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 U_{n}, the sequence <Sn> of partial sums is bounded and if <V_{n}> is a positive monotonically decreasing sequence with lim V_{n} = 0, then the series Sigma U_{n}V_{n} converges.

In other words, Dirichlet’s test can be stated as: If <V_{n}> is a positive monotonically decreasing sequence with lim V_{n} = 0, then the series V_{1} – V_{2} + …(-1)^n-1 V_{n} 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.

1)

2)

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:

In other words, Dirichlet’s test can be stated as: If <V

Dirichlet’s Theorem

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