scholarly journals Some Riemann Hypotheses from random walks over primes

2018 ◽  
Vol 20 (07) ◽  
pp. 1750085 ◽  
Author(s):  
Guilherme França ◽  
André LeClair
Keyword(s):  
Functional Equation ◽  
Critical Line ◽  
Arithmetic Function ◽  
Prime Numbers ◽  
Cusp Forms ◽  
Euler Product ◽  
Critical Strip ◽  
One Dimensional ◽  
The Right ◽  
Multiplicative Characters

The aim of this paper is to investigate how various Riemann Hypotheses would follow only from properties of the prime numbers. To this end, we consider two classes of [Formula: see text]-functions, namely, non-principal Dirichlet and those based on cusp forms. The simplest example of the latter is based on the Ramanujan tau arithmetic function. For both classes, we prove that if a particular trigonometric series involving sums of multiplicative characters over primes is [Formula: see text], then the Euler product converges in the right half of the critical strip. When this result is combined with the functional equation, the non-trivial zeros are constrained to lie on the critical line. We argue that this [Formula: see text] growth is a consequence of the series behaving like a one-dimensional random walk. Based on these results, we obtain an equation which relates every individual non-trivial zero of the [Formula: see text]-function to a sum involving all the primes. Finally, we briefly mention important differences for principal Dirichlet [Formula: see text]-functions due to the existence of the pole at [Formula: see text], in which the Riemann [Formula: see text]-function is a particular case.

scholarly journals ON THE RANKIN–SELBERG ZETA FUNCTION

2012 ◽  
Vol 93 (1-2) ◽  
pp. 101-113
Author(s):  
ALEKSANDAR IVIĆ
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Critical Line ◽  
Critical Strip ◽  
Approximate Functional Equation ◽  
Selberg Zeta Function

AbstractWe obtain the approximate functional equation for the Rankin–Selberg zeta function in the critical strip and, in particular, on the critical line $\operatorname {Re} s= \frac {1}{2}$.

scholarly journals The moments of the zeta-function on the line σ = 1

1994 ◽  
Vol 135 ◽  
pp. 113-120 ◽  
Author(s):  
Aleksandar Ivić
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
List Type ◽  
Critical Strip ◽  
Comprehensive Account ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

The evaluation of the integral(1.1) represents one of the fundamental problems of the theory of the Riemann zeta-function (see [4] for a comprehensive account). In view of the functional equationit is clear that one has to distinguish between the following three principal cases: a)σ = 1/2 (“the critical line”),b)1/2 < σ < 1 (“the critical strip”),c)σ = 1.

scholarly journals ANALISIS JENIS LAYANAN PENDIDIKAN DAN NON PENDIDIKAN YANG PENTING BAGI PENINGKATAN KEPUASAN MAHASISWA (STUDI KASUS: FAKULTAS TEKNIK UNIVERSITAS DIPONEGORO)

2018 ◽  
Vol 13 (1) ◽  
pp. 7
Author(s):  
Aries Susanty ◽  
Haryo Santoso ◽  
Pramudiastuti Nursyachbani
Keyword(s):  
Taguchi Method ◽  
Student Satisfaction ◽  
Educational Services ◽  
Optimal Result ◽  
One Dimensional ◽  
The Right ◽  
Right Order ◽  
Item Item

AbstrakPenelitian ini memiliki dua tujuan. Pertama, penelitian ini bertujuan untuk mengindentifikasi item layanan pendidikan dan non pendidikan yang dianggap penting untuk peningkatan kepuasan mahasiswa Fakultas Teknik Univesitas Diponegoro (UNDIP). Kedua, penelitian ini bertujuan untuk menyusun sejumlah rekomendasi untuk perbaikan atas item layanan pendidikan  dan non pendidikan yang dianggap penting tersebut. Terdapat 7 dimensi dan 28 item layanan yang digunakan untuk mengindentifikasi jenis layanan pendidikan dan non Pendidikan yang diterima oleh mahasiswa Fakultas Teknik UNDIP. Penelitian ini menggunakan Metode Kano dan Taguchi untuk mengindentifikasi item layanan pendidikan dan non pendidikan yang dianggap paling penting. Dalam hal ini, Metode Kano digunakan untuk memilih sejumlah item layanan pendidikan dan non pendidikan yang termasuk dalam kelompok attractive dan one-dimentional. Adapun Metode Taguchi digunakan untuk dua hal, yang pertama yaitu memverifikasi hasil pengelompokan dari Metode Kano sehingga diperoleh hasil yang lebih optimal dan mengurutkan  prioritas perbaikan dari item-item layanan yang termasuk dalam kelompok  attractive dan one-dimentional. Data untuk penelitian ini diperoleh dari hasil pengisian kuesioner oleh 120 responden untuk kuesioner Kano dan 60 responden untuk kuesioner Taguchi. Hasil pengolahan data dengan menggunakan Metode Kano menunjukkan bahwa terdapat 6 item layanan yang termasuk dalam kategori one-dimensional dan terdapat 2 item layanan .yang termasuk dalam kategori attractive. Selanjutnya, pengolahan data dengan menggunakan  Metode Taguchi diperoleh bahwa terdapat 2 item layanan yang perpindah dari one-dimensional ke attractive dan 1 item layanan yangberpindah dari attractive ke one-dimensional. AbstractAnalysis of the type of educational and non-educational services that are important for the enhancement of student satisfaction (case study Faculty of Engineering, Diponegoro University)] This research has two objective. First this study aims to identify the type of educational and non-educational services that are important for the satisfaction’s enhancement of the student of Faculty Engineering, Diponegoro University. Second, this study aims to formulate some recommendation for improving the type of educational and non-educational services that are important for the satisfaction’s enhancement of the student of Faculty Engineering. There are 7 dimensions and 28 indicators used to identify the type of educational and non-educational received by the student. This research uses the Kano and Taguchi method to identify the type of educational and non-educational services that are important for student. In this case, the Kano method is used to identify educational and non-educational services that are include attractive and one-dimensional categories. Whereas the Taguchi method is used to verify Kano’s categorize result for getting more optimal result than Kano method and to put priorities in the right order of those services that are include attractive and one-dimensional categories. Data for this research is got from questionnaires that were distributed to 120 respondents for Kano method and 60 respondents for Taguchi method. Kano method’s result showed that there are 6 type of services that are include in one-dimensional category and 2 type of services that are include in attractive category.  However, based on validation result that is using Taguchi method showed that there are 5 type of services that are include in one-dimensional category and 3 type of services that are include in attractive category.Keywords: Educational and Non-educational Services; Satisfaction’s Enhancement of Student; Faculty of Engineering Diponegoro Univesity; Kano Method; Taguchi Method.

Nonvanishing of derivatives of Hecke $$L$$L-functions associated to cusp forms inside the critical strip

2018 ◽  
Vol 51 (2) ◽  
pp. 319-327
Author(s):  
Winfried Kohnen ◽  
Jyoti Sengupta ◽  
Miriam Weigel
Keyword(s):  
Cusp Forms ◽  
Critical Strip ◽  
Derivatives Of

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

scholarly journals An Analytical and Numerical Detour for the Riemann Hypothesis

Information ◽  
2021 ◽  
Vol 12 (11) ◽  
pp. 483
Author(s):  
Michel Riguidel
Keyword(s):  
Riemann Hypothesis ◽  
Meromorphic Functions ◽  
Critical Line ◽  
Numerical Approach ◽  
Product Formula ◽  
Critical Strip ◽  
Associated Functions ◽  
Poles And Zeros ◽  
Riemann’S Zeta Function ◽  
Insight Into

From the functional equation of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function and its family of associated functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the function. This family is a mathematical and numerical tool which makes it possible to estimate the value of the function at a point in the critical strip from a point on the critical line .Generating estimates of at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the and functions over finite fields.

Hanna Yablonka, Yeladim beseder gamur: biyografiyah dorit shel yelidei haaretz 1948–1955 (Children by the Book, Biography of a Generation: The First Native Israelis Born 1948–1955). Rishon Letziyon: Miskal, 2018. 430 pp.

2021 ◽  
pp. 310-312
Keyword(s):  
Educational System ◽  
Historical Consciousness ◽  
The State ◽  
Historical Experience ◽  
One Dimensional ◽  
The People ◽  
Native Identity ◽  
The One ◽  
State Educational ◽  
The Right

This chapter examines Hanna Yablonka's Children by the Book, Biography of a Generation: The First Native Israelis Born 1948–1955 (2018). This book is unique in that it is neither politically committed to nationalist political slogans that are thrown daily into the arena of Israeli politics in the days of Netanyahu nor connected to the one-dimensional, sweeping condemnation of critics of the Israeli enterprise on the Right and Left. Instead, it suggests to set aside, even if only for a moment, what Yablonka calls “the current Israeli discourse, which furiously shatters everything that has happened in the state since it was established, brutally erasing all the achievements of Little Israel.” Yabonka is guided by Karl Mannheim's concept of a “historical generation”: a group in which there is a shared historical consciousness derived from historical experience. She shows how the state educational system fashioned the image of the new Israeli, endowing children with a local, native identity and imbuing them with the consciousness of belonging both to the people and to the land.

On the Non-Existence of Certain Euler Products

1980 ◽  
Vol 23 (3) ◽  
pp. 371-372
Author(s):  
M. V. Subbarao
Keyword(s):  
Euler Product ◽  
Product Decomposition ◽  
Euler Products ◽  
Image Position ◽  
The Right

In a paper with the above title, T. M. Apostol and S. Chowla [1] proved the following result:Theorem 1.For relatively prime integers h and k, l ≤ h ≤ k, the seriesdoes not admit of an Euler product decomposition, that is, an identity of the form1except when h = k = l; fc = 1, fc = 2. The series on the right is extended over all primes p and is assumed to be absolutely convergent forR(s)>1.

UNBOUNDED SETS OF ATTRACTION

2000 ◽  
Vol 10 (06) ◽  
pp. 1437-1469 ◽  
Author(s):  
GIAN-ITALO BISCHI ◽  
CHRISTIAN MIRA ◽  
LAURA GARDINI
Keyword(s):  
Functional Equation ◽  
Two Dimensional ◽  
Elementary Functions ◽  
Theoretical Methods ◽  
One Dimensional ◽  
Zero Measure ◽  
Analytical Expressions

In this paper we show that unbounded chaotic trajectories are easily observed in the iteration of maps which are not defined everywhere, due to the presence of a denominator which vanishes in a zero-measure set. Through simple examples, obtained by the iteration of one-dimensional and two-dimensional maps with denominator, the basic mechanisms which are at the basis of the existence of unbounded chaotic trajectories are explained. Moreover, new kinds of contact bifurcations, which mark the transition from bounded to unbounded sets of attraction, are studied both through the examples and by general theoretical methods. Some of the maps studied in this paper have been obtained by a method based on the Schröoder functional equation, which allows one to write closed analytical expressions of the unbounded chaotic trajectories, in terms of elementary functions.

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