scholarly journals On the distribution of values of the argument of the Riemann zeta-function

2019 ◽  
Vol 200 ◽  
pp. 96-131
Author(s):  
Aleksandar P. Ivić ◽  
Maxim A. Korolev
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Distribution Of Values ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals Value-Distribution of the Riemann Zeta-Function Along Its Julia Lines

2020 ◽  
Vol 20 (3-4) ◽  
pp. 389-401
Author(s):  
Jörn Steuding ◽  
Ade Irma Suriajaya
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Complex Number ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Distribution Of Values ◽  
Arbitrary Complex ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Average Behaviour

AbstractFor an arbitrary complex number $$a\ne 0$$ a ≠ 0 we consider the distribution of values of the Riemann zeta-function $$\zeta $$ ζ at the a-points of the function $$\Delta $$ Δ which appears in the functional equation $$\zeta (s)=\Delta (s)\zeta (1-s)$$ ζ ( s ) = Δ ( s ) ζ ( 1 - s ) . These a-points $$\delta _a$$ δ a are clustered around the critical line $$1/2+i\mathbb {R}$$ 1 / 2 + i R which happens to be a Julia line for the essential singularity of $$\zeta $$ ζ at infinity. We observe a remarkable average behaviour for the sequence of values $$\zeta (\delta _a)$$ ζ ( δ a ) .

scholarly journals On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
WonTae Hwang ◽  
Kyunghwan Song
Keyword(s):  
Zeta Function ◽  
Rational Number ◽  
Riemann Zeta Function ◽  
Rational Numbers ◽  
Critical Strip ◽  
Integer Part ◽  
Integral Points ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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.

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