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We believe that Euler constant is not just the "renormalized" value of theRiemann zeta function in 1. In a sense that we shall clarify it is in fact thenormal and natural value of zeta of 1. In this paper we first propose a limitdefinition of a function whose values coincide everywhere with those of theRiemann zeta function, save in 1, where our limit definition yields the Eulerconstant. Since in the literature one can find more than one way to regularizethe value of the zeta function at s=1, we give asymptotic expansions where, bydint of some extended analogies, Euler constant appears to be the true"renormalized" value. As a striking example of such analogies, we propose anexpansion of the logarithm function based on Euler constant and on all valuesof the zeta function at odd positive integers, in which all these presumablyirrational numbers are accompanied by Harmonic numbers of corresponding orders.The other aim of this paper is to show how sequences of rationals, often thesame, arise in computations related to Dirichlet L-functions. Here, aconnection with the Liouville lambda function appears to have been found. Thuswe raise the question about the possible usefulness of an extension of theLiouville lambda function to rationals. .

Euler constant as a renormalized value of Riemann zeta function at its pole. Rationals related to Dirichlet L-functions