共查询到20条相似文献,搜索用时 15 毫秒
1.
Mathematical Notes - In 2007, H. Mishou proved the universality theorem on the joint approximation of a pair of analytic functions by the shifts $$(zeta(s+itau),zeta(s+itau,alpha))$$ of the... 相似文献
2.
Lithuanian Mathematical Journal - In this paper, we prove the universality theorem for the iterated integrals of the logarithm of the Riemann zeta-function on a line parallel to the real axis. 相似文献
3.
It is well known that the multiplicity of a complex zero =ß+iof the zeta-function is O(log||). This may be proved by meansof Jensen's formula, as in Titchmarsh [7, Chapter 9]. It mayalso be seen from the formula for the number N(T) of zeros suchthat 0<<T,
(1) due to Backlund [1], in which E(T) is a continuous functionsatisfying E(T)=O(1/T) and
(2) We assume here that T is not the ordinate of a zero; with appropriatedefinitions of N(T) and S(T) the formula is valid for all T.We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT),(Cramér [2]), and on the Riemann Hypothesis
(Littlewood [5]). These results are over 70 years old. Because the multiplicity problem is hard, it seems worthwhileto see what can be said about the number of distinct zeros ina short T-interval. We obtain the following result, which isindependent of any unproved hypothesis. 相似文献
4.
Goldston D. A.; Gonek S. M.; Ozluk A. E.; Snyder C. 《Proceedings London Mathematical Society》2000,80(1):31-49
To study the distribution of pairs of zeros of the Riemann zeta-function,Montgomery introduced the function
where is real and T 2, and ' denote the imaginary parts ofzeros of the Riemann zeta-function, and w(u) = 4/(4 + u2). Assumingthe Riemann Hypothesis, Montgomery proved an asymptotic formulafor F() when || 1, and made the conjecture that F() = 1 + o(1)as T for any bounded with || 1. In this paper we use anapproximation for the prime indicator function together witha new mean value theorem for long Dirichlet polynomials andtails of Dirichlet series to prove that, assuming the GeneralizedRiemann Hypothesis for all Dirichlet L-functions, then for any > 0 we have
uniformlyfor and all T T0(). 1991Mathematics Subject Classification: primary 11M26; secondary11P32. 相似文献
5.
Simple Zeros of the Riemann Zeta-Function 总被引:1,自引:0,他引:1
Assuming the Riemann Hypothesis, Montgomery showed by meansof his pair correlation method that at least two-thirds of thezeros of Riemann's zeta-function are simple. Later he and Taylorimproved this to 67.25 percent and, more recently, Cheer andGoldston increased the percentage to 67.2753. Here we proveby a new method that if the Riemann and Generalized LindelöofHypotheses hold, then at least 70.3704 percent of the zerosare simple and at least 84.5679 percent are distinct. Our methoduses mean value estimates for various functions defined by Dirichletseries sampled at the zeros of the Riemann zeta-function. 1991Mathematics Subject Classification: 11M26. 相似文献
6.
In this paper, we prove a joint limit theorem for the Riemann zeta-function in the space of analytic functions in the sense of weak convergence of probability measures. 相似文献
7.
Let E2(T) denote the error term in the asymptotic formula forT0|(+it)|4dt. It is proved that there exist constants A>0,B>1 such that for TT0>0 every interval [T, BT] containspoints T1, T2 for which
and that T0|E2(t)|adt>>T1+(a/2) for any fixed a1. Theseresults complement earlier results of Motohashi and Ivi thatT0E2(t)dt<<T3/2 and that T0E22(t)dt<<T2logCT forsome C>0. Omega-results analogous to the above ones are obtainedalso for the error term in the asymptotic formula for the Laplacetransform of |(+it)|4. 相似文献
8.
A. Laurinčikas R. Macaitienė D. Mochov D. Šiaučiūnas 《Siberian Mathematical Journal》2018,59(5):894-900
The periodic Hurwitz zeta-function, a generalization of the classical Hurwitz zeta-function, is defined by a Dirichlet series with periodic coefficients and depends on a fixed parameter. We show that a wide class of analytic functions is approximated by shifts of a periodic zeta-function with rational parameter. 相似文献
9.
Antanas Laurin?ikas 《Journal of Number Theory》2010,130(10):2323-2331
In 1975, S.M. Voronin proved the universality of the Riemann zeta-function ζ(s). This means that every non-vanishing analytic function can be approximated uniformly on compact subsets of the critical strip by shifts ζ(s+iτ). In the paper, we consider the functions F(ζ(s)) which are universal in the Voronin sense. 相似文献
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11.
By J. Steuding 《Abhandlungen aus dem Mathematischen Seminar der Universit?t Hamburg》2001,71(1):113-121
We consider the value distribution of Hurwitz zeta-functions
at the nontrivial zeros ϱ= β + iγ of the Riemann zeta-function ζ (s):= ζ (s, 1). Using the method of Conrey, Ghosh and Gonek we prove for fixed 0< α< 1 andH ≤T that
with some absolute constantC > 0 (a similar result was first proved by Fujii [4] under assumption of the Riemann hypothesis). It follows that
is an entire function if and only if α = 1/2 or α = l. Further, we prove for α ≠ 1/2, 1 the existence of zeros ϱ = β +iγ withT < γ ≤T + T3/4, 1/2 β ≤ 9/10+ ε and ζ(ϱ,α)≠0. 相似文献
12.
We consider the periodic zeta-function introduced by B. C. Berndt and L. Schoenfeld. We obtain the asymptotics of the mean square and prove limit theorems in the space of analytic functions for this function. 相似文献
13.
O. M. Fomenko 《Journal of Mathematical Sciences》2004,122(6):3679-3684
Let
, and let
be Epstein's zeta-function of the form Q. It is proved that for |t| > C > 0 one has the estimate
Bibliography: 9 titles. 相似文献
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15.
C. R. Putnam 《Archiv der Mathematik》1960,11(1):346-349
16.
杨明顺 《数学的实践与认识》2009,39(10)
由Riemannζ函数的函数方程得到Hurwitzζ函数的Hermite公式,再从Hermite公式得到Γ(s)的Binet′s第二表达式,从而由ζ函数推得Γ(s)的性质. 相似文献
17.
Mathematical Notes - In this paper, we define the Dirichlet series $$ \zeta_{u_T j} (s)$$ , $$ j = 1, \dots, r$$ , absolutely converging in the half-plane $$ \operatorname{Re} s> 1/2 $$ and... 相似文献
18.
A discrete limit theorem for the Lerch zeta-function with an integer parameter in the space of meromorphic functions is proved. 相似文献
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In the paper, the explicit form of the limit measure in a joint limit theorem for the Riemann zeta-function in the space of analytic functions is given. 相似文献