- Length: 5 pages
- Subject: Education - Mathematics
- Type: Term Paper
- Paper: #25120721

The Number theory is one of the oldest and largest branches of pure mathematics, which relates to questions and problems based on numbers that are either whole numbers or rational numbers.

The number theory is a branch of mathematics that relates to the properties of the integers comprised of the numbers 0, 1, -1, 2, -2, 3, -3, etc. One of the most essential areas in number theory concerns the analysis of prime numbers. In mathematics, A prime number is defined as an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that can be divided by more than one divisors are called composite numbers; these numbers include some of the following 4, 6, 8, 9, 10, 12.

The principle theorem of arithmetic, is the factorization theorem, which states that any positive integer a is a product (a = p _(1) p _(2) p _(3) p _(n)) of primes that are unique, regardless of the order they are listed in; for example, the number 20 is the product 20 = 2-2-5, and it is unique regardless of the order the numbers are put in because 20 only has this product of primes. The Greek mathematician Euclid, who proved that there are many primes to a great extent, discovered this theorem. Analytic number theory is substantially a step ahead of the theorem proposed by Euclid.

The analytic number theory determines a function that measures how many times the prime numbers are distributed among all integers. Twin primes are prime numbers that have a difference of 2, examples include (3,5) and (11,13). This modern theory derived from the number theory is still an important area of mathematical research that is being studied using mathematical tools.

Analytic Number Theory is purely based on the study of the Riemann zeta function and other similar functions such as Dirichlet series. The zeta function is defined on half the complex plane as the sum 1 + 1/2s + 1/3s + 1/4s +...; it is connected with the number theory from the results of factorization as a product Prod (1-1/p^s)^(-1), the product taken over all primes p. Therefore, the distribution of the primes among the integers can be produced from the behavior of zeta(s). The Riemann Hypothesis states that zeta(s) is never zero except along the line Re(s)=1/2 (or at the negative even integers). This is the most important area of research in mathematics.

The multiplicative number theory or analytic number theory consist of the following topics:

1. Elementary theory of multiplicative functions. Convolutions 2. Summatory function. Counting square free numbers and primes

3. Analytic theory. Dirichlet series, Euler products, applications

4. Oscillations

5. Mean values. Elementary theory, Halasz theorem

6. Numbers having only small prime factors

An arithmetic function f

0 is called multiplicative if it satisfies the relation f (mn) = f (m) f (n) for all relatively prime positive integers m, n. This unique requirement creates a significant structure for an arithmetic function, where many of the functions are either multiplicative or are closely so.

Under this branch of mathematics, theoretic questions such as the prime number theorem, the Dirichlet divisor problem, and the distribution of square-free numbers are studied as multiplicative functions.

Multiplicative functions are explained in terms of convolutions and exponentials of arithmetic functions. Primarily, these functions are closely shown under convolution.

Linked with each arithmetic function is a Dirichlet series. This series provides important analytic information about the functions. In multiplicative functions the Dirichlet series contains a representation in factored form, called the Euler product. Examples of this also include the world's most famous Dirichlet series, called the Riemann zeta function.

Another area of the multiplicative function is the summatory function that is used in estimating mean values. For this function, theorems of Delange and Halasz are used to find the mean value.

Multiplicative Number Theory and Problems about Primes

This topic is about the properties of the positive integers under the operations of multiplication and division. The key problems crop up when the quotient of two integers is not an integer. This can be explained in the following manner: "Given two integers a and b, we say that b divides a, written a|b, if there is another integer c such that a=bc. If b divides a, then b…