Shintani–Barnes cocycles and values of the zeta functions of algebraic number fields

In this paper, we construct a new Eisenstein cocycle called the Shintani-Barnes which specializes in uniform way to values of zeta functions general number fields at positive integers. Our basic strategy is generalize construction presented work Vlasenko and Zagier by using some recent techniques developed Bannai, Hagihara, Yamada, Yamamoto their study polylogarithm for totally real fields. We also closely follow Charollois, Dasgupta, Greenberg. fact, one key ingredients paper enables us deal with introduction technique ``exponential perturbation'' slight modification Q-perturbation studied work.