Showing papers on "Arithmetic zeta function published in 1978"
TL;DR: In the case of the Riemann zeta function, the proofs are based on applications of classical function-theoretic theorems, together with mean value theorem for Dirichlet polynomials or series.
Abstract: This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer.
%This is of great interest, for little has been known on the mean value of $\vert\zeta(\frac{1}{2}+it)\vert^k$ for odd $k$, say $k=1$; for even $k$, see the book by E. C. Titchmarsh [The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951, Theorem 7.19].
The proofs are based on applications of classical function-theoretic theorems, together with mean value theorems for Dirichlet polynomials or series.
%In the case of the zeta function, the principle is to write $\vert\zeta(s)\vert^k=\vert\zeta(s)^{k/2}\vert^2$, where $\zeta(s)^{k/2}$ is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of $\zeta(s)^{k/2}$, convergent in the half-plane $\sigma>1$.
90 citations
TL;DR: In this article, the Riemann hypothesis for the Selberg zeta function is almost true, in the sense that any possible exceptional zeros are all located in the real segment (0, 1).
Abstract: The methods depend on the fact that the Riemann hypothesis for the Selberg zeta-function is almost true, in the sense that any possible exceptional zeros are all located in the real segment (0, 1). I have recently learned that Pierre Berard of the University of Paris 7 has succeeded in showing that the O(T'/2/log T) half of the estimate is also valid for surfaces of variable negative curvature [1]. His methods, which are based on [3], are radically different from those of this paper, and the two somewhat complement each other. Both techniques extend to some higherdimensional cases, in the case of this paper, via the zeta-functions of Selberg's
53 citations
TL;DR: In this article, it was shown that for every odd prime #, the f-adic absolute values l(n) can be expressed in the following way as 6tale Euler characteristics of the scheme X=Spec((9)-{points above f}, where (9 is the ring of integers of K:
Abstract: The values of the zeta function (~(s) of a totally real number field K on the negative integers s = n are, by a theorem of Siegel, rational numbers; they are zero iff n is even. S. Lichtenbaum has made the remarkable conjecture that, for every odd prime #, the f-adic absolute values l(~(-n)[~ can be expressed in the following way as 6tale Euler characteristics of the scheme X=Spec((9)-{points above f}, where (9 is the ring of integers of K:
40 citations
Journal Article•
TL;DR: In this paper, the uniformity of the distribution of the zeros of the Riemann zeta function ζ (s) and of DirichletL-functionsL(X χ) was studied.
Abstract: In this paper we shall deepen and generalize our previous works [2], [4] and [6] on the uniformity of the distribution of the zeros of the Riemann zeta function ζ (s) and ofDirichletL-functionsL(X χ). Το explain our problems and our results, we need the following notion which is a generalization of WeyFs uniform distribution mod l and was introduced by LeVeque (cf.[15]orp.4of[13]).Let 91 = {an ; n = l , 2, 3, . . . } be a sequence of non-negative real numbers and 93 = {bn; n = l, 2, 3, . . . } be a sequence of increasing non-negative real numbers. For each α e (0, 1], let A (χ, α) be the number of a„ < χ such that
29 citations
Journal Article•
TL;DR: In this paper, the problem of defining a formal Dirichlet group over a discrete valuation ring of characteristic 0 has been studied in the context of algebraic number fields and L-functions with characters.
Abstract: We discuss one-dimensional formal groups in section 2 and higher-dimensional ones in section 3. In section 2, we give several examples of one-dimensional formal groups attached to formal Dirichlet series over an integral domain R of characteristic 0. The fact that the majority of formal groups constructed from formal Dirichlet series over R are not defined over R gives us a number-theoretic problem to work on, i.e. what kind of formal Dirichlet series gives us a formal group defined over 7?? We discuss this problem in the case that R is a discrete valuation ring of characteristic 0. In section 3, we determine, up to isomorphism, the formal groups attached to the Dedekind zeta functions of algebraic number fields and also the formal groups associated to several kinds of L-functions (series) with characters. It is rather interesting that the Dedekind zeta function of a non-Abelian algebraic number field yields a commutative formal group (see Theorem 16 below).
3 citations
1 Jan 1978
3 citations
1 Jan 1978
TL;DR: In this article, it was shown that the Riemann zeta function £(s) has exactly 70,000,000 zeros a + it in the region 0 < t < 30,549,654.
Abstract: We describe a computation which shows that the Riemann zeta function £(s) has exactly 70,000,000 zeros a + it in the region 0 < t < 30,549,654. Moreover, all these zeros are simple and lie on the line a = h. (A similar result for the first 3,500,000 zeros was established by Rosser, Yohe and Schoenfeld.) Counts of the number of Gram blocks of various types and the number of failures of "Rossers rule" up to Gram number 70,000,000 are given. AMS (MPS) Subject Classifications (1970) Primary 10H05 ; Secondary 10-04 , 65E05 , 30-04.
1 citations
1 citations
TL;DR: In this article, it was shown that the Dedekind zeta function of any finite normal extension of a number field K can be removed if Artin's conjecture is true, and that the normality assumption can also be removed by using all Schwartz-Bruhat functions as test functions.
Abstract: In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as "test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin's conjecture is true. Introduction. We shall first review the definition of the zeta distribution associated with a number field K: let AK resp. A' denote the adele resp. idele groups of K, d Xx a Haar measure on A', and S (AK) the Schwartz-Bruhat space of the locally compact additive group AK; the topological dual S (AK)' of S (AK) is then the space of tempered distributions on AK. Let {xl denote the modulus of an idele x; then