distinct solutions of the congruence […] having […]. A special case of this function arises in connection with a certain visibility problem. A typical result is that […].

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(PHD, 1956)

Abstract:

NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

Let A(x) be a monotone non-decreasing function of x, and let […]. It is possible that T(x)~ ax log x, but […] = 0, […]. If T(x) = ax log x + 0(x), then […], […], but A(x) ~ ax is in general false. If T(x) = ax log x + bx + […] then A(x) ~ ax.

The prime number theorem is the special case A(x) = ….

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(PHD, 1956)

Abstract: The Poisson summation formula, which gives, under suitable conditions on f(x), and expression for sums of the form
^(n_2)Σ_(n=n_1) f(n) 1 ≤ n_1 < n_2 ≤ ∞
can be derived from the functional equation for the Riemann zeta-function (s). In this thesis a class of Dirichlet series is defined whose members have properties analogous to those of s(s); in particular, each series in the class, written in the form
Ø(s) = ^∞Σ_(n=1) a(n) λ ^{(-s)n
defines a meromorphic function Ø(s) which satisfies a relation analogous to the functional equation of s(s). From this relation an identity for sum of the form
Σ(}λn^(≤x) a(n) (x - λ_n)^q
is derived. This identity in turn leads, in a quite simple fashion, to summation formulas which give expressions for sums of the form
^(n_2)Σ_(n=n_1) a(n) f(λ_n) 1 ≤ n_1 ≤ n_2
The summation formulas thus derived include the Poisson and other well-known summation formulas as special cases and in addition embrace many expressions that are new. The formulas are not only of interest in themselves, but also provide a tool for investigating many problems that arise in analytic number theory.

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