L Functions and the Riemann Hypothesis

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The most famous L function is perhaps the Riemann Zeta function which has a Dirichlet series (sum of integers) and an Euler product of primes.





Where


is a complex number.

Furthermore, Riemann zeta function is convergent for all x  >1 but Riemann also showed the amazing symmetry it has about the vertical line x =1/2 which allows us to find values everywhere except the singularity at x =1.


Riemann hypothesis claims that all non-trivial Zeros lie on the line of symmetry at x = ½. Mathematicians have been trying to prove that since the 19th century as the L function encodes crucial information about the distribution of prime numbers.

One way is to study different L functions since they all have a sum of integers, an Euler product of primes and the line of symmetry just like the Riemann Zeta function. The hope is that by studying several of these L functions some pattern will emerge which will help prove the Riemann Hypothesis.

The late Indian genius Ramanujan studied the following series


By expanding the above series, he got


By noticing, several unique properties for this series like by taking the product of the second and third term powers of q gives us 6( 2 times 3) and the coefficient of the 6th term(-6048) is equal to the product of the coefficient of the second(-24) and third term (252) and so on one is able to get an L function which has a Dirichlet series (sum of integers) and an Euler product of primes and the symmetry properties and a Riemann hypothesis just like the Riemann Zeta function.



Where FP (T) is a polynomial of degree 2




         

           
            …………………………………………..
                                                          
An excellent resource for different L functions is lmfdb.org. Perhaps you could be the one who spots that elusive pattern between these L functions! Good luck! 

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