scholarly journals Ω-Theorems for the real part of the Riemann zeta function

1973 ◽  
Vol 43 (1) ◽  
pp. 123-127
Author(s):  
Norman Levinson
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

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.

On the zeros of the Riemann zeta-function

1924 ◽  
Vol 22 (3) ◽  
pp. 295-318 ◽  
Author(s):  
J. E. Littlewood
Keyword(s):  
Zeta Function ◽  
Imaginary Part ◽  
Riemann Zeta Function ◽  
The Real ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Image Position

Let N (T) denote, as usual, the number of zeros of ζ (s) whose imaginary part γ satisfies 0 < γ < T, and N (σ, T) the number of these for which, in addition, the real part is greater than σ. In this definition we suppose, in the first place, that no zero actually lies on the line t = T: if the line contains zeros we define

scholarly journals A Purely Mathematical Proof of Riemann Hypothesis

2020 ◽  
pp. 1-5
Author(s):  
Robert Deloin
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Riemann Hypothesis ◽  
Mathematical Proof ◽  
The Real ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Primary 11A41

Riemann hypothesis (RH) is the conjecture that the real part of every non-trivial zero of the Riemann zeta function is 1/2. The main contribution of this paper is to achieve the proof of Riemann hypothesis. The key idea to do it is to choose a counter-hypothesis to RH and show that it leads to a (double) contradiction. MSC 2010 classification numbers: Primary 11A41, 11M06, 11M26. Keywords: Riemann Hypothesis, Prime, Conjecture.

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

2016 ◽  
Vol 12 (06) ◽  
pp. 1703-1723 ◽  
Author(s):  
K. Paolina Koutsaki ◽  
Albert Tamazyan ◽  
Alexandru Zaharescu
Keyword(s):  
Asymptotic Formula ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Linear Combinations ◽  
The Real ◽  
Number Of Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Derivatives Of

In this paper, the zeros of linear combinations of the Riemann zeta function and its derivatives are studied. We establish an asymptotic formula for the number of zeros in a rectangle of height [Formula: see text]. We also find a sharp asymptotic formula for the supremum of the real parts of zeros of such combinations in a certain family.

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