Vinogradov's Integral and Bounds for the Riemann Zeta Function
By Kevin Ford. 2002
159 citations recorded by the source corpus.
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 Wooley. An auxiliary result is the exponential sum bound S(N,t)⩽9.463N1−1/(133.66λ2) , where N is a positive integer, t is a real number, λ = log (t)/(log N) and S(N,t)=max0
Subfields: Algebra and Number Theory, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/e6e56faf01a8a2a62d825153c2c5f73df80a331d
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In Algebra and Number Theory
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Citations in the corpus
- Nested efficient congruencing and relatives of Vinogradov's mean value theorem (2017, background): “mann zeta function, and the investigation of equidistribution modulo 1 of polynomial sequences, all depend for their success on estimates associated with Vinogradov’s mean value theorem (see [6, 48], [15], [3, 56], respectively). We direct the reader to [50] for an overview of several other applicat”
- Vinogradov's mean value theorem via efficient congruencing, II (2011, methodology): “ere may be some scope for improvement in the upper bound presented in Theorem 1.3 by exploiting the sharpest bounds available from Vinogradov’s mean value theorem for smaller moments (see [37], [39], [10] and [6]). In this way, one might hope to improve even the coefficient of kin the upper bound for ”