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Analytic Number Theory

Number theory is a branch of mathematics which deals with the system of numbers like integers, prime numbers etc. and their related extended properties with the application of calculus and the four major operations of mathematics (+, -, *, /).
Analytic Number Theory: It is a theory which is related to the solution of number systems like integers, prime numbers etc. by applying different analysis methods of calculus. It is divided into two parts:

1) Multiplicative Number Theory: It includes various concepts like primes and the Fundamental Theorem of Arithmetic, Arithmetic functions and elementary theory and their asymptotic estimates, distribution of primes and their elementary results, Dirichlet series and Euler products, distribution of primes with proof of the Prime Number Theorem and primes in arithmetic progressions including Dirichlet's Theorem.

2) Additive Number Theory: It is concerned with the additive property of integers, like Goldbach's conjecture which states that every even number greater than 2 is the sum of two primes. One of the important solution in additive number theory is the solution to Waring's problem.
Basic Concepts in Analytic Number Theory:
An arithmetic function is defined as any real- or complex-valued function defined on the set N of positive integers.
Constant function is a function defined by y(n) = c for all n, where c is a constant; for example, 1 denotes the function that is equal to 1 for all n.
Unit function is denoted as e(n), defined by e(1) = 1 and e(n) = 1 for $n\geq 2$.
Identity function is denoted as id(n), defined by id(n) = n for all n.
Logarithm is defined as log n, the (natural) logarithm, restricted to N and regarded as an arithmetic function.

Dirichlet’s Test:
If for a series Sigma Un, the sequence <Sn> of partial sums is bounded and if <Vn> is a positive monotonically decreasing sequence with lim Vn = 0, then the series Sigma UnVn converges.
In other words, Dirichlet’s test can be stated as: If <Vn> is a positive monotonically decreasing sequence with lim Vn = 0, then the series V1 – V2 + …(-1)^n-1 Vn converges. This is also known as Leibnitz’s test, which is a particular case of Dirichlet’s test.

Dirichlet’s Theorem
: Any series obtained from an absolutely convergent series by a rearrangement of terms is absolutely convergent, and has the same sum as the original series.

Solved Examples:
Example: Find whether the series $\sum \frac{(-1)^{n-1}}{n}$ is absolutely convergent or not.
Solution: The given series is not absolutely convergent, but by Leibnitz’s test the above series is convergent, this example shows that the conclusion of the Dirichlet’s theorem may not hold true if a series is not absolutely convergent.

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