Vinogradov's Integral and Bounds for the Riemann Zeta Function

@article{Ford2002VinogradovsIA,
  title={Vinogradov's Integral and Bounds for the Riemann Zeta Function},
  author={Kevin Ford},
  journal={Proceedings of the London Mathematical Society},
  year={2002},
  volume={85}
}
  • Kevin Ford
  • Published 1 November 2002
  • Mathematics
  • Proceedings of the London Mathematical Society
The main result is an upper bound for the Riemann zeta function in the critical strip: ζ(σ+it)⩽A|t|B(1−σ)3/2log2/3⁡|t| with A = 76.2 and B = 4.45, valid for ½ ⩽ σ ⩽ 1 and |t| ⩾ 3. The previous best constant B was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of… 
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The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(s) = 1/2. The first non-trivial zeros can be seen at Im(s) = ±14.135, ±21.022 and ±25.011. In
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