On the Absoluate Value of Ramanuja n's ...-Function.
Under the Generalized Riemann Hypothesis, it is proved that for any there is depending on only such that every even integer is a sum of two odd primes and powers of .
We give several examples of classes of trace forms for which the ideal of annihilating polynomials is principal. We prove, that in general, the annihilating ideal is not a principal ideal.
In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.