scholarly journals The Energy Spectral Density of the Mertens Function

2021 ◽  
Author(s):  
Darrell Cox ◽  
Eldar Sultanow ◽  
Sourangshu Ghosh
Keyword(s):  
Spectral Density ◽  
Upper Bound ◽  
Fourier Transforms ◽  
Möbius Function ◽  
Recursive Definition ◽  
Absolute Value ◽  
Energy Spectral Density ◽  
Mobius Function ◽  
The Absolute

The Mertens function is the summatory Mobius function but the Mertens function can be generated recursively without using this definition. This recursive definition is the basis of autocorrelations that can be done on sequences of Mertens function values. Fourier transforms of the autocorrelations result in the energy spectral density. A likely upper bound of the absolute value of the Mertens function is determined.

scholarly journals On an exponential sum involving the Möbius function

2005 ◽  
Vol Volume 28 ◽  
Author(s):  
H. Maier ◽  
A Sankaranarayanan
Keyword(s):  
Upper Bound ◽  
Exponential Sum ◽  
Möbius Function ◽  
Absolute Value ◽  
Mobius Function ◽  
International Audience ◽  
The Absolute

International audience In this paper we study the upper bound for the absolute value of the exponential sum related to the Möbius function unconditionally and present some interesting applications also.

Transforms of Deterministic Continuous-Time Signals

2021 ◽  
pp. 599-673
Author(s):  
Stevan Berber
Keyword(s):  
Spectral Density ◽  
Continuous Time ◽  
Communication Systems ◽  
Fourier Transforms ◽  
Linear Time ◽  
Linear Time Invariant ◽  
Energy Spectral Density ◽  
Time Signals ◽  
Analysis And Synthesis ◽  
The Fourier Transform

Chapter 12 presents a detailed analysis of continuous-time signals and systems in the frequency domain, including the theory of Fourier series and Fourier transforms, and key examples relevant for the analysis and synthesis of signals processed in the digital transceiver blocks of a communication system. The amplitude, magnitude, phase, and power spectra are defined and calculated for typical signals. In particular, the Fourier transform of periodic signals is presented, due to its importance in communication systems theory and practice. Using a unique notation that distinguishes energy and power signals, the correlation, power, and energy spectral density functions are inter-related by proving the Wiener–Khintchine theorem. A comprehensive analysis of a linear-time-invariant system, using the concepts of impulse response, system correlation function, and power spectral density, both for power signals and energy signals, is presented. In addition, Parseval’s theorem and the Rayleigh theorem are proven.

scholarly journals Thue Inequalities With Few Coefficients

2020 ◽  
Author(s):  
Paloma Bengoechea
Keyword(s):  
Upper Bound ◽  
Binary Form ◽  
Number Of Solutions ◽  
Absolute Value ◽  
Integer Coefficients ◽  
The Absolute

Abstract Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$, and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll s m^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.

scholarly journals Mean-square upper bound of Hecke $L$-functions on the critical line.

2003 ◽  
Vol Volume 26 ◽  
Author(s):  
A Sankaranarayanan
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Critical Line ◽  
Congruence Subgroup ◽  
Mean Square ◽  
Absolute Value ◽  
International Audience ◽  
The Mean ◽  
The Absolute

International audience We prove the upper bound for the mean-square of the absolute value of the Hecke $L$-functions (attached to a holomorphic cusp form) defined for the congruence subgroup $\Gamma_0 (N)$ on the critical line uniformly with respect to its conductor $N$.

Evaluation of a Fourier-Transform-Based Pattern-Recognition Algorithm for Two-Dimensional Fluorescence Data

1987 ◽  
Vol 41 (3) ◽  
pp. 496-502 ◽  
Author(s):  
Chou-Pong Pau ◽  
Isiah M. Warner
Keyword(s):  
Pattern Recognition ◽  
Correlation Function ◽  
Frequency Domain ◽  
Fourier Transforms ◽  
Recognition Algorithm ◽  
Two Dimensional ◽  
Pattern Recognition Algorithm ◽  
Absolute Value ◽  
The Absolute ◽  
Matching Criteria

A pattern-recognition algorithm for two-dimensional fluorescence data previously reported is critically evaluated. The three spectral matching criteria—sum of the absolute value of the imaginary coefficients of the frequency-domain correlation function, sum of the absolute value of the real-negative coefficients of the frequency-domain correlation function, and the intervector distance between the abbreviated Fourier transforms of two spectra—are calculated. Spectra simulated with a computer as well as data acquired with a video fluorometer are examined. Results indicate that all three parameters are sensitive to changes in peak position, peak width, relative peak height, and intensity of background noises.

scholarly journals Decay of Mean Values of Multiplicative Functions

2003 ◽  
Vol 55 (6) ◽  
pp. 1191-1230 ◽  
Author(s):  
Andrew Granville ◽  
K. Soundararajan
Keyword(s):  
Upper Bound ◽  
Multiplicative Function ◽  
Mean Value ◽  
Multiplicative Functions ◽  
Mean Values ◽  
Absolute Value ◽  
New Type ◽  
The Absolute

AbstractFor given multiplicative function f , with |f(n)| ≤ 1 for all n, we are interested in how fast its mean value (1/x) Σn≤xf(n) converges. Halász showed that this depends on the minimum M (over y ∈ ℝ) of Σp≤x (1 – Re(f(p)p–iy )/p, and subsequent authors gave the upper bound ⪡ (1 + M)e–M. For many applications it is necessary to have explicit constants in this and various related bounds, and we provide these via our own variant of the Halász-Montgomery lemma (in fact the constant we give is best possible up to a factor of 10). We also develop a new type of hybrid bound in terms of the location of the absolute value of y that minimizes the sum above. As one application we give bounds for the least representatives of the cosets of the k-th powers mod p.

scholarly journals Bounds of the Mertens Function

2020 ◽  
Author(s):  
Darrell Cox ◽  
Sourangshu Ghosh
Keyword(s):  
Upper Bound ◽  
Riemann Hypothesis ◽  
Sufficient Condition ◽  
Absolute Value ◽  
The Absolute

In this paper we derive new properties of Mertens function and discuss about a likely upper bound of the absolute value of the Mertens function √log(𝑥!) > |𝑀(𝑥)| when 𝑥 > 1. Using this likely bound we show that we have a sufficient condition to prove the Riemann Hypothesis.

ADJACENCY AND DEGREE OF GRAPH OF MOBIUS FUNCTION

2016 ◽  
Vol 5 (1) ◽  
pp. 31
Author(s):  
SRIMITRA K.K ◽  
BHARATHI D ◽  
SAJANA SHAIK ◽  
◽  
◽  
...  
Keyword(s):  
Möbius Function ◽  
Mobius Function

scholarly journals Incentive Function of Audit Opinion for the Increase of Regional Operational Expenditure and Own-Source Revenues Through Sensitivity Analysis in Indonesia

2020 ◽  
Vol 11 (1) ◽  
pp. 20
Author(s):  
Muhammad Ikbal Abdullah ◽  
Andi Chairil Furqan ◽  
Nina Yusnita Yamin ◽  
Fahri Eka Oktora
Keyword(s):  
Sensitivity Analysis ◽  
Significant Influence ◽  
Provincial Government ◽  
Sensitivity Testing ◽  
Audit Opinion ◽  
Absolute Value ◽  
Operating Expenditure ◽  
The Absolute ◽  
Operational Expenditure

This study aims to analyze the sensitivity testing using measurements of realization of regional own-source revenues and operating expenditure and to analyze the extent of the effect of sample differences between Java and non-Java provinces by using samples outside of Java. By using sensitivity analysis, the results found the influence of audit opinion on the performance of the provincial government mediated by the realization of regional operating expenditure. More specifically, when using the measurement of the absolute value of the realization of regional operating expenditure it was found that there was a direct positive and significant influence of audit opinion on the performance of the Provincial Government. However, no significant effect of audit opinion was found on the realization value of regional operating expenditure and the effect of the realization value of regional operating expenditure on the performance of the Provincial Government. This result implies that an increase in audit opinion will be more likely to be used as an incentive for the Provincial Government to increase the realization of regional operating expenditure.

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