共查询到20条相似文献,搜索用时 15 毫秒
1.
Lithuanian Mathematical Journal - We generalize our zero-free regions of the integral derivatives for the Riemann zeta function to the general fractional derivatives case, and then we apply them to... 相似文献
2.
3.
Michael O. Rubinstein 《The Ramanujan Journal》2012,27(1):29-42
In this paper, we obtain several expansions for ζ(s) involving a sequence of polynomials in s, denoted by α
k
(s). These polynomials can be regarded as a generalization of Stirling numbers of the first kind and our identities extend some
series expansions for the zeta function that are known for integer values of s. The expansions also give a different approach to the analytic continuation of the Riemann zeta function. 相似文献
4.
Sergio Albeverio 《Bulletin des Sciences Mathématiques》2007,131(1):12
A formula first derived by Müntz which relates the Riemann zeta function ζ times the Mellin transform of a test function f and the Mellin transform of the theta transform of f is exploited, together with other analytic techniques, to construct zero free regions for ζ(s) with s in the critical strip. Among these are regions with a shape independent of Res. 相似文献
5.
6.
D. Masser 《Journal of Number Theory》2011,131(11):2037-2046
We prove the existence of a constant C such that for any D?3 there are at most rational numbers s with 2<s<3 and denominator at most D such that ζ(s) is also rational with denominator at most D. This is done by combining elements of the works of Bombieri-Pila, Pila, and Surroca with a new zero estimate. 相似文献
7.
8.
Science China Mathematics - Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-Pólya... 相似文献
9.
10.
The flow of the Riemann zeta function, , is considered, and phase portraits are presented. Attention is given to the characterization of the flow lines in the neighborhood of the first 500 zeros on the critical line. All of these zeros are foci. The majority are sources, but in a small proportion of exceptional cases the zero is a sink. To produce these portraits, the zeta function was evaluated numerically to 12 decimal places, in the region of interest, using the Chebyshev method and using Mathematica.
The phase diagrams suggest new analytic properties of zeta, of which some are proved and others are given in the form of conjectures.
11.
H. Mishou 《Lithuanian Mathematical Journal》2007,47(1):32-47
In this paper, we investigate the joint value-distribution for the Riemann zeta function and Hurwitz zeta function attached
with a transcendental real parameter. Especially, we establish the joint universality theorem for these two zeta functions.
Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 39–57, January–March, 2007. 相似文献
12.
A study is made of the function H(s, z) defined by analytic continuation of the Dirichlet series H(s, z) = Σn=1∞n?sΣm=1nm?z, where s and z are complex variables. For each fixed z it is shown that H(s, z) exists in the entire s-plane as a meromorphic function of s, and its poles and residues are determined. Also, for each fixed s ≠ 1 it is shown that H(s, z) exists in the entire z-plane as a meromorphic function of z, and again its poles and residues are determined. Two different representations of H(s, z) are given from which a reciprocity law, H(s, z) + H(z, s) = ζ(s) ζ(z) + ζ(s + z), is deduced. For each integer q ≥ 0 the function values H(s, ?q) and H(?q, s) are expressed in terms of the Riemann zeta function. Similar results are also obtained for the Dirichlet series T(s, z) = Σn=1∞n?sΣm=1nm?z (m + n)?1. Applications include identities previously obtained by Ramanujan, Williams, and Rao and Sarma. 相似文献
13.
Machiel van Frankenhuijsen 《Journal of Number Theory》2005,115(2):360-370
If the Riemann zeta function vanishes at each point of the finite arithmetic progression {D+inp}0<|n|<N (D?1/2, p>0), then N<13p if D=1/2, and N<p1/D-1+o(1) in general. 相似文献
14.
Lithuanian Mathematical Journal - In this short note, we prove the following result: If a completely multiplicative function f : ? → [?1, 1] is small on average in the sense that... 相似文献
15.
16.
17.
Djurdje Cvijovic Jacek Klinowski 《Proceedings of the American Mathematical Society》1997,125(9):2543-2550
It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.
18.
Let ζ be the Riemann zeta function and δ(x)=1/(2x-1). For all x>0 we have
(1-δ(x))ζ(x)+αδ(x)<ζ(x+1)<(1-δ(x))ζ(x)+βδ(x),