scholarly journals Some local-convexity theorems for the zeta-function-like analytic functions

1988 ◽  
Vol Volume 11 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Lower Bounds ◽  
Exponential Function ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Local Convexity ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience In this paper we investigate lower bounds for $$I(\sigma)= \int^H_{-H}\vert f(\sigma+it_0+iv)\vert^kdv,$$ where $f(s)$ is analytic for $s=\sigma+it$ in $\mathcal{R}=\{a\leq\sigma\leq b, t_0-H\leq t\leq t_0+H\}$ with $\vert f(s)\vert\leq M$ for $s\in\mathcal{R}$. Our method rests on a convexity technique, involving averaging with the exponential function. We prove a general lower bound result for $I(\sigma)$ and give an application concerning the Riemann zeta-function $\zeta(s)$. We also use our methods to prove that large values of $\vert\zeta(s)\vert$ are ``rare'' in a certain sense.

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 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.

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 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 UNIVERSALITY OF ZETA-FUNCTIONS OF CUSP FORMS AND NON-TRIVIAL ZEROS OF THE RIEMANN ZETA-FUNCTION

2021 ◽  
Vol 26 (1) ◽  
pp. 82-93
Author(s):  
Aidas Balčiūnas ◽  
Violeta Franckevič ◽  
Virginija Garbaliauskienė ◽  
Renata Macaitienė ◽  
Audronė Rimkevičienė
Keyword(s):  
Zeta Function ◽  
Wide Class ◽  
Riemann Zeta Function ◽  
Analytic Functions ◽  
Pair Correlation ◽  
Zeta Functions ◽  
Weak Form ◽  
Cusp Forms ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

It is known that zeta-functions ζ(s,F) of normalized Hecke-eigen cusp forms F are universal in the Voronin sense, i.e., their shifts ζ(s + iτ,F), τ R, approximate a wide class of analytic functions. In the paper, under a weak form of the Montgomery pair correlation conjecture, it is proved that the shifts ζ(s+iγkh,F), where γ1 < γ2 < ... is a sequence of imaginary parts of non-trivial zeros of the Riemann zeta function and h > 0, also approximate a wide class of analytic functions.

scholarly journals IRRATIONALITÉ AUX ENTIERS IMPAIRS POSITIFS D'UN q-ANALOGUE DE LA FONCTION ZÊTA DE RIEMANN

2010 ◽  
Vol 06 (05) ◽  
pp. 959-988 ◽  
Author(s):  
FRÉDÉRIC JOUHET ◽  
ELIE MOSAKI
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Zeta Function ◽  
Recent Result ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Positive Integers

Dans cet article, nous nous intéressons à un q-analogue aux entiers positifs de la fonction zêta de Riemann, que l'on peut écrire pour s ∈ ℕ* sous la forme ζq(s) = ∑k≥1qk∑d|kds-1. Nous donnons une nouvelle minoration de la dimension de l'espace vectoriel sur ℚ engendré, pour 1/q ∈ ℤ\{-1; 1} et A entier pair, par 1, ζq(3), ζq(5), …, ζq(A - 1). Ceci améliore un résultat récent de Krattenthaler, Rivoal et Zudilin ([13]). En particulier notre résultat a pour conséquence le fait que pour 1/q ∈ ℤ\{-1; 1}, au moins l'un des nombres ζq(3), ζq(5), ζq(7), ζq(9) est irrationnel. In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s ∈ ℕ* by ζq(s) = ∑k≥1qk∑d|kds-1. We give a new lower bound for the dimension of the vector space over ℚ spanned, for 1/q ∈ ℤ\{-1; 1} and an even integer A, by 1, ζq(3), ζq(5), …, ζq(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin ([13]). In particular, a consequence of our result is that for 1/q ∈ ℤ\{-1; 1}, at least one of the numbers ζq(3), ζq(5), ζq(7), ζq(9) is irrational.

scholarly journals A note on Hardy's theorem

2017 ◽  
Vol Volume 39 ◽  
Author(s):  
Usha K. Sangale
Keyword(s):  
Zeta Function ◽  
Simple Proof ◽  
Riemann Zeta Function ◽  
Hardy’S Theorem ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
Complex Zeros

International audience Hardy's theorem for the Riemann zeta-function ζ(s) says that it admits infinitely many complex zeros on the line (s) = 1 2. In this note, we give a simple proof of this statement which, to the best of our knowledge, is new.

scholarly journals On a problem of Ivi\'c.

2000 ◽  
Vol Volume 23 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Upper Bound ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Implicit Constant ◽  
International Audience ◽  
Riemann Zeta

International audience Let $\gamma$ denote the imaginary parts of the nontrivial zeros of the Riemann zeta-function $\zeta(s)$. For sufficiently large $T$ and $\varepsilon>0$, Ivi\'c proved that $\sum_{T<\gamma\leq2T} \vert\zeta(\frac{1}{2}+i\gamma)\vert^2 <\!\!\!<_{\varepsilon} (T(\log T)^2\log\log T)^{3/2+\varepsilon},$ where the implicit constant depends only on $\varepsilon$. In this paper, this result is improved by (i) replacing $\vert\zeta(\frac{1}{2}+i\gamma)\vert^2$ by $\max\vert\zeta(s)\vert^2$, where the maximum is taken over all $s=\sigma+it$ in the rectangle $\frac{1}{2}-A/\log T\leq\sigma\leq2,\, \vert t-\gamma\vert\leq B(\log\log T)/\log T$ with some fixed positive constants $A, B,$ and (ii) replacing the upper bound by $T(\log T)^2\log\log T$. The method of proof differs completely from Ivi\'c's approach.

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