Hardy–Littlewood–Riesz type equivalent criteria for the Generalized Riemann hypothesis

Generalized Riemann Hypothesis

(Generalized) Riemann Hypothesis (that all non-trivial zeros of the (Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems ([BRW], [SA])...

Institute for Mathematical Physics the Nyman{beurling Equivalent Form for the Riemann Hypothesis the Nyman-beurling Equivalent Form for the Riemann Hypothesis

A “good” problem equivalent to the Riemann hypothesis

We exhibit a sequence cn such that the convergence of ∑ n≥1 cnz n for |z| < 1 is equivalent to the Riemann Hypothesis. We argue that this particular RH-equivalent problem is “better” than most, or perhaps every, other RH-equivalent problem devised so far, in the sense that (we prove) there is a tremendous gap in behaviors of the cn if the RH is true versus if the RH is false.

A Note on Nyman’s Equivalent Formulation of the Riemann Hypothesis

A certain subspace of L2((0, 1), dt) has been considered by Nyman, Beurling, and others, with the result that the constant function 1 belongs to it if and only if the Riemann Hypothesis holds. I show in this note that the product ∏ ζ(ρ)=0,Re(ρ)> 1 2 ∣∣1−ρ ρ ∣∣ is the norm of the projection of 1 to this subspace. This provides a quantitative refinement to Nyman’s theorem. 1 Nyman’s criterion for...

An Elementary Problem Equivalent to the Riemann Hypothesis

The problem is: Let Hn = n ∑ j=1 1 j be the n-th harmonic number. Show, for each n ≥ 1, that