There is only one pair of non-real zeros of ζ′′(s), and of ζ′′′(s), in the left half-plane. The Riemann Hypothesis implies that ζ′′(s) and ζ′′′(s) have no zeros in the strip 0 ≤ <s < 1 2 . It was shown by Speiser [3] that the Riemann Hypothesis is equivalent to ζ′(s) having no zeros in 0 < σ < 12 (as usual we write s = σ + it). Levinson and Montgomery [2] gave a different proof; moreover, they ...

This note introduces a new general conjecture correlating the dimensionality dT of an infinite lattice with N nodes to the asymptotic value of its Wiener Index W(N). In the limit of large N the general asymptotic behavior W(N)≈Ns is proposed, where the exponent s and dT are related by the conjectured formula s=2+1/dT allowing a new definition of dimensionality dW=(s-2)-1. Being related to the t...

In this note, we give some estimate of the generalized quadrature formula of Gauss-Jacobi$$underset{a}{overset{a+eta left( b,aright) }{int }}left( x-aright)^{p}left( a+eta left( b,aright) -xright) ^{q}fleft( xright) dx$$in the cases where $f$ and $left| fright| ^{lambda }$ for $lambda >1$, are $s$-preinvex functions in the second sense.