Power mean values of the Riemann zeta‐function

Mathematika ◽  
1982 ◽  
Vol 29 (2) ◽  
pp. 202-212 ◽  
Author(s):  
J.-M. Deshouillers ◽  
H. Iwaniec
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Power Mean ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals VINOGRADOV'S INTEGRAL AND BOUNDS FOR THE RIEMANN ZETA FUNCTION

2002 ◽  
Vol 85 (3) ◽  
pp. 565-633 ◽  
Author(s):  
KEVIN FORD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Auxiliary Result ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Secondary 11D72 ◽  
Explicit Bounds ◽  
Riemann Zeta

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 over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

scholarly journals On mean value results for the Riemann zeta-function in short intervals.

2009 ◽  
Vol Volume 32 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Upper And Lower Bounds ◽  
Mean Values ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience We discuss the mean values of the Riemann zeta-function $\zeta(s)$, and analyze upper and lower bounds for $$\int_T^{T+H} \vert\zeta(\frac{1}{2}+it)\vert^{2k}\,dt~~~~~~(k\in\mathbb{N}~{\rm fixed,}~1<\!\!< H \leq T).$$ In particular, the author's new upper bound for the above integral under the Riemann hypothesis is presented.

Sign in / Sign up

Export Citation Format

Share Document