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1.
We investigate here a new numerical method, base on the Laguerre inequalities, for determining lower bounds for the de Bruijn-Newman constant ∧, which is related to the Riemann Hypothesis. (Specifically, the truth of the Riemann Hypothesis would imply that ∧≦0.) Unlike previous methods which involved either finding nonreal zeros of associated Jensen polynomials or finding nonreal zeros of a certain real entire function, this new method depends only on evaluating, in real arithmetic, the Laguerre difference $$L_1 (H_\lambda (x))\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = (H'_\lambda (x))^2 - H_\lambda (x) \cdot H''_{_\lambda } (x){\text{ (}}x,{\text{ }}\lambda \in \mathbb{R}{\text{)}}$$ where \((H_\lambda (z)\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = \int_0^\infty {e^{\lambda t^2 } \Phi (t)}\) cos(tz)dt is a real entire function. We apply this method to obtain the new lower bound for ∧, -0.0991 < ∧ which improves all previously published lower bounds for ∧.  相似文献   

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The explicit formulas of Riemann and Guinand-Weil relate the set of prime numbers with the set of nontrivial zeros of the zeta function of Riemann. We recall Alain Connes’ spectral interpretation of the critical zeros of the Riemann zeta function as eigenvalues of the absorption spectrum of an unbounded operator in a suitable Hilbert space. We then give a spectral interpretation of the zeros of the Dedekind zeta function of an algebraic number field K of degree n in an automorphic setting.

If K is a complex quadratic field, the torical forms are the functions defined on the modular surface X, such that the sum of this function over the “Gauss set” of K is zero, and Eisenstein series provide such torical forms.

In the case of a general number field, one can associate to K a maximal torus T of the general linear group G. The torical forms are the functions defined on the modular variety X associated to G, such that the integral over the subvariety induced by T is zero. Alternately, the torical forms are the functions which are orthogonal to orbital series on X.

We show here that the Riemann hypothesis is equivalent to certain conditions bearing on spaces of torical forms, constructed from Eisenstein series, the torical wave packets. Furthermore, we define a Hilbert space and a self-adjoint operator on this space, whose spectrum equals the set of critical zeros of the Dedekind zeta function of K.  相似文献   


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We obtain necessary and sufficient conditions for the Riemann hypothesis for the Riemann zeta-function, in terms of the functional distribution of quadratic Dirichlet L-functions. Received: 29 November 2004  相似文献   

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LetV(t) be the even function on (–, ) which is related to the Riemann xi-function by (x/2)=4 exp(ixtV(t))dt. In a proof of certain moment inequalities which are necessary for the validity of the Riemann Hypothesis, it was previously shown thatV'(t)/t is increasing on (0, ). We prove a stronger property which is related to the GHS inequality of statistical mechanics, namely thatV' is convex on [0, ). The possible relevance of the convexity ofV' to the Riemann Hypothesis is discussed.Communicated by Richard Varga.  相似文献   

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We show that, for a listable set P of polynomials with integer coefficients, the statement “for all roots θ of all polynomials in P, the generalized Riemann hypothesis for Q(θ) holds” is Diophantine. That is, the statement is equivalent to the unsolvability of a particular Diophantine equation. This is achieved by finding a decidable property P such that the aforementioned statement may be written in the form “P holds for all natural numbers”.  相似文献   

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For every positive integer n, let be the set of primitive Dirichlet characters modulo n. We show that if the Riemann hypothesis is true, then the inequality holds for all k?1, where nk is the product of the first k primes, γ is the Euler-Mascheroni constant, C2 is the twin prime constant, and φ(n) is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many k for which the same inequality holds and infinitely many k for which it fails to hold.  相似文献   

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In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL(2), i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis (GRH) for the above L-functions.  相似文献   

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The Ramanujan Journal - Let $$\sigma (n)=\sum _{d\mid n}d$$ be the sum of divisors function and $$\gamma =0.577\ldots $$ the Euler constant. In 1984, Robin proved that, under the Riemann...  相似文献   

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Moment inequalities for the discrete-time bulk service queue   总被引:1,自引:0,他引:1  
For the discrete-time bulk service queueing model, the mean and variance of the steady-state queue length can be expressed in terms of moments of the arrival distribution and series of the zeros of a characteristic equation. In this paper we investigate the behaviour of these series. In particular, we derive bounds on the series, from which bounds on the mean and variance of the queue length follow. We pay considerable attention to the case in which the arrivals follow a Poisson distribution. For this case, additional properties of the series are proved leading to even sharper bounds. The Poisson case serves as a pilot study for a broader range of distributions.  相似文献   

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We prove that the Riemann hypothesis on zeros of the zeta function (s) is equivalent to the equality
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G. Donadze 《Mathematical Notes》2012,91(3-4):334-338
We prove a statement equivalent to the Riemann hypothesis.  相似文献   

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