2018/11/14 by S. K. Sekatskiǐ, Sekatskii, Sergey, Stefano Beltraminelli +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1811.05861
openalex publication_date 2018/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes kn,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true, and proved the relation kn,a=n*(1-2a)/(n-1)!*dn/dzn((z-a)^(n-1)*ln(ξ(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0Re(a)>1/2+/Delta+delta0, where /delta0 is an arbitrary small fixed positive number, one has dn/dsn(ln(/zeta(s))=Sum_(m<=N)((-1)n*/Lambda(m)*ln^(n-1)(m)/ma) + Int_(0)^(N)(x^(-a)*ln^(n-1)(x)*dx)+O(N^(1/2+Delta-a)*ln^(n-1)(N)); derivative is taken at s=a. In particular, d(ln(/zeta(a))/da=-Sum_(m<=N)(/Lambda(m)/ma+N^(1-a)/(1-a)+O(N^(1/2+/Delta-a)). Numerical verifications of these equalities are also presented.