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Standard Models of Abstract Intersection Theory for Operators in Hilbert Space

Banaszak G., Uetake Y.

Bulletin of the Polish Academy of Sciences. Mathematics

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2015

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Vol. 63, no. 2

149--175

EN

For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper (2011) we constructed a GNS (Gelfand–Naimark–Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection theory for a given operator if and only if the Riemann hypothesis and semisimplicity hold for that operator. (For the definition of semisimplicity of an operator in Hilbert space, see the Introduction.) We show this result under a condition for a given operator which is much weaker than the condition in the previous paper. An operator satisfying this condition can be constructed by using the method of automorphic scattering of Uetake (2009). Combining this with a result from Uetake (2009), we can show that a Dirichlet L-function, including the Riemann zeta-function, satisfies the Riemann hypothesis and its all nontrivial zeros are simple if and only if there is a corresponding standard model of abstract intersection theory. Similar results can be proven for GNS models since the same technique of proof for standard models can be applied.

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Some remarks on the Báez-Duarte criterion for the Riemann hypothesis

Wolf M.

Computational Methods in Science and Technology

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2014

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Vol. 20, No. 2

39--47

EN

In this paper we are going to describe the results of the computer experiment, which in principle can rule out validity of the Riemann Hypothesis (RH). We use the sequence ck appearing in the Báez-Duarte criterion for the RH and compare two formulas for these numbers. We describe the mechanism of possible violation of the Riemann Hypothesis. Next we calculate c100000 with a thousand digits of accuracy using two different formulas for ck with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996th decimal place (accuracy of 10-996). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.

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Evidence in Favor of the Baez-Duarte Criterion for the Riemann Hypothesis

Wolf M.

Computational Methods in Science and Technology

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2008

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Vol. 14, nr 1

47-54

EN

We present the results of the numerical experiments in favor of the Baez-Duarte criterion for the Riemann Hypothesis. We give formulae allowing calculation of numerical values of the numbers ck appearing in this criterion for arbitrary large k. We present plots of ck for element of (1, 10 9).

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Li's criterion for the Riemann hypothesis - numerical approach

Maślanka K.

Opuscula Mathematica

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2004

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Vol. 24, no. 1

103-114

EN

There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li [9]. The present paper reports on a numerical computation of the first 3300 of Li's coefficients which appear in this criterion. The main empirical observation is that these coefficients can be separated in two parts. One of these grows smoothly while the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true.

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On differences of primes in short intervals under the Riemann hypothesis

Kaniecki L.

Demonstratio Mathematica

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1998

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Vol. 31, nr 1

121-124