A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis
By Luis Báez-Duarte. 2002
101 citations recorded by the source corpus.
Let $\rho(x)=x-[x]$, $\chi=\chi_{(0,1)}$. In $L_2(0,\infty)$ consider the subspace $\B$ generated by $\{\rho_a | a \geq 1\}$ where $\rho_a(x):=\rho(\frac{1}{ax})$. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement $\chi\in\bar{\B}$. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that $\chi\in\bar{\Bnat}$ where $\Bnat$ is the much smaller subspace generated by $\{\rho_a | a\in\Nat\}$.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/14dfc77b1cec91e0a3bdac434b2084af42254ef4
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Étude de l’autocorrélation multiplicative de la fonction ‘partie fractionnaire’ (2005, background): “[4] et [2]), cela revient simplement à exiger que χ , la fonction indicatrice de [1, +∞[, soit limite dans H de combinaisons linéaires des eα , α ≥ 1.”
- On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis (2006, background): “(However, as already stated, the implication (i ) =⇒ (ii) of this theorem is its only part which explicitly occurs in [2].”
- Factorial ratios, hypergeometric series, and a family of step functions (2007, unknown): “10 [18,4,3] [9,8,6,2] 6 [ 1 18 , 5 18 , 7 18 , 11 18 , 13 18 , 17 18 ] [ (1)8 , 3 8 , 1 2 , 5 8 , 7 8 ] 48”