An Elementary Problem Equivalent to the Riemann Hypothesis

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

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.

An Elementary Dyadic Riemann Hypothesis

The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula chi(G)=sum_x (-1)^dim(x) = p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negati...

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...

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