Showing papers on "Riemann zeta function published in 1971"

Chebyshev approximations for the Riemann zeta function

Abstract: This paper presents well-conditioned rational Chebyshev approximations, involving at most one exponentiation, for computation of either i(s) or ?(s) 1, .5 ? s : 55, for up to 20 significant figures. The logarithmic error is required in one case. An algorithm for the Hurwitz zeta function, and an example of nearly double degeneracy are also given.

Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half

Abstract: The roots of Bessel functions of order one-half are special cases of roots of transcendental equations of the form tan z = A(z)/B(z), where A(z), B(z) are polynomials and A(z)/B(z) is odd. We prove that the function f(z) = B(z) sin z - A(z) cos z, f(z) even or odd, satisfies the conditions of Hadamard's factorization theorem, and derive recurrences for sums of the form SI = *- 1 aT 2t, 1 = 1, 2, * * *, where the ak's are the nonzero roots of f(z). We also prove under what conditions on A(z) and B(z) is S1 = ir212r(2l + 2) or S1 = Wv-212r(21 + 2)(221+2 - 1), where r is the Riemann zeta function. We prove that, although Bessel functions of positive half-order, JI+1/2, have only real roots, perturbation of

Applications of the theory of dirichlet series to the superposition of an infinite number of regge poles

Abstract: A theorem (Theorem 3) is given, which proves, under certain general conditions, that the trajectoriesα n (t) in the Chew-Frautschi diagram, are, asymptotically inn, parallel and spaced by one unit.

The functional equation of some Dirichlet series. II

Abstract: We derive the functional equation of a class of Dirichlet series. A particular case of our result was first given by Rademacher. For any positive integer k, Rademacher [2] showed that the Dirichlet series k 2 Z(s) = n-s + n ifs1 (o = Re s > 1) 1=1 n >0; n-1(k:) n > ;n=_-I(k) J has an analytic continuation to the entire complex s-plane that is analytic except for a double pole at s = 1, and satisfies the functional equation (1) (ir/k)-sF2(s/2)Z(s) = (-lk)s-l2({l-s}/2)Z(l s). Rademacher's proof used a familiar representation of the Hurwitz zetafunction. The purpose of this note is to show that a simpler proof of (1) as well as a considerable generalization can be given by employing Epstein zeta-functions rather than the Hurwitz zeta-function. For g and h real and a > 1 let Z(s; g, h) = Zt e27ihn in + gl-s, n where the dash ' indicates that the summation is over all integers n except in the possibility that n + g = 0. Z(s; g, h) has an analytic continuation to the entire complex plane and is entire if h is not an integer and is analytic everywhere except at s = 1 where there is a simple pole with residue 2 when h is an integer [1]. Furthermore, [1, p. 207] we have the functional equation (2) T-s12F(s12)Z(s; g, h) = e-2Tigh7T(s-l)I2f({j s}/2)Z(l s; h, -g). Received by the editors November 13, 1970. AMS 1969 subject classifications. Primary 1041.

Zeta functions of parabolic forms

Abstract: It is shown that the zeta functions of parabolic forms for the multiplicative group of a simple algebra over a function field can be continued meromorphically, and a functional equation is obtained for such forms. Bibliography: 13 citations.