Research Projects

Euler's formula for even zeta values is known to be transcendental, but the arithmetic nature of odd zeta values remains mysterious. Roger Apery in 1979 showed that zeta(3) is an irrational number, but the question remains whether zeta(3) is transcendental. Mathematicians believe that all odd zeta values must be transcendental, and it is widely conjectured that pi and odd zeta values are algebraically independent. Ramanujan wrote a formula for odd zeta values in his notebook before going to England. Recently, researchers have established an interesting one variable generalization of Ramanujan's identity and found an extension for the Hurwitz zeta function. Ramanujan's identity has attracted attention from many mathematicians, including Emil Grosswald. Mutty, Rath, Murty, Smith, Wang, Chourasiya, and Jamal defined a polynomial as "Ramanujan's polynomial," which has only unimodular roots with multiplicity 1 apart from four real roots. They also established an analog of Ramanujan's formula for the Dirichlet L-function and derived an identity analog to Grosswald's identity. These identities motivated the definition of new Ramanujan-type polynomials. A more general version of Lalin and Rogers's result might be true, with the Ramanujan-type polynomial having only real zeros of multiplicity $2 and simple non-real zeros. The main objective of this proposal is to prove these conjectures.

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