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

International Audience ◽

Riemann Zeta ◽

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 $\delta$.

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

10.46298/hrj.2009.164 ◽

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 ◽

International Audience ◽

Riemann Zeta ◽

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.

Some remarks on the mean value of the riemann zeta-function and other Dirichlet series-II

10.46298/hrj.1980.88 ◽

1980 ◽

Vol Volume 3 ◽

Author(s):

K Ramachandra

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Convergent Series ◽

Mean Value ◽

Dirichlet Polynomials ◽

Riemann Zeta ◽

The Mean ◽

Real Truth

International audience This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer. %This is of great interest, for little has been known on the mean value of $\vert\zeta(\frac{1}{2}+it)\vert^k$ for odd $k$, say $k=1$; for even $k$, see the book by E. C. Titchmarsh [The theory of the Riemann zeta function, Clarendon Press, Oxford, 1951, Theorem 7.19]. The proofs are based on applications of classical function-theoretic theorems, together with mean value theorems for Dirichlet polynomials or series. %In the case of the zeta function, the principle is to write $\vert\zeta(s)\vert^k=\vert\zeta(s)^{k/2}\vert^2$, where $\zeta(s)^{k/2}$ is related to a rapidly convergent series which is essentially a partial sum of the Dirichlet series of $\zeta(s)^{k/2}$, convergent in the half-plane $\sigma>1$.

The mean value theorem for the Riemann zeta‐function

Mathematika ◽

10.1112/s0025579300009414 ◽

1978 ◽

Vol 25 (2) ◽

pp. 177-184 ◽

Cited By ~ 16

Author(s):

D. R. Heath-Brown

Keyword(s):

Zeta Function ◽

Riemann Zeta Function ◽

Mean Value ◽

Mean Value Theorem ◽

Riemann Zeta ◽

Non-vanishing of Dirichlet series without Euler products

10.46298/hrj.2018.4027 ◽

2018 ◽

Vol Volume 40 ◽

Author(s):

William D. Banks

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Half Plane ◽

Riemann Zeta Function ◽

Euler Product ◽

International Audience ◽

Euler Products ◽

Riemann Zeta

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

Notes on the Riemann zeta Function-III

10.46298/hrj.1999.139 ◽

1999 ◽

Vol Volume 22 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra ◽

A Sankaranarayanan ◽

K Srinivas

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

International Audience ◽

Riemann Zeta

International audience For a good Dirichlet series $F(s)$ (see Definition in \S1) which is a quotient of some products of the translates of the Riemann zeta-function, we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also, we study the gaps between the ordinates of the consecutive poles of $F(s)$.

On Riemann zeta-function and allied questions-II

10.46298/hrj.1995.131 ◽

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 ◽

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 $T$.

A Lemma in complex function theory I

10.46298/hrj.1989.108 ◽

1989 ◽

Vol Volume 12 ◽

Author(s):

R Balasubramanian ◽

K Ramachandra

Keyword(s):

Zeta Function ◽

Dirichlet Series ◽

Riemann Zeta Function ◽

Function Theory ◽

Complex Function ◽

Complex Function Theory ◽

Closed Disc ◽

International Audience ◽

Riemann Zeta

International audience Continuing our earlier work on the same topic published in the same journal last year we prove the following result in this paper: If $f(z)$ is analytic in the closed disc $\vert z\vert\leq r$ where $\vert f(z)\vert\leq M$ holds, and $A\geq1$, then $\vert f(0)\vert\leq(24A\log M) (\frac{1}{2r}\int_{-r}^r \vert f(iy)\vert\,dy)+M^{-A}.$ Proof uses an averaging technique involving the use of the exponential function and has many applications to Dirichlet series and the Riemann zeta function.