scholarly journals Notes on the Riemann zeta-function-IV

1999 ◽  
Vol Volume 22 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra ◽  
A Sankaranarayanan
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience For ``good Dirichlet series'' $F(s)$ 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 numbers $p_2$ arranged in the non-decreasing order.

scholarly journals Non-vanishing of Dirichlet series without Euler products

2018 ◽  
Vol Volume 40 ◽  
Author(s):  
William D. Banks
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Half Plane ◽  
Riemann Zeta Function ◽  
Euler Product ◽  
The Riemann Zeta Function ◽  
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).

scholarly journals Notes on the Riemann zeta Function-III

1999 ◽  
Vol Volume 22 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra ◽  
A Sankaranarayanan ◽  
K Srinivas
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
The 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 $F(s)$.

scholarly journals A Lemma in complex function theory I

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 ◽  
The Riemann Zeta Function ◽  
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.

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 Alternating Euler sums at the negative integers.

2009 ◽  
Vol Volume 32 ◽  
Author(s):  
Khristo Boyadzhiev ◽  
H. Gopalkrishna Gadiyar ◽  
R Padma
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Complex Plane ◽  
Riemann Zeta Function ◽  
Euler Numbers ◽  
Euler Sums ◽  
The Riemann Zeta Function ◽  
Entire Complex ◽  
International Audience ◽  
Riemann Zeta

International audience We study three special Dirichlet series, two of them alternating, related to the Riemann zeta-function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.

On the zeros of linear combinations of derivatives of the Riemann zeta function, II

2018 ◽  
Vol 14 (02) ◽  
pp. 371-382
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Unique Integer ◽  
Riemann Zeta ◽  
Derivatives Of

The relevant number to the Dirichlet series [Formula: see text], is defined to be the unique integer [Formula: see text] with [Formula: see text], which maximizes the quantity [Formula: see text]. In this paper, we classify the set of all relevant numbers to the Dirichlet [Formula: see text]-functions. The zeros of linear combinations of [Formula: see text] and its derivatives are also studied. We give an asymptotic formula for the supremum of the real parts of zeros of such combinations. We also compute the degree of the largest derivative needed for such a combination to vanish at a certain point.

scholarly journals Disproof of some conjectures of K.Ramachandra

1999 ◽  
Vol Volume 22 ◽  
Author(s):  
johan anderson
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Universality Theorem ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
Voronin Universality

International audience In a recent paper K. Ramachandra states some conjectures, and gives consequences in the theory of the Riemann zeta function. In this paper we will present two different disproofs of them. The first will be an elementary application of the Szasz-M\"unto theorem. The second will depend on a version of the Voronin universality theorem, and is also slightly stronger in the sense that it disproves a weaker conjecture. An elementary (but more complicated) disproof has been given by Rusza-Lazkovich.

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