# 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},
pages={565-633}
}
• K. Ford
• Published 2002
• Mathematics
• Proceedings of The London Mathematical Society
The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. 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… Expand
114 Citations
Explicit $L^2$ bounds for the Riemann $\zeta$ function
• Mathematics
• 2019
Bounds on the tails of the zeta function $\zeta(s)$, and in particular explicit bounds, are needed for applications, notably for integrals involving $\zeta(s)$ on vertical lines or other paths goingExpand
Mean square of the Hurwitz zeta-function and other remarks
• Mathematics
• 2004
The Hurwitz zeta-function associated with the parameter $a\,(0 1)$$and its analytic continuation. %In fact$$\zeta(s,a)=\sum_{n=0}^{\infty}\left((n+a)^{-s}-\int_{n}^{n+1}\frac{du}{(u+a)^s}Expand On the multiplicites of zeros of$\zeta(s)$and its values over short intervals We investigate bounds for the multiplicities$m(\beta+i\gamma)$, where$\beta+i\gamma\,$($\beta\ge \1/2, \gamma>0)$denotes complex zeros of$\zeta(s)$. It is seen that the problem can be reduced toExpand An Explicit Upper Bound for$|\zeta(1+it)|$In this paper we provide an explicit bound for$|\zeta(1+it)|$in the form of$|\zeta(1+it)|\leq \min\left(\log t, \frac{1}{2}\log t+1.93, \frac{1}{5}\log t+44.02 \right)$. This improves on theExpand On a Hybrid Fourth Moment Involving the Riemann Zeta-Function • Mathematics • 2014 For each integer 1 ≤ j ≤ 6, we provide explicit ranges for σ for which the asymptotic formula $$\displaystyle{\int _{0}^{T}\left \vert \zeta \left (\frac{1} {2} + it\right )\right \vert ^{4}\vertExpand On the zeta function on the line Re(s) = 1 We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) asExpand A new kth derivative estimate for exponential sums via Vinogradov’s mean value We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov’s mean value. Coupling this with the recent works of Wooley, and ofExpand Zero-free regions for the Riemann zeta function We improve existing explicit bounds of Vinogradov-Korobov type for zero-free regions of the Riemann zeta function, both for large height t and for every t. A primary input is an explicit bound of theExpand Proof of the the Riemann hypothesis from the density and Lindelof hypotheses via a power sum method • Yuanyou Cheng • Mathematics • 2008 The Riemann hypothesis is equivalent to the \varpi-form of the prime number theorem as \varpi(x) =O(x\sp{1/2} \log\sp{2} x), where \varpi(x) =\sum\sb{n\le x}\ \bigl(\Lambda(n) -1\big) with theExpand On the Balasubramanian-Ramachandra method close to Re(s)=1 We study the problem on how to get good lower estimates for the integral$$ \int_T^{T+H} |\zeta(\sigma+it)| dt,$$when$H \ll 1$is small and$\sigma$is close to$1\$, as well as related integralsExpand