On some mean values for the divisor function and the Riemann zeta-function

2017 ◽  
Vol 296 (1) ◽  
pp. 142-153
Author(s):  
Kar-Lun Kong ◽  
Kai-Man Tsang
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Divisor Function ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals The generalized divisor problem and the Riemann hypothesis

1991 ◽  
Vol 122 ◽  
pp. 149-159 ◽  
Author(s):  
Hideki Nakaya
Keyword(s):  
Zeta Function ◽  
Natural Number ◽  
Complex Number ◽  
Riemann Zeta Function ◽  
Multiplicative Function ◽  
Riemann Hypothesis ◽  
Divisor Problem ◽  
Divisor Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Let dz(n) be a multiplicative function defined bywhere s = σ + it, z is a. complex number, and ζ(s) is the Riemann zeta function. Here ζz(s) = exp(z log ζ(s)) and let log ζ(s) take real values for real s > 1. We note that if z is a natural number dz(n) coincides with the divisor function appearing in the Dirichlet-Piltz divisor problem, and d-1(n) with the Möbious function.

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 the Selberg integral of the k-divisor function and the 2k-th moment of the Riemann zeta-function

2010 ◽  
Vol 88 (102) ◽  
pp. 99-110 ◽  
Author(s):  
Giovanni Coppola
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Divisor Function ◽  
Elementary Approach ◽  
Selberg Integral ◽  
Reverse Link ◽  
Recent Approach ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Simple Average

In the literature one can find links between the 2k-th moment of the Riemann zeta-function and averages involving dk(n), the divisor function generated by ?k(s). There are, in fact, two bounds: one for the 2k-th moment of ?(s) coming from a simple average of correlations of the dk; and the other, which is a more recent approach, for the Selberg integral involving dk(n), applying known bounds for the 2k-th moment of the zeta-function. Building on the former work, we apply an elementary approach (based on arithmetic averages) in order to get the reverse link to the second work; i.e., we obtain (conditional) bounds for the 2k-th moment of the zeta-function from the Selberg integral bounds involving dk(n).

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 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.

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