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

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We show that Li’s criterion equivalent to the Riemann hypothesis, i.e., the statement that the sums
over zeros of the Riemann xi-function and the derivatives
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
for any real

$$ {k}_n={\Sigma}_{\uprho}\left(1-{\left(1-\frac{1}{\uprho}\right)}^n\right) $$

$$ \begin{array}{ccc}\hfill {\uplambda}_n\equiv \frac{1}{\left(n-1\right)!}\frac{d^n}{d{z}^n}{\left.\left({z}^{n-1} \ln \left(\upxi (z)\right)\right)\right|}_{z=1},\hfill & \hfill \mathrm{where}\hfill & \hfill n=1,2,3,\dots, \hfill \end{array} $$

*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$$ {k}_{n,a}={\Sigma}_{\uprho}\left(1-{\left(\frac{\uprho -a}{\uprho +a-1}\right)}^n\right) $$

*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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