Mittelwerte von Hecke-Polynomen

Abstract

In generalization of a classical result in number theory, we derive an estimate for the mean value of the Hecke polynomial $$S(t, \lambda ) = \sum {a(\mathfrak{a})} \lambda (\mathfrak{a})N\mathfrak{a}^{ - it} .$$ Here the sum runs over the integral ideals a of an algebraic number fieldK with normsNa≤X, and the mean value is defined via integration of |S(t, λ)|2 with respect tot∈[−T,T], and summation of the result over the Hecke Groessencharacters λ modulo some ideal f ofK having exponents <T. From this we derive an upper bound for the mean value of the fourth power of Hecke's zeta functions on the half-line Res=1/2.