We show explicit expressions for an inverse power series over the gaps values of numerical semigroups generated by two and three integers. As an application, a set of identities of the Hurwitz zeta functions is derived.
For an arbitrary complex number a = 0 we consider the distribution of values of the Riemann zeta-function ζ at the a-points of the function which appears in the functional equation ζ(s) = (s)ζ (1 − s). These a-points δ a are clustered around the critical line 1/2 + iR which happens to be a Julia line for the essential singularity of ζ at infinity. We observe a remarkable average behaviour for the sequence of values ζ(δ a).
Fujii obtained a formula for the average number of Goldbach representations with lower order terms expressed as a sum over the zeros of the Riemann zeta-function and a smaller error term. This assumed the Riemann Hypothesis. We obtain an unconditional version of this result, and obtain applications conditional on various conjectures on zeros of the Riemann zeta-function.
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