Showing papers on "Riemann zeta function published in 1972"

Analytic continuation of Eisenstein series

Abstract: The classical Eisenstein series are essentially of the form Em n((m + rl)z + n + r2) s, m, n ranging over integer values, Imz > O, r1, r2 rational and s an integer > 2. In this paper we show that if s is taken to be complex the series, with rl, r2 any real numbers, defines an analytic function of (z, s) for Imz > 0, Re s > 2. Furthermore this function has an analytic continuation over the entire s plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when z is subjected to a modular transformation is obtained and the special case of s an integer is studied in detail. Introduction. In this paper we are concerned with the function G (z, s, r 1' r2) defined as the sum of the series Y, n 1/((m + r )z + n + r2Y), for z having positive imaginary part, s an arbitrary complex number and r1, r2 arbitrary real numbers. m, n range over all integer values. In ?I we define which branch of the 'multi-valued' complex power should be taken, and show that although the series converges only for Res > 2, G has an analytic continuation to all values of s. This is done by exhibiting a Fourier expansion for G convergent for all s. For r1, r2 rational numbers and s an integer > 3 one sees that G coincides essentially with the classical Eisenstein series; this then is the reason for the title. A few more details on this will be found in ?III. In ?II we determine how G transforms when z is subjected to a transformation z (az + b)/(cz + d) belonging to the modular group. To put this paper in proper perspective we remark that its significance is not just in the formulas in ? H but also in that considering G as a function of the variable s, several classical functions are obtained for special values of s and r1, r2, each of whose functional equations has always been derived separately while here they all appear as specializations of one general formula. Thus, besides the above mentioned Eisenstein series for s = k > 3, rl, r2, rational, we also obtain in ?III Hecke's generalized Eisenstein series for s = 2, while the case s = r = r2= 0 gives us the transformation formula for logqr, ri being Dedekind's 71 function. Received by the editors September 30, 1971. AMS 1970 subject classifications. Primary 10D05; Secondary 30A14, 30A16, 30A58.

On meromorphic functions commuting with elements of a function group

Abstract: The problem of whether there always exist meromorphic functions commuting with all substitutions of a function group is solved in the affirmative.

On the arithmetic of Pfaffians

Abstract: In this paper, we shall supply proofs to the results announced in [2], pp. 74-75: we shall prove the Siegel formula for the Pfaffian of degree n over an algebraic number field and also determine the zeta function of the Pfaffian. In the appendix, we shall briefly discuss the non-split case where the Pfaffian is replaced by the norm form of the simple Jordan algebra of quaternionic hermitian matrices of degree n .

Instability in ^{} (³) and the nongenericity of rational zeta functions

Abstract: In the search for an easily-classified Baire set of diffeomor- phisms, all the studied classes have had the property that all maps close enough to any diffeomorphism in the class have the same number of periodic points of each period. The author constructs an open subset U of Diffr(T3) with the prop- erty that if f is in U there is a g arbitrarily close to f and an integer n such that ff and gn have a different number of fixed points. Then, using the open set U, he illustrates that having a rational zeta function is not a generic prop- erty for diffeomorphisms and that Q-conjugacy is an ineffective means for classi- fying any Baike set of diffeomorphisms.

The lowest zero of sections of the zeta function.

Abstract: N Lei fjv(s) = Σ ~~*> s = a-\\-it.lt is shown that for N sufficiently large ζ N (s) has a simple zero with n=l least positive t. Close asymptotic bounds are given for the location of this zero.

On convolutions with the Möbius function

Abstract: By using the results of [6], it is r5roved that for an extensive class of increasing functions h, (*) E ?~?x [t(d)h(x) xh'(x) as x o l

Complex zeros of two incomplete Riemann zeta functions

Abstract: The computation of the complex zeros of an incomplete Riemann zeta function defined in an earlier paper is extended and new zero trajectories are given. A second in- complete Riemann zeta function is denned and its zero trajectories are investigated numer- ically as functions of the upper limit X of the definition integral. It becomes apparent that there exist three different classes of zero trajectories for this function, distinguished by their behaviour for X —► . were presented, X > 0 being a real parameter. These results were obtained by a systematic numerical investigation. It became apparent that not all, but only some, of the zero trajectories s(X), defined in the s-plane by A(s(\), X) = 0, reach a zero of the Riemann zeta function on the line °°. The remaining curves j(X) approach the zero trajectories s(X) of the incomplete gamma function P(s, X), which are denned by r°(s(X), X) = 0. It is the aim of this paper to present further solutions s(X) which again have been obtained by numerical calculation. Because of the fact that there exists the relation

On zeta functions and Anosov diffeomorphisms

Abstract: This thesis considers some problems in Dynamical Systems concerned with zeta functions and with Anosov diffeomorphisms. In chapter 1 Bowen's method of expressing a basic set of an Axiom A diffeomorphism as a quotient of a subshift of finite type is used ,to calculate the numbers of periodic points of the diffeomorphism and show that its zeta function is ration31 which gives an affirmative answer to a question of Smale. The rest of the thesis is concerned with Anosov diffeomorphisms of nilmanifolds.Chapter 2 contains some facts about nilmanifolds describing them as twisted products of tori. Anilmanifold has a maximal torus factor. A hyperbolic nilmanifold automorphism projects onto an automorphism of this torus and we , say it has the toral automorphism as a factor. In chapter 3 we generalize this situation to show that many diffeomorphisms of other manifolds have toral automorphisms as factors and give some examples. In the last chapter we use a spectral sequence associated to another decomposition of a nilrnanifold into tori to calculate the Lefschetz number of any diffeomorphism of the nilmanifold. This enables us to prove a necessary condition on the map induced by an Anosov, diffeomorphism of a nilmanifold on its fundamental group. Then we consider the question of finding hyperbolic automorphisms of nilmanifolds from the decomposition into tori. Fin311y we calculate the zeta function of such an automorphism.

Class number in constant extensions of function fields

Abstract: Let F/K be a function field in one variable of genus g having the finite field K as exact field of constants. Suppose p is a rational prime not dividing the class number of F. In this paper an upper bound is derived for the degree of a constant extension E necessary to have p occur as a divisor of the class number of the field E. Throughout this paper the term "function field" will mean a function field in one variable whose exact field of constants is a finite field with q elements. Let F/K be a function field. The order of the finite group of divisor classes of degree zero is the class number hf. For F/K of genus g, we use the notation of [2] and denote by L(u) the polynomial numerator of the zeta function of F. It follows from the functional equation of the zeta function that (1) L(u)1? + a1u + a2u2 + * * + ag u + qag_lu9l + * + qg-lalu2g-l + q U2g and L(u) & Z[u], Z the rational integers. Furthermore the class number hBF=L(1). If E/F is a constant field extension of degree n, then the polynomial numerator L,(u) of the zeta function for E is given by (2) L,(u) = 1 + b1u + * * * + bg u9 + qnbg_1u9l + * * * + q nu2g where the coefficients by (j= 1, ... ,g) are, with appropriate sign, the elementary symmetric functions of the nth powers of the reciprocals of the roots of (1). The genus of E is the same as that of F because F is conservative. In this paper we give an upper bound for the degree of a constant extension E of F necessary to have a predetermined prime p occur as a Received by the editors September 10, 1971 and, in revised form, March 12, 1972. AMS 1969 subject classifications. Primary 1078; Secondary 1278, 1435.

Methods of computing vocabulary size for the two-parameter rank distribution

Abstract: A summation method is described for computing the vocabulary size for given parameter values in the 1- and 2-parameter rank distributions. Two methods of determining the asymptotes for the family of 2-parameter rank-distribution curves are also described. Tables are computed and graphs are drawn relating paris of parameter values to the vocabulary size. The partial product formula for the Riemann zeta function is investigated as an approximation to the partial sum formula for the Riemann zeta function. An error bound is established that indicates that the partial product should not be used to approximate the partial sum in calculating the vocabulary size for the 2-parameter rank distribution.

Complex Zeros of Two Incomplete Riemann

Abstract: The computation of the complex zeros of an incomplete Riemann zeta function defined in an earlier paper is extended and new zero trajectories are given. A second in- complete Riemann zeta function is defined and its zero trajectories are investigated numer- ically as functions of the upper limit X of the definition integral. It becomes apparent that there exist three different classes of zero trajectories for this function, distinguished by their behaviour for X -- c. ( ) ( ) ~~~~~r(s) (nex + I where