Analytic Number Theory

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Prime Number Theorem Basics

The Analytic Number Theory is a major branch of number theory and is just as relevant as the algebraic number theory. The difference between the analytic number theory and the algebraic number theory is that the former uses the Mathematical analysis method while the other uses normal computations reminiscent of more popular mathematics.

The aim of the Analytical Number Theory, which is abbreviated as ANT, is a type of mathematics that uses mathematical analysis computation to solve Integers problems. This concept studies the behaviors of integers by using mathematical analysis and computations on them. ANT is popular with respect to its results on the Prime Numbers and the respective Prime Number Theorem.

The Riemann Zeta Function is another important branch of ANT that plays an important role in mathematical and real analysis.

Analytic Number Theory Branches

Analytic Number Theory is divided into two main parts. The two parts are divided into smaller parts that focus on the fundamental problems they solve.

The two major categories are:

Multiplicative Number Theory

The multiplicative number theory includes mathematical concepts like the Distribution of Prime Numbers in an interval, Asymptotic Analysis, the Prime Number Theorem & Dinchlets Theorem of Primes in Arithmetic Progressions

Additive Number Theory

The Additive Number Theory include structures like Goldbach’s Conjecture and Waring’s Problem

This article will focus on the Prime Numbers, Asymptotic Analysis, and Prime Number Theorem, which are among the basics problems of the Analytic Number Theory

Prime Numbers, Asymptotic Analysis, and Prime Number Theorem

Prime Number

Understanding the Prime Number theorem requires knowing what prime numbers are.

A prime number is a positive number that is divisible by just two numbers. Itself and 1. The prime number must be divisible by two numbers. Nothing more and nothing less. Any number than can be divided by itself and 1 and any other number cannot be called a prime number. In essence, the division in this context must give a whole number.

For example, 7 is a prime number. This is because it can only be divided by 1 and itself. I.e 7/1 = 7 and 7/7 = 1. These two options are the only things that can divide 7 without it ending in decimals

On the other hand, 6 is not a prime number. This is because 6 can be divided by 1, 2, 3, and 6 and give a whole number.

This is an explanation.

6/1 = 6

6/2 = 3

6/3 = 2

6/6 = 1

The above clearly shows that 6 is not a prime number.

For the first 10 integers in the number theory; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, there are 4 prime numbers and 6 non-prime numbers.

1 is not a prime number because it is just 1.

2, 3, 5 and 7 are the prime numbers because they are the only numbers that can be divided by themselves and by 1.

4, 6, 8, 9 and 10 are not prime numbers just as 1. However, the reason why this is so is because of the fact that they can be divided themselves, 1 and other numbers failing the condition of only two divisible numbers.

Asymptotic Analysis

Asymptotic analysis is the mathematical analysis of limiting behaviour that describes how a function will resemble as it moves toward infinity.

Asymptotic in mathematics is a way to describe that a function will start to resemble just a part of its element as it tends to infinity.

An example is this function which is given as

F(x) = x2 + 2x

Let x = 2

F(x) = 22 + 2(2) = 4 + 4 = 8

Clearly, looking at the right side x2 holds the same stake as 2x. However, as x starts to become larger in number or tends to infinity, x2 will start to hold more stake in the final answer while 2x will only depreciate.

Now if x = 3

F(x) = 32 + 2(3) = 9 + 6 = 15 we can see that x2 is contributing +3 more than 2x

Now let x = 100000

F(x) = 100002 + 2(10,000) = 100,000,000 + 20,000 = 100,020,000

Obviously, x2 is now representing much more of the entire function of f(x) while 2x continues to be insignificant. As x still continues to tend towards infinity, it will only get more evident that f(x) will almost = x2, with 2x becoming less insignificant

The function f(x) can be said to be asymptotically equivalent to x2 because 2x will only become insignificant over time. this will be denoted as f(x) ~ 2x, where ~ is the asymptotic sign

Also, since f(x)~2x as x tends to infinity, we can claim that f(x)/2(x) will give us an answer that can be approximated to 1. This is because any value divided by itself is 1. But since f(x) and 2(x) are not the same but will only be very close in the asymptotic analysis, they can be represented as

Asymptotic analysis is used in both Mathematical Analysis and Computer science. It is also important in describing the Prime Number Theorem.

Prime Number Theorem

While the prime numbers are one of the basic arithmetic skills that even high school students, the Prime Number Theorem, actually considered the advanced nature of primes. Many students do not often get what the prime Number theorem is. Fortunately, this section will consider it by breaking it into small sections so that it becomes as easy as possible. It is important to also understand that the Prime Number Theorem is very important in the advanced study of the Analytic Theorem.

The Prime Number Theorem is a formalization of the intuition that it becomes much more difficult to find Integer prime numbers as counting moves up the scale. Now, 2 is a prime number with 1 and itself being the primes. 7 is a prime number. However, 8, 9, and 10 are not prime numbers, and this is roughly attributed to the fact that they are larger and will likely have numbers below them that can divide them apart from 1. Now, the Prime Number theorem roughly states that the larger an integer is, the less likely it will be a prime.

Formally, the Prime Number Theorem states that a prime counting function given as π(x) is asymptotic to x/logx. Meaning that the prime counting function will be almost equivalent to the value x/logx

Now π(x) as a prime counting function means the number of primes in an integer x

Now let x = 10, π(x) = 4. This is because four prime numbers before 10 are given as 2, 3, 5 and 7.

The prime number theorem states that the larger an integer x, the more likely its number of primes will be equal to the integer / its natural logarithm i.e. x/logx

Alternatively;

The prime number theory generally states that the function π(x) is asymptotic to x/logx this is denoted in the form

The theorem is generally helpful. However, proving it with numbers is difficult because of the challenge of counting out all of the primes in a very large integer. There are several analytical proofs, though, that confirms that the Prime Number Theorem is indeed correct.

Conclusion

Analytical Number Theory is an advanced set of mathematical concepts and calculations based on the number theory. This piece has considered this concept by highlighting its usefulness in the form of the Prime Number Theorem and the Distribution that follows. This concept highlighted the relationship between integers by considering them in the form of prime numbers. There is quite more to the Analytical Number Theory. However, this article has considered the basics to give students an appropriate understanding of what it is about.

Analytic Number Theory

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