Riemann zeta function

Riemann Zeta function, denoted by Greek letter , is defined as

Elsewhere is defined by Analytic Continuation. It is a Meromorphic function on or Holomorphic everywhere except only simple pole at with Residue .

Integral Representation

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The zeta function can be represented as

Where is Gamma Function

Contour integral representation:

Where is Hankel Contour.

Euler Product

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In 1937, Euler found a marvellous equality between infinite sum and an infinite product. It is the important bridge connecting prime numbers and the riemann zeta function in Analytic Number Theory. This formula is known as Euler Product:

Where ranges over all positive numbers and ranges over all primes.

Each factor on the right of is the sum of a Geometric Series:

Thus formula can be rewritten as,

Euler product is also the easy way to prove the Infinitude of Primes.

Bernoulli Numbers

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Bernoulli Numbers has many applications in mathematics such as summing powers of integers, finding asymptotics of Stirling’s formula, and estimating the Harmonic Series, … And it also frequently pop up in considerations involving the zeta function.

Definition

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The Bernoulli Numbers are defined to be the coefficients in the series expansion

Replacing with its Taylor Series to we can see that

Comparing the coefficients of power of both sides of , we get

Replacing with , we have formula

Where is Iverson bracket.

The first few Bernoulli numbers are

Relationship to Zeta Function

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Euler also discovered the formula for exact values of for

By Euler’s reflection formula for Gamma Function we have

Replacing

Also for Legendre duplication formula

Replacing

Dividing both sides from to

By Functional Equation for Riemann Zeta Function

Substituting to , we finally get the Unsymmetric Functional Equation for Riemann Zeta Function

Now for with , and , substitute the value of to

References

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1. https://www.whitman.edu/documents/academics/majors/mathematics/2019/Larson-Balof.pdf
2. https://en.wikipedia.org/wiki/Bernoulli_number
3. https://math.osu.edu/sites/math.osu.edu/files/Bernoulli_numbers.pdf
4. https://arxiv.org/abs/1812.02574
5. https://mikespivey.wordpress.com/2015/03/09/proof-of-the-recursive-identity-for-the-bernoulli-numbers/
6. https://en.wikipedia.org/wiki/Euler_product
7. https://en.wikipedia.org/wiki/Riemann_zeta_function
8. https://proofwiki.org/wiki/Unsymmetric_Functional_Equation_for_Riemann_Zeta_Function
9. https://people.reed.edu/~jerry/361/lectures/bernoulli.pdf
10. https://www.cut-the-knot.org/proofs/AfterEuler.shtml