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1.
Deux extensions de Théorèmes de Hamburger (portant sur l’équation fonctionnelle de la fonction zêta)
Jean-François Burnol 《Expositiones Mathematicae》2012,30(3):295-308
We propose two types of extensions to Hamburger’s theorems on the Dirichlet series with a functional equation like the one of the Riemann zeta function, under weaker hypotheses. This builds upon the dictionary between the moderate meromorphic functions with the functional equation and the tempered distributions with an extended S-support condition. 相似文献
2.
Tsukasa Hayashi 《Journal of Number Theory》2007,125(2):473-527
In this paper, we study the zeta function, named non-abelian zeta function, defined by Lin Weng. We can represent Weng's rank r zeta function of an algebraic number field F as the integration of the Eisenstein series over the moduli space of the semi-stable OF-lattices with rank r. For r=2, in the case of F=Q, Weng proved that it can be written by the Riemann zeta function, and Lagarias and Suzuki proved that it satisfies the Riemann hypothesis. These results were generalized by the author to imaginary quadratic fields and by Lin Weng to general number fields. This paper presents proofs of both these results. It derives a formula (first found by Weng) for Weng's rank 2 zeta functions for general number fields, and then proves the Riemann hypothesis holds for such zeta functions. 相似文献
3.
Nicolas Bouleau 《Potential Analysis》1992,1(4):379-384
The average of the values of a function f on the points of an equidistributed sequence in [0, 1]
s
converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions. 相似文献
4.
Lejla Smajlovi? 《Journal of Number Theory》2010,130(4):828-851
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In this paper, we shall prove a generalization of Li's positivity criterion for the Riemann hypothesis for the extended Selberg class with an Euler sum. We shall also obtain two arithmetic expressions for Li's constants , where the sum is taken over all non-trivial zeros of the function F and the indicates that the sum is taken in the sense of the limit as T→∞ of the sum over ρ with |Imρ|?T. The first expression of λF(n), for functions in the extended Selberg class, having an Euler sum is given terms of analogues of Stieltjes constants (up to some gamma factors). The second expression, for functions in the Selberg class, non-vanishing on the line , is given in terms of a certain limit of the sum over primes.Video
For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EwDtXrkuwxA. 相似文献5.
Xian-Jin Li 《Journal of Number Theory》2005,113(1):175-200
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the “explicit formula” of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence subgroups Γ0(N). The nonnegativity of these coefficients gives a criterion for the Riemann hypothesis for all these L-functions at once. 相似文献
6.
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
7.
Udayan B. Darji Michael J. Evans 《Journal of Mathematical Analysis and Applications》2008,347(2):381-390
Benedetto Bongiorno constructed a certain class of improperly Riemann integrable functions on [0,1] which are not first-return integrable. He asked if all improper Riemann integrable functions which are not Lebesgue integrable are not first-return integrable. Recently David Fremlin provided a clever example to show that this is not the case. It remains open as to which functions are first-return integrable. We prove two general theorems which imply the existence of a large class of improperly Riemann integrable functions which are not first-return integrable. As a corollary we obtain that there is an improperly Riemann integrable function which is C∞ on (0,1] yet fails to be first-return integrable. 相似文献
8.
Takashi Nakamura 《Journal of Number Theory》2007,125(2):424-441
In this paper, we show the following theorems. Suppose 0<al<1 are algebraically independent numbers and 0<λl?1 for 1?l?m. Then we have the joint t-universality for Lerch zeta functions L(λl,al,s) for 1?l?m. Next we generalize Lerch zeta functions, and obtain the joint t-universality for them. In addition, we show examples of the non-existence of the joint t-universality for Lerch zeta functions and generalized Lerch zeta functions. 相似文献
9.
D. Azagra J.B. Seoane-Sepúlveda 《Journal of Mathematical Analysis and Applications》2009,354(1):229-233
If f is continuous on the interval [a,b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [α,β] and g([α,β])⊂[a,b], then f○g is Riemann integrable (resp. measurable) on [α,β]. A well-known fact, on the other hand, states that f○g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a c2-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f∈V?{0} and g∈W?{0}, f○g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g∈W?{0} there exists a c-dimensional space V of measurable functions such that f○g is not measurable for all f∈V?{0}. 相似文献
10.
Masatoshi Suzuki 《Journal of Number Theory》2011,131(10):1770-1796
In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions. 相似文献
11.
Xia Gao 《Journal of Number Theory》2011,131(6):994-1019
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The goal of this note is to generalize a formula of Datskovsky and Wright on the zeta function associated with integral binary cubic forms. We show that for a fixed number field K of degree d, the zeta function associated with decomposable forms belonging to K in d−1 variables can be factored into a product of Riemann and Dedekind zeta functions in a similar fashion. We establish a one-to-one correspondence between the pure module classes of rank d−1 of K and the integral ideals of width <d−1. This reduces the problem to counting integral ideals of a special type, which can be solved using a tailored Moebius inversion argument. As a by-product, we obtain a characterization of the conductor ideals for orders of number fields.Video
For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=RePyaF8vDnE. 相似文献12.
Hidehiko Mishou 《Archiv der Mathematik》2008,90(3):230-238
In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function
attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these
two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta
function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for
these two zeta functions.
Received: 17 Apri1 2007 相似文献
13.
The Bernoulli convolution νλ measure is shown to be absolutely continuous with L2 density for almost all , and singular if λ−1 is a Pisot number. It is an open question whether the Pisot type Bernoulli convolutions are the only singular ones. In this paper, we construct a family of non-Pisot type Bernoulli convolutions νλ such that their density functions, if they exist, are not L2. We also construct other Bernoulli convolutions whose density functions, if they exist, behave rather badly. 相似文献
14.
Hidehiko Mishou 《Journal of Number Theory》2011,131(12):2352-2367
In this paper we investigate the joint functional distribution for a pair of Hurwitz zeta functions ζ(s,αj) (j=1,2) in the case that real transcendental numbers α1 and α2 satisfy α2∈Q(α1). Especially we establish the joint universality theorem for these zeta functions. 相似文献
15.
We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however, a major difficulty for this problem is the existence of non-integrable singularities of the function q y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann–Hilbert problem to an equivalent modified Riemann–Hilbert problem, we show that the solution can be expressed in terms of a 2×2 matrix Riemann–Hilbert problem whose “jump matrix” depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h(λ). The determination of the function h remains open. 相似文献
16.
17.
Somjit Dutt 《Journal of Number Theory》2011,131(9):1547-1552
We prove two identities involving Dirichlet series, in the denominators of whose terms sums of two, three and four squares appear. These follow from two classical identities of Jacobi involving the four Jacobian Theta Functions θ1(z;q), θ2(z;q), θ3(z;q) and θ4(z;q), by the application of the Mellin transform. These results motivate the well-known correspondence between the set of the four Jacobian Theta Functions and the set of four classical zeta functions of which the Riemann Zeta Function is the third, and the Dirichlet Beta Function is the first. 相似文献
18.
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In this paper, using the fermionic p -adic integral on Zp, we define the corresponding p-adic Log Gamma functions, so-called p-adic Diamond–Euler Log Gamma functions. We then prove several fundamental results for these p-adic Log Gamma functions, including the Laurent series expansion, the distribution formula, the functional equation and the reflection formula. We express the derivative of p-adic Euler L -functions at s=0 and the special values of p-adic Euler L-functions at positive integers as linear combinations of p-adic Diamond–Euler Log Gamma functions. Finally, using the p-adic Diamond–Euler Log Gamma functions, we obtain the formula for the derivative of the p -adic Hurwitz-type Euler zeta function at s=0, then we show that the p-adic Hurwitz-type Euler zeta functions will appear in the studying for a special case of p -adic analogue of the (S,T)-version of the abelian rank one Stark conjecture.Video
For a video summary of this paper, please click here or visit http://youtu.be/DW77g3aPcFU. 相似文献19.
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
20.
Yoichi Uetake 《Integral Equations and Operator Theory》2008,60(2):271-288
We construct a Lax-Phillips scattering system on the arithmetic quotient space of the Poincaré upper half-plane by the full
modular group, based on the Eisenstein transform. We identify incoming and outgoing subspaces in the ambient space of all
functions with finite energy-form for the non-Euclidean wave equation. The use of the Eisenstein transform along with some
properties of the Eisenstein series of two variables enables one to work only on the space corresponding to the continuous
spectrum of the Laplace-Beltrami operator. It is shown that the scattering matrix is the complex function appearing in the
the functional equation of the Eisenstein series of two variables. We obtain a compression operator constructed from the Laplace-Beltrami
operator, whose spectrum consists of eigenvalues that coincide, counted with multiplicities, with the non-trivial zeros of
the Riemann zeta-function. For this purpose we construct and use a scattering model on the one-dimensional Euclidean space.
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