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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.
  • 8 June 2012
TLDR
This paper contains new explicit upper bounds for the number of zeroes of Dirichlet L-functions and Dedekind zeta-funfunctions in rectangles. Expand
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Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function
Abstract We prove that the Riemann zeta-function ζ ( σ + i t ) has no zeros in the region σ ≥ 1 − 1 / ( 5.573412 log ⁡ | t | ) for | t | ≥ 2 . This represents the largest known zero-free regionExpand
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On the first sign change of θ(x) - x
TLDR
We show that θ(x) < x for 2 < x < 1.39 ·10 . Expand
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  • 5
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An improved upper bound for the argument of the Riemann zeta-function on the critical line
  • T. Trudgian
  • Computer Science, Mathematics
  • Math. Comput.
  • 25 August 2011
TLDR
This paper concerns the function S(T), the argument of the Rie- mann zeta-function along the critical line. Expand
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  • 5
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A Log-Free Zero-Density Estimate and Small Gaps in Coefficients of L-Functions
Research of the authors is partially supported by NSERC. The second author research is also partially supported by ARC.
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Improvements to Turing's method
  • T. Trudgian
  • Computer Science, Mathematics
  • Math. Comput.
  • 11 March 2009
TLDR
This paper refines the argument of Lehman by reducing the size of the constants in Turing's method. Expand
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Linear relations of zeroes of the zeta-function
TLDR
This article considers linear relations between the non-trivial zeroes of the Riemann zeta-function. Expand
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BETWEEN THE PROBLEMS OF PÓLYA AND TURÁN
Abstract We investigate the behaviour of the function $L_{\alpha }(x) = \sum _{n\leq x}\lambda (n)/n^{\alpha }$, where $\lambda (n)$ is the Liouville function and $\alpha $ is a real parameter. TheExpand
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  • 2
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A new upper bound for $|\zeta(1+ it)|$
It is known that $|\zeta(1+ it)|\ll (\log t)^{2/3}$. This paper provides a new explicit estimate, viz.\ $|\zeta(1+ it)|\leq 3/4 \log t$, for $t\geq 3$. This gives the best upper bound on $|\zeta(1+Expand
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A still sharper region where π(x)-li(x) is positive
TLDR
In this article, we derive a stronger version of Lehman’s theorem involving a different weight function. Expand
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