Exponential Sums and the Riemann Zeta Function V

By M. Huxley. 2005

197 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9821ec7cf4095836f74bbfda2fca30dc42e9baaf

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In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Value-Distribution of L-Functions (2007, background): “The best known upper bound for the Riemann zeta-function is µζ(12) 6 89570 due to Huxley [11].”
Euler's constant: Euler's work and modern developments (2013, background): “ons of ∆(n) is that of Soundararajan [278] in 2003. Concerning upper bounds, a long improvements have resulted decreasing exponent down to θ= 131 416 ≈ 0.31490, a result following from Huxley ([165], [166]) in 2005. The exponent θ= 1/4 is conjectured to be the correct answer, and would correspond to”
Decoupling, exponential sums and the Riemann zeta function (2014, methodology): “ Ar(δ,Hδ),δ = R2 N2 (3.5) and B = B(Q) is a quantity related to the so-called ‘second spacing problem’ mentioned in the Introduction. At 12 this point, we invoke the treatment and estimate (8.8) from [H1] B ≪ Q R 2/3 M R14 3 N20 3 . (3.6) Huxley sets r = 5 and relies on the bound A5(δ,δH) ≪ δH7 obt”
Three-dimensional exponential sums with monomials (2006, background): “This is a particular case of [9], lemma 2.”
Zero-free regions for the Riemann zeta function (2019, unknown): “Better upper bounds are known for |ζ(1/2 + it)| for large t, the best being Oε(|t|) due to Huxley [10].”
Piatetski-Shapiro sequences (2012, unknown): “ any exponent pair (κ,λ) we derive that S≪ (H′K)1/2L+(HK)2+22+κκ (H′KγLγ) κ 2+2κ L 1+κ+λ 2+2κ logx. Examining the ‘worst’ case in the proof of [5, Theorem 2] leads us to choose the exponent pair (see [19]) (κ,λ) = BA4( 32 205 +ε,1 2 + 32 205 +ε) = (3843 8480 ,4304 8480 )+O(ε). Noting that the bound ”