FP//LINSPACE computability of Riemann zeta function in Ko-Friedman model
نویسنده
چکیده
In the present paper, we construct an algorithm for the evaluation of real Riemann zeta function ζ(s) for all real s, s > 1, in polynomial time and linear space on Turing machines in Ko–Friedman model. The algorithms is based on a series expansion of real Riemann zeta function ζ(s) (the series globally convergents) and uses algorithms for the evaluation of real function (1 + x) h and hypergeo-metric series in polynomial time and linear space. The algorithm (modified in an obvious way to work with the complex numbers) from the present paper can be used to evaluate complex Riemann zeta function ζ(s) for s = σ +it, σ > 1, in polynomial time and linear space in n wherein 2 −n is a precision of the computation (but the modified algorithm will be exponential in time and space in ⌈log 2 (t)⌉).
منابع مشابه
A more accurate half-discrete Hardy-Hilbert-type inequality with the best possible constant factor related to the extended Riemann-Zeta function
By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the rever...
متن کاملUnprovability, phase transitions and the Riemann zeta-function
Unprovability Theory started with Kurt Gödel’s incompleteness theorems in 1931 but only gained mathematical significance since the late 1970s when Jeff Paris and Harvey Friedman discovered the first few families of interesting combinatorial statements that cannot be proved using the axioms of Peano Arithmetic or even some stronger axiomatic systems. In this survey article we briefly introduce t...
متن کاملRiemann Zeta Function with Odd Arguments
Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in ther...
متن کاملOn the cyclicity of the group of Fp-rational points of non-CM elliptic curves
ABSTRACT: Let E be an elliptic curve defined over Q and without complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. Assuming that certain Dedekind zeta functions have no zeros in Re(s) > 3/4, we determine how often E(Fp) is a cyclic group. This result was previously obtained by J. -P. Serre using the full Generalized Riemann Hypothesis for the same Ded...
متن کاملNewman’s Conjecture in Function Fields
De Bruijn and Newman introduced a deformation of the completed Riemann zeta function ζ, and proved there is a real constant Λ which encodes the movement of the nontrivial zeros of ζ under the deformation. The Riemann hypothesis is equivalent to the assertion that Λ ¤ 0. Newman, however, conjectured that Λ ¥ 0, remarking, “the new conjecture is a quantitative version of the dictum that the Riema...
متن کامل