ContentsIntroduction I. Metrical theory of approximation on manifolds ??1. The basic problem ??2. Brief survey of results ??3. The principal conjecture II. Metrical theory of transcendental numbers ??1. Mahler's classification of numbers ??2. Metrical characterization of numbers with a given type of approximation ??3. Further problems III. Approximation of algebraic numbers by rationals ??1. Simultaneous approximations ??2. The inclusion of p-adic metrics ??3. Effective improvements of Liouville's inequality IV. Estimates of linear forms in logarithms of algebraic numbers ??1. The basic method ??2. Survey of results ??3. Estimates in the p-adic metric V. Diophantine equations ??1. Ternary exponential equations ??2. The Thue and Thue-Mahler equations ??3. Equations of hyperelliptic type ??4. Algebraic-exponential equations VI. The arithmetic structure of polynomials and the class number ??1. The greatest prime divisor of a polynomial in one variable ??2. The greatest prime divisor of a polynomial in two variables ??3. Square-free divisors of polynomials and the class number ??4. The general problem of the size of the class number Conclusion References

http://iopscience.iop.org/article/10.1070/RM1980v035n04ABEH001861/pdf

Achievements and Problems in Diophantine Approximation Theory

It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the special values of continued fractions that represent modular functions, especially the Rogers-Ramanujan continued fraction. These give analogues of the simple continued fraction expansions of units in real quadratic fields. My primary motivation is to furnish a coherent treatment of this topic, around which an air of mystery seems to linger. Another is to provide an inviting and non-standard introduction to the classical theory of modular functions. This is largely an expository paper; most of the ideas I discuss are well known. Yet it is hoped that the elaboration given here combines these ideas in a novel way. Although this paper is not intended to be comprehensive, its later sections contain more material than is likely needed to gain a clear impression of the main themes, which the first six sections should provide. These will take the general reader through a proof of the first main result, Theorem 1, introducing the needed concepts along the way. The sections that follow these assume a somewhat greater background in number theory.

/pdf/continued-fractions-and-modular-functions-3rwbrb31lh.pdf

Continued fractions and modular functions

We point out that although Baker [2, 3] and Stark [18, 19] succeeded in solving the class number 1 and 2 problems, their methods did not extend to the higher cases. It was not until the work of Goldfeld, Gross, and Zagier [12, 13] that a general method was developed. This new method, however, does not allow one to solve the even class number problems without a good deal of further work. In this regard, we note that Oesterlé [17] finished the class number 3 problem, while the class number 8 problem still appears to be intractable.

The imaginary quadratic fields of class number 4

Let N be a non-abelian normal CM-eld of degree 4p; p any odd prime. Note that the Galois group of N is either the dicyclic group of order 4p; or the dihedral group of order 4p: We prove that the (relative) class number of a dicyclic CM-eld of degree 4p is always greater then one. Then, we determine all the dihedral CM-elds of degree 12 with class number one: there are exactly nine such CM-elds.

/pdf/the-class-number-one-problem-for-some-non-abelian-normal-cm-51kvh0paao.pdf

The class number one problem for some non-abelian normal CM-fields

This thesis is concerned with behaviour of some famous arithmetic functions. The first part of the thesis deals with prime number races. Rubinstein-Sarnak [62] developed a technique to study primes in arithmetic progressions. This thesis studies prime numbers that are described by Chebotarev’s density theorem using the techniques developed in [62]. Let L/K be a normal extension with Galois group G. Consider conjugacy classes C1, . . . , Cr and the set PL/K;1,2,...,r = {x ≥ 2 | |G| |C1| πC1(x) ≥ |G| |C2| πC2(x) ≥ · · · ≥ |G| |Cr| πCr(x)} . Following Rubinstein-Sarnak and by applying effective versions of Chebotarev’s density theorem we do the following: 1. A limiting distribution μL/K;1,2,...,r attached to the set PL/K;1,2,...,r is constructed. 2. The Fourier transform of μL/K;1,2,...,r is calculated. It can be expressed as an infinite product of J0(x) Bessel functions evaluated at zeros of the corresponding Artin L-functions. 3. Logarithmic densities of some specific examples of the sets PL/K;1,2,...,r are computed. This computation requires many zeros of Artin L-functions. Some of these were computed using programs written in Fortran and C. Others were provided by Robert Rumely. 4. An explanation of Chebyshev’s bias in the Galois group setting is given. In addition, the algebraic bias coming from possible zeros of an L-function at the centre of the critical is considered. Two examples of quaternion Galois groups were studied. The Dedekind zeta function of one of these fields has a zero at s = 1 2 . 5. The analogous problems for class groups are also considered. A simple explanation of when a bias occurs in two-way races for complex quadratic fields is presented. We also compute some logarithmic densities and derive a central limit theorem in this setting. The second part of the thesis studies the summatory function of the Möbius function M(x) = ∑ n≤x μ(n) . Assuming conjectures due to Gonek and Hejhal concerning the reciprocal of the zeta function, the following results are shown: 1. The weak Mertens conjecture is true. Precisely, ∫ X 2 ( M(x) x )2 dx logX .

/pdf/limiting-distributions-and-zeros-of-artin-l-functions-2rqi8d73c4.pdf

Limiting distributions and zeros of artin l-functions

Let d < 0 be the discriminant of a complex quadratic field of class-number h(d). In a previous paper the author has effectively shown how to find all d with h(d) = 2. In this paper, it is proved that, if h(d) = 2, then Id l < 427.

/pdf/on-complex-quadratic-fields-wth-class-number-two-3i7y0yymzr.pdf

On complex quadratic fields wth class-number two

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On complex quadratic fields wth class-number two