scholarly journals On the Evaluation of Riemann Zeta Functions for Even Integers

2021 ◽  
Author(s):  
Ribhu Paul
Keyword(s):  
Fourier Series ◽  
Zeta Function ◽  
Closed Form ◽  
Series Expansion ◽  
Recursion Relation ◽  
Zeta Functions ◽  
Fourier Series Expansion ◽  
Recursive Method ◽  
Riemann Zeta

A recursive method for obtaining the Zeta function for even integers is obtained starting from the Fourier series expansion of the function f(x) = x. Repeating the method after term by term integration yields the final, simplified closed form that happens to be a recursion relation. Using the obtained recursion relation, one can successively evaluate the values of zeta functions for even integers.

A SIMPLE MODEL FOR ORIENTATIONAL ORDERING OF A SILVER MONOLAYER ELECTRODEPOSITED ON A C(0001) SURFACE

1995 ◽  
Vol 02 (04) ◽  
pp. 489-494 ◽  
Author(s):  
E.E. MOLA ◽  
A.G. APPIGNANESSI ◽  
J.L. VICENTE ◽  
L. VAZQUEZ ◽  
R.C. SALVAREZZA ◽  
...  
Keyword(s):  
Experimental Data ◽  
Fourier Series ◽  
Simple Model ◽  
Series Expansion ◽  
Unit Cell ◽  
Fourier Series Expansion ◽  
Crystal Formation ◽  
Substrate Potential ◽  
Weber Type ◽  
Orientational Ordering

The model for the angular orientational energy (AOE) has been extended to hexagonal submonolayer domains of Ag electrodeposited at a constant overpotential on a C(0001) surface. These domains which are characterized by an epitaxy angle θ=15±5° and an Ag−Ag distance d Ag−Ag =0.330± 0.016 nm, can be considered as precursors of 3D Ag crystal formation, according to a Volmer-Weber type mechanism. Calculations are based upon a simple Hamiltonian evaluated by introducing the concept of the commensurable unit cell. A Fourier series expansion for the substrate potential was used. Results from the model predict the existence of a commensurable cell in agreement with the experimental data derived from STM imaging.

scholarly journals Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 754 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Zeta Function ◽  
Zeta Functions ◽  
Formal Proof ◽  
Hurwitz Zeta Function ◽  
Mean Square ◽  
Novel Approach ◽  
Starting Point ◽  
Lindelöf Hypothesis ◽  
Riemann Zeta ◽  
The Mean

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.

Predicting Apple Firmness and Soluble Solids Content Based on Hyperspectral Scattering Imaging Using Fourier Series Expansion

2017 ◽  
Vol 60 (4) ◽  
pp. 1053-1062
Author(s):  
Wei Wang ◽  
Min Huang ◽  
Qibing Zhu
Keyword(s):  
Fourier Series ◽  
Least Squares ◽  
Series Expansion ◽  
Fourier Coefficients ◽  
Expansion Method ◽  
Soluble Solids ◽  
Fourier Series Expansion ◽  
Support Vector ◽  
Soluble Solids Content ◽  
Solids Content

Abstract. This article reports on using a Fourier series expansion method to extract features from hyperspectral scattering profiles for apple fruit firmness and soluble solids content (SSC) prediction. Hyperspectral scattering images of ‘Golden Delicious’ (GD), ‘Jonagold’ (JG), and ‘Delicious’ (RD) apples, harvested in 2009 and 2010, were acquired using an online hyperspectral imaging system over the wavelength region of 500 to 1000 nm. The moment method and Fourier series expansion method were used to analyze the scattering profiles of apples. The zeroth-first order moment (Z-FOM) spectra and Fourier coefficients were extracted from each apple, which were then used for developing fruit firmness and SSC prediction models using partial least squares (PLS) and least squares support vector machine (LSSVM). The PLS models based on the Fourier coefficients improved the standard errors of prediction (SEP) by 4.8% to 19.9% for firmness and by 2.4% to 13.5% for SSC, compared with the PLS models using the Z-FOM spectra. The LSSVM models for the prediction set of Fourier coefficients achieved better SEP results, with improvements of 4.4% to 11.3% for firmness and 2.8% to 16.5% for SSC over the LSSVM models for the Z-FOM spectra data and 3.7% to 12.6% for firmness and 5.4% to 8.6% for SSC over the PLS models for the Fourier coefficients. Experiments showed that Fourier series expansion provides a simple, fast, and effective means for improving Keywords: Apples, Firmness, Fourier series expansion, Hyperspectral scattering imaging, Least squares support vector machine, Partial least squares, Soluble solids content.

scholarly journals Analysis and combinatorics of partition zeta functions

2020 ◽  
pp. 1-10
Author(s):  
Robert Schneider ◽  
Andrew V. Sills
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Complete Information ◽  
Combinatorial Proof ◽  
Fixed Length ◽  
The Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

Texture Evolution and Earing in Aluminium Can Sheet

2005 ◽  
Vol 495-497 ◽  
pp. 1565-1572 ◽  
Author(s):  
Jürgen Hirsch
Keyword(s):  
Fourier Series ◽  
Cold Rolling ◽  
Series Expansion ◽  
Deep Drawing ◽  
Recrystallization Texture ◽  
Texture Evolution ◽  
Processing Parameters ◽  
New Method ◽  
Fourier Series Expansion ◽  
Hot And Cold Rolling

The texture evolution during hot and cold rolling of AlMg1Mn1 can body sheet is described and the related anisotropy effects during deep drawing are analysed quantitatively. The typical textures of rolled aluminium show the transition between ß-fibre orientations and cube recrystallization texture, depending on rolling temperature and strain. These correlate with transitions between 45° and 0°/90° ear heights in deep drawn cups which are described by a new method of Fourier series expansion. Processing parameters to achieve low anisotropy are discussed.

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