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.

Are you an author of this paper? Check the Twitter handle we have for you is correct.

Sergey Sekatskii | (add twitter) | |

Stefano Beltraminelli | (add twitter) |

Ask the authors of this paper a question or leave a comment.

User:

None
(add)

Repo:

None
(add)

Stargazers:

0

Forks:

0

Open Issues:

0

Network:

0

Subscribers:

0

Language:

None

Sample Sizes (N=):

Inserted:

Words Total:

Words Unique:

Source:

Abstract:

MathPaper:
Approximation of the derivatives of the logarithm of the Riemann zeta-function in the critical strip. https://t.co/kOiLfG62YF