Abstract: For a totally real algebraic number field k, it is known that every (partial) zeta function of k is a finite sum of Dirichlet series which are regarded as natural generalizations of the Hurwits zeta function (see [1] and [2]). In this note we show that the similar result holds for arbitrary (not necessarily totally real) algebraic number field. At the time of the Bombay Colloquium (1979), H. M. Stark orally communicated to the author that he has obtained such a result for non-real cubic fields. His oral communication was an initial impetus to the present work. The author wishes to express his gratitude to Stark.

Abstract: One obtains, under certain restrictions, an estimate for the Hecke L -functions on the critical line, similar to the estimate for the Riemann zeta-function

Abstract: : Nonlinear diffusion equations of the form u sub t = delta zeta (u) where zeta is a given nondecreasing function occur in many situations. Existence and uniqueness of solutions of the initial-value problem for this type of equation have been studied by many authors. The regularity of the solutions, i.e. how smooth or continuous they are, is less well understood, although many results have recently been obtained in this direction. In this paper we contribute to the study of regularity by proving estimates of the general form zeta (u) sub t or = - c(zeta(u) + a)/t on nonnegative solutions in all of space. That is, one can bound below the time derivative of zeta(u) in a pointwise fashion by zeta (u) itself. Those results generalize those for the special case zeta(r) = r to the mth power obtained by Aronson and Benilan. (Author)

Abstract: It is proved that the zeta function in m complex variables of a system of positive-definite forms of degree gd ≥ 2 with real coefficients extends meromorphically to the entire space ℂm.

Abstract: In this paper the author obtains a formula representing the free term of the Laurent expansion of the zeta functions of absolute and ray classes of a real quadratic field at the point 1, as well as the value of Hecke's L-function corresponding to the signature character. The formula contains Dedekind sums and sums analogous to them in which functions depending on the same arguments as in ordinary Dedekind sums appear. Bibliography: 4 titles.

Abstract: In this paper we will consider the functions E(z, ρ) obtained by setting the complex variable s in the Eisenstein series E(z, s) equal to a zero of the Riemann zeta-function and will show that these functions satisfy a number of remarkable relations. Although many of these relations are consequences of more or less well known identities, the interpretation given here seems to be new and of some interest. In particular, looking at the functions E(z, ρ) leads naturally to the definition of a certain representation of SL2(R) whose spectrum is related to the set of zeroes of the zeta-function.

Abstract: and the other is the inner product $\langle f, g\rangle$ of $f$ and $g$ , when they have the same weight. These have been treated in our previous papers [12], [13], and [14] for the forms $f$ and $g$ of integral weight. Therefore the present investigation concerns the cases in which either or both of $f$ and $g$ have half-integral weight. To describe the nature of the problems as well as of the results, we let $m^{\prime}$ and $m$ denote the weights of $f$ and $g$ , respectively, which are positive elements of $2^{-1}Z$. If $m=m^{\prime}$ , there is a well known relation between $\langle f, g\rangle$ and the residue of $D(s, f, g)$ at $s=m$ , and therefore the second object has a character similar to the first one. Setting this residue aside, we restrict, for some natural reasons, the study of the values of $D(s, f, g)$ to the case $m