Numerical computations concerning the GRH
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Two new algorithms for the efficient and rigorous computation of Dirichlet L-functions are described and their use to verify the Generalised Riemann Hypothesis for all such L-Functions associated with primitive characters of modulus q<=400,000 is described.Abstract:
We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q<=400,000. For even q, we check to height t_0=max(1e8/q,7.5e7/q+200) and for odd q to height t_0=max(1e8/q,3.75e7/q+200).read more
Citations
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Journal ArticleDOI
Explicit bounds for primes in arithmetic progressions
TL;DR: In this paper, explicit upper bounds for various counting functions for primes in arithmetic progressions were derived for the special case of quadratic characters and for exceptional zeros, but only for much larger values of $x.
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Isolating some non-trivial zeros of Zeta
TL;DR: A rigorous algorithm to compute Riemann’s zeta function on the half line and its use to isolate the non-trivial zeros of zeta with imaginary part to an absolute precision of ±2−102 is described.
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Explicit bounds for primes in arithmetic progressions
TL;DR: In this paper, the authors derived explicit upper bounds for various functions counting primes in arithmetic progressions, and showed that these upper bounds can be improved for much larger values of $x.
Posted ContentDOI
The ternary Goldbach problem
TL;DR: The ternary Goldbach conjecture, or three-primes problem, was proved by Hardy, Littlewood and Vinogradov in 2013 as mentioned in this paper, using the circle method, the large sieve and exponential sums.
Journal Article
An explicit lower bound on moduli of Dirichlet L-functions at s = 1
TL;DR: In this article, the explicit lower bound on |L(1, χ) was improved to |L (1 + χ), where χ is a non-quadratic primitive Dirichlet character.
References
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