Explicit upper bounds for the Stieltjes constants

TextLet χ be a primitive Dirichlet character modulo q   and let (−1)nγn(χ)/n!(−1)nγn(χ)/n! (for n larger than 0) be the n  -th Laurent coefficient around z=1z=1 of the associated Dirichlet L-series. When χ   is non-principal, (−1)nγn(χ)(−1)nγn(χ) is simply the value of the n  -th derivative of L(z,χ)L(z,χ) at z=1z=1. In this paper we give an explicit upper bounds for |γn(χ)||γn(χ)| for q⩽π2e(n+1)/2n+1. In particular, when q=1q=1 the explicit upper bound we get improves on earlier work. We conclude this paper by showing that we can altogether dispense in these proofs with the functional equation of L(z,χ)L(z,χ).VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=q340UciEvAA.