Fujii’s development on Chebyshev’s conjecture

@article{Platt2019FujiisDO,
  title={Fujii’s development on Chebyshev’s conjecture},
  author={Dave Platt and Tim Trudgian},
  journal={International Journal of Number Theory},
  year={2019}
}
Chebyshev presented a conjecture after observing the apparent bias towards primes congruent to [Formula: see text]. His conjecture is equivalent to a version of the Generalized Riemann Hypothesis. Fujii strengthened this conjecture; we strengthen it still further using detailed computations of zeroes of Dirichlet [Formula: see text]-functions. 
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References

SHOWING 1-10 OF 16 REFERENCES
Chebyshev's Bias
TLDR
It is shown that the bias to the distribution to primes in ideal classes in number fields, and to prime geodesics in homology classes on hyperbolic surfaces, can be characterized exactly those moduli and residue classes for which the bias is present. Expand
Chebyshev's conjecture and the prime number race
We survey results about prime number races, that is, results about the relative sizes of prime counting functions $\pi_{q,a}(x)$, with $q$ fixed and $a$ varying. In particular, we describe recentExpand
Artin's Conjecture, Turing's Method, and the Riemann Hypothesis
  • A. Booker
  • Mathematics, Computer Science
  • Exp. Math.
  • 2006
TLDR
A group-theoretic criterion under which one may verify the Artin conjecture for some Galois representations, up to finite height in the complex plane is presented and a rigorous algorithm for computing general L-functions on the critical line via the fast Fourier transform is developed. Expand
An improved upper bound for the error in the zero-counting formulae for Dirichlet L-functions and Dedekind zeta-functions
  • T. Trudgian
  • Computer Science, Mathematics
  • Math. Comput.
  • 2015
This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-functions in rectangles.
THE PRIME NUMBER RACE AND ZEROS OF DIRICHLET L-FUNCTIONS OFF THE CRITICAL LINE: PART III
We show, for any q > 3 and distinct reduced residues a,b (mod q), the existence of certain hypothetical sets of zeros of Dirichlet L-functions lying off the critical line implies that �(x;q,a) <Expand
Arb: Efficient Arbitrary-Precision Midpoint-Radius Interval Arithmetic
TLDR
This work discusses the low-level number representation, strategies for precision and error bounds, and the implementation of efficient polynomial arithmetic with interval coefficients in C Arb. Expand
Über einige ältere Vermutungen und Behauptungen in der Primzahltheorie
----------------------------------------------------Nutzungsbedingungen DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf NutzungExpand
  • 1974
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