Computations concerning the conjecture of Mertens.

@article{Riele1979ComputationsCT,
  title={Computations concerning the conjecture of Mertens.},
  author={Herman J. J. te Riele},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={1979},
  volume={1979},
  pages={356 - 360}
}
  • H. Riele
  • Published 1979
  • Physics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
A self-contained data acquisition device or probe capable of monitoring any physical function that can be translated into an analog or digital signal. The data acquisition probe is coupled to a separate translator whereat the stored analog or digital voltage signals are transferred to the translator to be coupled to a display device for displaying the voltage signals. 
View via Publisher
8 Citations
Highly Influential Citations
1
Methods Citations
1

Tables from this paper

8 Citations

Citation Type

Citation Type

UNRESTRICTED ALGORITHMS FOR ELEMENTARY AND SPECIAL FUNCTIONS Invited Paper 1
TLDR
High-level languages support only a small number of floating-point precisions, and those which support an arbitrary number usually demand that the precision be determinable at compile time.
Computational number theory at CWI in 1970--1994
TLDR
A concise survey of the research in Computational Number Theory, carried out at CWI in the period 1970 to 1994, with updates to the present state-of-the-art of the various subjects, if necessary.
Unrestricted Algorithms for Elementary and Special Functions
  • R. Brent
  • Mathematics, Computer Science
    IFIP Congress
  • 1980
TLDR
The topics include: power series methods, use of halving identities, asymptotic expansions, continued fractions, recurrence relations, Newton's method, numerical contour integration, and the arithmetic-geometric mean.
Disproof of the Mertens conjecture.
The Mertens conjecture states that  M(x)  1, where M(x) = n ≤ x Σ μ(n) , and μ(n) is the Mo bius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of…
The Mertens conjecture
The Mertens conjecture states that |M(x)|x−1/2 < 1 for x > 1, where M(x) = ∑ 1≤n≤x μ(n) and where μ(n) is the Möbius function defined by: μ(1) = 1, μ(n) = (−1)k if n is the product of k distinct…
The Mertens Conjecture Revisited
TLDR
New numerical evidence is presented for the conjecture that M(x)/\sqrt{x}=\Omega_{\pm}(\sqrt\log\log x}) and the explicit upper bound of Pintz on the smallest number for which the Mertens conjecture is false is reduced.
On the History of the Function $$ M\left( \chi \right)/\sqrt \chi $$ Since Stieltjes
Yet another machine experiment in support of Riemann's conjecture

References

SHOWING 1-7 OF 7 REFERENCES
An Algorithm with Guaranteed Convergence for Finding a Zero of a Function
TLDR
An algorithm is presented for finding a zero of a function which changes sign in a given interval using linear interpolation and inverse quadratic interpolation with bisection and ALGOL 60 procedures.
A constructive approach to Kronecker approximations and its application to the Mertens conjecture.
holds if the Riemann hypothesis is true and if <Γ(ρν)Φθ ([14], 14. 27). The series in (1) is convergent in the following sense: There is a sequence Tn —> oo (independent of Λ:) such that j^iyv the…
Zeros of sections of the zeta function. II
We write s = c + it, and take 1M6 ? 3. Turain [1], [2] showed that the Riemann hypothesis would be true provided the zeros of DM(s) all had real parts ? 1 + k1 M"'2, for some positive k. The author…
Table of the Riemann Zeta Function
Übereine zahlentheoretische Funktion
  • Sitzungsberichte Akad. Wien 106, Abt. 2a
  • 1897
rewi, Computation of the zeros of the Riemann zeta function in the critical strip, Report CMU-CS-78-148, Dept
  • 1978
Linear relations connecting the imaginary parts of the zeros of the zeta function
  • Computers in number theory,
  • 1971