EIGENVALUES FOR HIGH ORDER ELLIPTIC OPERATORS IN A FRACTAL STRING

Fractals ◽  
1999 ◽  
Vol 07 (03) ◽  
pp. 267-275 ◽  
Author(s):  
HONG DENG ◽  
GONGWEN PENG
Keyword(s):  
Zeta Function ◽  
Elliptic Operator ◽  
Riemann Zeta Function ◽  
Counting Function ◽  
High Order ◽  
Minkowski Dimension ◽  
Fractal Boundary ◽  
Open Set ◽  
Riemann Zeta ◽  
Fractal String

In this paper, we study the spectrum of order 2m (m≥1) elliptic operator A in a bounded open set Ω∈R1, with fractal boundary Γ=∂Ω and Minkowski dimension D∈(0, 1), thus proving the corresponding modified Weyl-Berry conjecture to be true, namely [Formula: see text] where N(λ, A, Ω) is the counting function, [Formula: see text], C1, D =2-(1-D) π-D(1-D)(-ζ(D)), ζ(D) is the classical Riemann–zeta function, and ℳ(D, Γ) is the Minkowski measure of Γ.

scholarly journals Schemes over 𝔽1 and zeta functions

2010 ◽  
Vol 146 (6) ◽  
pp. 1383-1415 ◽  
Author(s):  
Alain Connes ◽  
Caterina Consani
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Counting Function ◽  
Toric Varieties ◽  
Zeta Functions ◽  
Global Field ◽  
Chevalley Groups ◽  
The Real ◽  
Riemann Zeta

AbstractWe determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, whose corresponding zeta function is the complete Riemann zeta function. We show that such a counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q=1 as expected from the infinite genus of C. Then, we develop a theory of functorial 𝔽1-schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of $\Mo $-schemes to interpret conceptually the spectral realization of zeros of L-functions.

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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