On Solé and Planat criterion for the Riemann Hypothesis

On Robin’s criterion for the Riemann Hypothesis

Robin’s criterion states that the Riemann Hypothesis (RH) is true if and only if Robin’s inequality σ(n) := ∑ d|n d < en log log n is satisfied for n ≥ 5041, where γ denotes the Euler(Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin’s criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, moreo...

On the Balazard-saias Criterion for the Riemann Hypothesis

Báez-Duarte’s Criterion for the Riemann Hypothesis and Rice’s Integrals

Criterion for the Riemann hypothesis found by Báez-Duarte involves certain real coefficients ck defined as alternating binomial sums. These coefficients can be effectively investigated using Nörlund-Rice’s integrals. Their behavior exhibits characteristic trend, due to trivial zeros of zeta, and fading oscillations, due to complex zeros. This method enables to calculate numerical values of ck f...

Complements to Li's Criterion for the Riemann Hypothesis

In a recent paper Xian-Jin Li showed that the Riemann Hypothesis holds if and only if n = P 1?(1?1==) n has n > 0 for n = 1; 2; 3; : : : where runs over the complex zeros of the Riemann zeta function. We show that Li's criterion follows as a consequence of a general set of inequalities for an arbitrary multiset of complex numbers and therefore is not speciic to zeta functions. We also give an a...

A General Strong Nyman-beurling Criterion for the Riemann Hypothesis

For each f : [0,∞) → C formally consider its co-Poisson or Müntz transform g(x) = ∑ n≥1 f(nx) − 1 x ∫∞ 0 f(t)dt. For certain f ’s with both f, g ∈ L2(0,∞) it is true that the Riemann hypothesis holds if and only if f is in the L2 closure of the vector space generated by the dilations g(kx), k ∈ N. Such is the case for example when f = χ(0,1] where the above statement reduces to the strong Nyman...