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Ukrainian Mathematical Journal

, Volume 66, Issue 3, pp 415–431 | Cite as

Generalized Bombieri–Lagarias’ Theorem and Generalized Li’s Criterion with its Arithmetic Interpretation

  • S. K. Sekatskii
Article
We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums
$$ {k}_n={\Sigma}_{\uprho}\left(1-{\left(1-\frac{1}{\uprho}\right)}^n\right) $$
over zeros of the Riemann xi-function and the derivatives
$$ \begin{array}{ccc}\hfill {\uplambda}_n\equiv \frac{1}{\left(n-1\right)!}\frac{d^n}{d{z}^n}{\left.\left({z}^{n-1} \ln \left(\upxi (z)\right)\right)\right|}_{z=1},\hfill & \hfill \mathrm{where}\hfill & \hfill n=1,2,3,\dots, \hfill \end{array} $$
are nonnegative if and only if the Riemann hypothesis is true, can be generalized and the nonnegativity of certain derivatives of the Riemann xi-function estimated at an arbitrary real point a, except a = 1/2, can be used as a criterion equivalent to the Riemann hypothesis. Namely, we demonstrate that the sums
$$ {k}_{n,a}={\Sigma}_{\uprho}\left(1-{\left(\frac{\uprho -a}{\uprho +a-1}\right)}^n\right) $$
for any real a such that a < 1/2 are nonnegative if and only if the Riemann hypothesis is true (correspondingly, the same derivatives with a > 1/2 should be nonpositive). The arithmetic interpretation of the generalized Li’s criterion is given. Similarly to Li’s criterion, the theorem of Bombieri and Lagarias applied to certain multisets of complex numbers is also generalized along the same lines.

Keywords

Nonnegative Integer Riemann Hypothesis Riemann Function Arbitrary Real Number Nontrivial Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • S. K. Sekatskii
    • 1
  1. 1.LPMV, Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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