scholarly journals An estimate of the lower bound of the real parts of the zeros of the partial sums of the Riemann zeta function

2015 ◽  
Vol 427 (1) ◽  
pp. 428-439 ◽  
Author(s):  
G. Mora
Keyword(s):  
Lower Bound ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals On The Tails of the Exponential Series

1994 ◽  
Vol 37 (2) ◽  
pp. 278-286 ◽  
Author(s):  
C. Yalçin Yildirim
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Partial Sums ◽  
Maclaurin Series ◽  
Exponential Series ◽  
Asymptotic Estimation ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

AbstractA relation between the zeros of the partial sums and the zeros of the corresponding tails of the Maclaurin series for ez is established. This allows an asymptotic estimation of a quantity which came up in the theory of the Riemann zeta-function. Some new properties of the tails of ez are also provided.

scholarly journals THE SECOND MOMENT OF THE RIEMANN ZETA FUNCTION WITH UNBOUNDED SHIFTS

2010 ◽  
Vol 06 (08) ◽  
pp. 1933-1944 ◽  
Author(s):  
SANDRO BETTIN
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Second Moment ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

We prove an asymptotic formula for the second moment (up to height T) of the Riemann zeta function with two shifts. The case we deal with is where the real parts of the shifts are very close to zero and the imaginary parts can grow up to T2-ε, for any ε > 0.

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 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 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.

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