Vinogradov's Integral and Bounds for the Riemann Zeta Function

  title={Vinogradov's Integral and Bounds for the Riemann Zeta Function},
  author={Kevin Ford},
  journal={Proceedings of The London Mathematical Society},
  • 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

Tables from this paper

Explicit $L^2$ bounds for the Riemann $\zeta$ function
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
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
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


On exponential sums over smooth numbers.
This paper is concerned with the theory and applications of exponential sums over smooth numbers. Despite the numerous applications stemming from suitable estimates for these exponential sums, thusExpand
An Explicit Zero-Free Region for the Riemann Zeta-Function
This paper gives an explicit zero-free region for the Riemann zeta-function derived from the VinogradovKorobov method. We prove that the Riemann zeta-function does not vanish in the region σ ≥ 1 −Expand
Topics in Multiplicative Number Theory
Three basic principles.- The large sieve.- Arithmetic formulations of the large sieve.- A weighted sieve and its application.- A lower bound of Roth.- Classical mean value theorems.- New mean valueExpand
Quasi-diagonal behaviour in certain mean value theorems of additive number theory
Of fundamental significance in many problems of additive number theory are estimates for mean values of exponential sums over polynomial functions. In this paper we shall show that the exponentialExpand
Approximate formulas for some functions of prime numbers
The Riemann zeta-function
The disclosed method and apparatus relate to the rejuvenation of a bed of granular filter medium which has accumulated particulate and lint contaminants during filtration flow of dirty liquid. TheExpand
A note on simultaneous congruences
Abstract We prove, by an essentially elementary argument, that the number of non-singular solutions of a system ofdsimultaneous congruences, to a prime power modulus, indvariables is at most theExpand
Some effective estimation in the theory of the Hurwitz zeta function, Funct
  • Approx. Comment. Math
  • 1994
Some effective estimation in the theory of the Hurwitz zeta function, Funct
  • Approx. Comment. Math
  • 1994
Some new estimates in the Dirichlet divisor problem