MAS430 Analytic number theory notes (2014/15)

Dirichlet's theorem states that there exist an infinite number of primes in an arithmetic progression a + mk when a and m are relatively prime and k runs over the positive integers. While a few special cases of Dirichlet's theorem, such as the arithmetic progression 2 + 3k, can be settled by elementary methods, the proof of the general case is much more involved. Analysis of the Riemann zeta-function and Dirichlet L-functions is used.

We look at the mathematical foundations of analytic number theory to prove Dirichlet's Theorem. The intent is to provide Teachers of advanced secondary math courses the mathematical knowledge required to use Analytic number theory as a vehicle for teaching various calculus topics.

Journal of Advances in Mathematics and Computer Science

The first theorem related to the denseness of the image of a vertical line Re s = σ0, σ0 > 1 by the Riemann Zeta function has been proved by Harald Bohr in 1911. We argue that this theorem is not really a denseness theorem. Later Bohr and Courant proved similar theorems for the case 1/2 < Re s ≤ 1. Their results have been generalized to classes of Dirichlet functions and are at the origin of a burgeoning field in analytic number theory, namely the universality theory. The tools used in this theory are mainly of an arithmetic nature and do not allow a visualization of the phenomena involved. Our method is based on conformal mapping theory and is supported by computer generated illustrations. We generalize and refine Bohr and Courant results.

provides an insight for undergraduates to tackle Dirichlet's problem at an undergraduate level understanding of Mathematics.

2013

Contents 6. Expression of the integral Chebyshev function via the Riemann function § 2.3. Riemann function: Analytic properties 1. Analytic extension of the Riemann function 2. Zeros of the Riemann function 3. Estimates of the logarithmic derivative 4. Proof of the Prime Number Theorem § 2.4. Problems Chapter 3. Dirichlet Theorem § 3.1. Finite abelian groups and groups of characters 1. Finite abelian groups 2. Characters 3. Characters modulo m § 3.2. Dirichlet series 1. Convergence of L-series 2. Landau Theorem 3. Proof of the Dirichlet Theorem Chapter 4. p-adic numbers § 4.1. Valuation fields 1. Basic properties 2. Valuations over rationals 3. The replenishment of a valuation field § 4.2. Construction and properties of p-adic fields 1. Ring of p-adic integers and its properties 2. The field of p-adic rationals is the replenishment of rationals in p-adic metric 3. Applications § 4.3. Problems Bibliography Glossary Index CHAPTER 1 Algebraic and transcendental numbers § 1.1. Field of algebraic numbers. Ring of algebraic integers 1. Preliminary information. Let us recall some basic notions from Abstract Algebra. Throughout we use the following notations:

数理解析研究所講究録, 1994

\S 2-1, we make an explicit determination of coeffcients of the main term in the Piltz type divisor problem and related constants, in \S 2-2, as a generalization of the M\"uller-Carlitz-Ayoub-Chowla-Redmond-Bemdt theorem, give a Bessel series expression for the associated summatory functions and explicit determination of coeffcienls in the main term, and fnally in \S 2-3 we refer to the general product of L-series. Finally, in \S 3 we give Chowla-Selberg type formulas in special cases.

Closure of Golomb&#39;s topology over the composite numbers provides a substantial condition for the infinitude of prime numbers in relatively prime arithmetic progressions.

HAL (Le Centre pour la Communication Scientifique Directe), 2018

This article presents a alternative and simple proof to the Dirichlet's Theorem and a proof for the Goldbach's Strong Conjecture, using the Complex Wave Model and its properties. The Complex Wave Model allows the definition of an α function based on a sines product. As a consequence of the properties of trigonometric functions, with emphasis to its periodicity and symmetry, some of the questions regarding the prime numbers distribution, k-tuples and prime numbers arithmetic progressions, including the Dirichlet's Theorem and Goldbach's Conjecture, can be solved.

Duke Mathematical Journal, 1993

In the study of Dirichlet series with arithmetic significance there has appeared (through the study of known examples) certain expectations, namely (i) if a functional equation and Euler product exists, then it is likely that a type of Riemann hypothesis will hold, (ii) that if in addition the function has a simple pole at the point s=1, then it must be a product of the Riemann zeta-function and another Dirichlet series with similar properties, and (iii) that a type of converse theorem holds, namely that all such Dirichlet series can be obtained by considering Mellin transforms of automorphic forms associated with arithmetic groups.

Mathematische Annalen, 1996