Showing papers on "Arithmetic zeta function published in 1985"
Book•
1 Jan 1985
TL;DR: In this article, the Voronoi Summation formula was used for the mean square problem, and the Dirichlet Divisor Problem was used to solve the problem.
Abstract: Elementary Theory Exponential Integrals and Exponential Sums The Voronoi Summation Formula The Approximate Functional Equations The Fourth Power Moment The Zero-Free Region Mean Value Estimates Over Short Intervals Higher Power Moments Omega Results Zeros on the Critical Line Zero-Density Estimates The Distribution of Primes The Dirichlet Divisor Problem Various Other Divisor Problems Atkinson's Formula for the Mean Square Appendix Author Index. Subject Index.
218 citations
1 Jun 1985
TL;DR: This paper showed that Riemann's zeta function has exactly 200,000,001 zeros of the form a + it in the region 0 < t < 81,702,130.19.
Abstract: We describe extensive computations which show that Riemann's zeta function t(s) has exactly 200,000,001 zeros of the form a + it in the region 0 < t < 81,702,130.19; all these zeros are simple and lie on the line a = . (This extends a similar result for the first 81,000,001 zeros, established by Brent in Math. Comp., v. 33, 1979, pp. 1361-1372.) Counts of the numbers of Gram blocks of various types and the failures of "Rosser's rule" are given.
94 citations
Journal Article•
TL;DR: In this article, the conditions générales d'utilisation (http://www.compositio.nl/) implique l'accord avec les conditions generales de utilisation, i.e., usage commerciale ou impression systématique, constitutive of an infraction pénale.
Abstract: © Foundation Compositio Mathematica, 1985, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
31 citations
26 citations
1 Jan 1985
TL;DR: In this article, the Lerch's zeta function was shown to be equivalent to the classical Eisenstein formula for n 1, where a is an integer and a $ 0 (mod m) and n > 1.
Abstract: For x not an an integer and Re(s) > 0, let oo 27rtkx F(x, s) E k8 k=1 be the Lerch's zeta function. In this note, we will show that ,e-2'iw/mF (15,1 -n = ]n B( t_[? ) m=1 where a is an integer and a $ 0 (mod m) and n > 1. For n 1, this formula is equivalent to the classical Eisenstein formula ak a 1 1 .2,7r-ac lr_=s_n cot m [ml 2 2m m n ly=1
20 citations
TL;DR: Several exact representations for the partial derivative delta zeta (z,q)/ delta z mod z −1 of the generalized Riemann zeta function zeta was given in this article.
Abstract: Several exact representations (as an integral and as an infinite series) for the partial derivative delta zeta (z,q)/ delta z mod z=-1 of the generalized Riemann zeta function zeta (z,q) are given.
18 citations
1 Jan 1985
14 citations
13 citations
5 citations
TL;DR: This article showed that no conclusions can be drawn about the significance of a false proof even when it comes from the work of first class mathematicians, and showed that such conclusions cannot be drawn even when the proof comes from first-class mathematicians.
Abstract: These examples show that no conclusions can be drawn about the significance of a false proof even when it comes from the work of first class mathematicians.
3 citations
TL;DR: In this article, it was shown that the Riemann zeta function can be used to derive the prime number theorem from the known zero-free regions for the Zeta function.
Abstract: It is shown that the function\(\zeta ^{1/k} \) (s) (k large) can be used to derive the prime number theorem from the known zero-free regions for the Riemann Zeta-function. For the proof no upper bound for |ζ′/ζ(s)| is required.
Dissertation•
1 Jan 1985
1 Jan 1985
TL;DR: In this paper, it is proved that the zeta functions of expanding mappings on compact manifolds are rational, i.e., they describe the behavior of periodic points of semi-dynamical systems.
Abstract: In this paper we discuss a sort of zeta function which describes the behavior of periodic points of semi-dynamical systems. It is proved that the zeta functions of expanding mappings on compact manifolds are rational.
1 Jan 1985
1 Jan 1985
TL;DR: In this paper, the authors studied the analogous zeta function and coefficients which arise for an order in a semi-simple F (X) -algebra, where F(X) is a field of rational functions over a q q finite field F.
Abstract: L. Solomon recently introduced a wide-ranging but concrete general- ization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solo- mon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi- simple F (X) -algebra, where F (X) is a field of rational functions over a q q finite field F. Use is made of the analogues for function fields of results q on his zeta functions which were first conjectured by Solomon, and later estab-
TL;DR: In this article, the authors studied the analogous zeta function and coefficients which arise for an order in a semi-simple Fq(X) -algebra, where Fq is a field of rational functions over a finite field Fq.
Abstract: L. Solomon recently introduced a wide-ranging but concrete generalization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solomon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi-simpleFq(X) -algebra, whereFq(X) is a field of rational functions over a finite fieldFq. Use is made of the analogues for function fields of results on his zeta functions which were first conjectured by Solomon, and later established by C J Bushnell and l Reiner.
TL;DR: In this article, the functional equation for c(s) is used to obtain formulas for all derivatives t(k) s. A closed form evaluation of t (k) 0 is given, and numerical values are computed to
Abstract: The functional equation for c(s) is used to obtain formulas for all derivatives t(k)(s). A closed form evaluation of t(k)(0) is given, and numerical values are computed to
Journal Article•
TL;DR: In this article, the same number of points were distributed at random (uniformly and independently on an interval of length Γ), and the ratio on the left would tend to A. This feature is most striking for small A.
Abstract: s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at random (uniformly and independently on an interval of length Γ), then the ratio on the left would tend to A. According to the conjecture, the zeros have on the average fewer near neighbors than they would have if they were distributed at random. This feature is most striking for small A.