Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip
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 k_n,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 k_n,a=n*(1-2a)/(n-1)!*d^n/dz^n((z-a)^(n-1)*ln(\xi(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0</Delta<1/2, using this relation together with the functional equation for the /xi-function and the explicit formula of Weil, we prove that in these conditions for n=1, 2, 3... and an arbitrary complex a with 1>Re(a)>1/2+/Delta+delta_0, where /delta_0 is an arbitrary small fixed positive number, one has d^n/ds^n(ln(/zeta(s))=Sum_(m<=N)((-1)^n*/Lambda(m)*ln^(n-1)(m)/m^a) + 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)/m^a+N^(1-a)/(1-a)+O(N^(1/2+/Delta-a)). Numerical verifications of these equalities are also presented.
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Sergey Sekatskii (add twitter)
Stefano Beltraminelli (add twitter)
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Mathematics - Number Theory

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11/14/18 06:03PM
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