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.

scholarly journals Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

1978 ◽  
Vol Volume 1 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Positive Constant ◽  
Riemann Zeta Function ◽  
Nonnegative Integer ◽  
Mean Value ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience The present paper is concerned with $\Omega$-estimates of the quantity $$(1/H)\int_{T}^{T+H}\vert(d^m/ds^m)\zeta^k(\frac{1}{2}+it)\vert dt$$ where $k$ is a positive number (not necessarily an integer), $m$ a nonnegative integer, and $(\log T)^{\delta}\leq H \leq T$, where $\delta$ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if $2k\geq1$ and $T\geq T_0(\delta)$, then $$(1/H)\int_{T}^{T+H}\vert \zeta(\frac{1}{2}+it)\vert^{2k}dt > (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.

scholarly journals On Riemann zeta-function and allied questions-II

1995 ◽  
Vol Volume 18 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Value ◽  
Dirichlet Polynomials ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience In this paper, two conjectures on the mean value of Dirichlet polynomials are given and are shown to imply good lower bound for $\int_H^{T+H}\vert\zeta(\frac{1}{2}+it)^k\vert^2\,dt$, uniform in $k$ and independent of $T$.

scholarly journals On the mean square of the Riemann zeta-function in short intervals

2009 ◽  
Vol 85 (99) ◽  
pp. 1-17 ◽  
Author(s):  
Aleksandar Ivic
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Mean Square ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean

It is proved that, for T? ? G = G(T)? ??T, ?T2T(I1(t+G,G)- I1(t,G))2 dt = TG ?aj logj (?T/G)+ O?(T1+? G1/2+ +T1/2?G? with some explicitly computable constants aj(a3>0)where, for fixed K ? N, Ik(t,G)= 1/?? ? ? -? |?(1/2 + it + iu)|2k e -(u/G)?du. The generalizations to the mean square of I1(t+U,G)-I1(t,G) over [T,T+H] and the estimation of the mean square of I2(t+ U,G) - I2(t,G) are also discussed.

scholarly journals Mean-value of the Riemann zeta-function and other remarks III

1983 ◽  
Vol Volume 6 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Continued Fraction ◽  
Mean Value ◽  
Continued Fraction Expansion ◽  
Mean Values ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
International Audience ◽  
Riemann Zeta

International audience The results given in these papers continue the theme developed in part I of this series. In Part III we prove $M(\frac{1}{2})>\!\!\!>_k (\log H_0/q_n)^{k^2}$, where $p_m/q_m$ is the $m$th convergent of the continued fraction expansion of $k$, and $n$ is the unique integer such that $q_nq_{n+1}\geq \log\log H_0 > q_nq_{n-1}$. Section 4 of part III discusses lower bounds of mean values of Titchmarsh series.

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