Multiplicative Number Theory the Number Theory Is

Multiplicative Number Theory 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 is a divisor of a. All integers are divisors of 0. All integers 'a' are divisible by and If a has any other divisors, then it is called a composite. Otherwise, it is called a prime, unless which are called units. Of the positive integers not greater than 20, the composite numbers are The prime numbers less than 20 are Primality testing and factorization The problem of an algorithm determining that all the divisors of a given integer are prime is very old.

Yet no solution has been found. A number with 100 digits can be proven to be prime or composite on the computer within a matter of seconds. Through this it can be shown that a number a is composite without actually displaying two factors b and c, other than 1 and a, such that a=bc. Since primality testing is easier in comparison to factoring, the U.S. security and military use them for security codes.

Distribution of primes There is still the question of how the primes are distributed among the integers. There can be many integers without primes, but also many with primes. This makes the distribution random. After several calculations, Gauss assumed the approximate size of the function defined to be the number of primes less than or equal to X.

A century ago, in the 1890's, Gauss' established the following limit from his hypothesis for the distribution of primes: This says that for very large values of X the number is very close to Here, is the natural logarithm (i.e. To the base e). This limit is known today as the "prime number theorem." Gauss discovered a function much closer to the value of the prime number counting function This is called the logarithmic integral where means the natural logarithm.

Riemann conjectured that the difference is just a shade above This is the equivalent form of Riemann's Hypothesis: for any This is largely agreed to be the most important unsolved problem in number theory. Problems involving congruences According to David J. Wright, "Given a positive integer m and any integers a, b, we say that a is congruent to b modulo m if m is a divisor of the difference a-b. This is written For example, since Every integer a is congruent modulo m to one of 0, 1, 2,..., m-1.

This is seen by adding.