The mean-square value of the divisor function

2016 ◽  
Vol 100 (548) ◽  
pp. 203-212
Author(s):  
Peter Shiu
Keyword(s):  
Counting Function ◽  
Mean Value ◽  
Mean Square ◽  
Divisor Function ◽  
Dirichlet’S Theorem ◽  
Fixed Power ◽  
The Mean ◽  
Dirichlet's Theorem ◽  
Image Position ◽  
Prime Counting Function

The behaviour of the divisor function d (n) is rather tricky. For a prime p, we have d(p) = 2, but if n is the product of the first k primes then, by Chebyshev's estimate for the prime counting function [1, Theorem 414], we have so thatfor such n then, d (n) is ‘unusually large’ — it can exceed any fixed power of log n, for example.In [2] Jameson gives, amongst other things, a derivation of Dirichlet's theorem, which shows that the mean-value of the divisor function in an interval containing n is log n. However, the result is somewhat deceptive because, for most n, the value of d (n) is substantially smaller than log n.

The mean value of the fluctuations in pressure and pressure gradient in a turbulent fluid

1938 ◽  
Vol 34 (4) ◽  
pp. 534-539
Author(s):  
A. E. Green
Keyword(s):  
Unit Volume ◽  
Isotropic Turbulence ◽  
Fluid Motion ◽  
Mean Value ◽  
Mean Square ◽  
Turbulent Fluid ◽  
Dissipation Of Energy ◽  
The Mean ◽  
Rate Of Dissipation ◽  
Image Position

1. Taylor has shown (1) that two characteristic lengthsλ and λη may be defined for turbulent fluid motion. The length λ, which is connected with the dissipation of energy, is, for isotropic turbulence, given bywhere is the mean rate of dissipation of energy per unit volume and represents the mean square value of any component of velocity. The length λη can be defined in terms of thuswhere For isotropic turbulence Taylor assumed thatwhere B is a constant. Since the turbulence is isotropic,and so, from (1), (2), (3) and (4) we have

The mean square of the Dedekind zeta function in quadratic number fields

1989 ◽  
Vol 106 (3) ◽  
pp. 403-417 ◽  
Author(s):  
Wolfgang Müller
Keyword(s):  
Zeta Function ◽  
Number Fields ◽  
Mean Value ◽  
Fourth Power ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Quadratic Number ◽  
Dedekind Zeta Function ◽  
The Mean ◽  
Image Position

Let K be a quadratic number field with discriminant D. The aim of this paper is to study the mean square of the Dedekind zeta function ζK on the critical line, i.e.It was proved by Chandrasekharan and Narasimhan[1] that (1) is at most of order O(T(log T)2). As they noted at the end of their paper, it ‘would seem likely’ that (1) behaves asymptotically like a2T(log T)2, with some constant a2 depending on K. Applying a general mean value theorem for Dirichlet polynomials, one can actually proveThis may be done in just the same way as this general mean value theorem can be used to prove Ingham's classical result on the fourth power moment of the Riemann zeta function (cf. [3], chapter 5). In 1979 Heath-Brown [2] improved substantially on Ingham's result. Adapting his method to the above situation a much better result than (2) can be obtained. The following Theorem deals with a slightly more general situation. Note that ζK(s) = ζ(s)L(s, XD) where XD is a real primitive Dirichlet character modulo |D|. There is no additional difficulty in allowing x to be complex.

scholarly journals On a sum analogous to Dedekind sum and its mean square value formula

2002 ◽  
Vol 32 (1) ◽  
pp. 47-55 ◽  
Author(s):  
Zhang Wenpeng
Keyword(s):  
Asymptotic Property ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Mean Square ◽  
Dedekind Sum ◽  
The Mean

The main purpose of this paper is using the mean value theorem of DirichletL-functions to study the asymptotic property of a sum analogous to Dedekind sum, and give an interesting mean square value formula.

scholarly journals Microstructure of the Galactic Magnetic Field

1958 ◽  
Vol 8 ◽  
pp. 952-954
Author(s):  
K. Serkowski
Keyword(s):  
Magnetic Field ◽  
Position Angle ◽  
Mean Value ◽  
Open Clusters ◽  
Electric Vector ◽  
Stokes Parameters ◽  
Galactic Magnetic Field ◽  
The Mean ◽  
Image Position

The polarization of the stars in open clusters, explained on the basis of the Davis-Greenstein theory, gives some information on the microstructure of the galactic magnetic field.The polarization is most conveniently described by the parameters Q, U, proportional to the Stokes parameters and defined by where p is the amount of polarization, θ is the position angle of the electric vector, and θ̄ is the mean value of θ for the region under consideration.

The theory of the mean square variation of a function formed by adding known functions with random phases, and applications to the theories of the shot effect and of light

1936 ◽  
Vol 32 (4) ◽  
pp. 580-597 ◽  
Author(s):  
E. N. Rowland
Keyword(s):  
Total Effect ◽  
Mean Square ◽  
Single Event ◽  
Random Events ◽  
Variation Of A Function ◽  
The Mean ◽  
Image Position

In the study of random events and associated fluctuations such as occur in the shot effect, a theorem first stated and discussed by Dr N. R. Campbell can often be employed. It applies on any occasion when there occur at random a number of events whose effects are additive. Let us suppose that a single event occurring at time tr causes at time t an effect f(t − tr) in some part of the observed system, and that the effects of different events are additive, so that the total effect or output is ϑ(t), given byWe may suppose that the same events cause another set of effects g(t − tr) with output ϑ(t), whereBoth the functions are assumed to be bounded and integrable in the Riemann sense, as are all the functions studied in physics.

scholarly journals On the mean value of the enumeration function for multiplicative partitions

1990 ◽  
Vol 41 (3) ◽  
pp. 407-410 ◽  
Author(s):  
Cao Hui-Zong ◽  
Ku Tung-Hsin
Keyword(s):  
Natural Number ◽  
Mean Value ◽  
The Mean ◽  
Image Position

Let g(n) denote the number of multiplicative partitions of the natural number n. We prove that

Dirichlet's theorem on diophantine approximation

1978 ◽  
Vol 83 (1) ◽  
pp. 37-59 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Diophantine Approximation ◽  
Euclidean Plane ◽  
Dirichlet’S Theorem ◽  
Dirichlet's Theorem ◽  
Image Position

Let N take the values 1, 2, … A theorem of Dirichlet asserts that for any (x, y) in the Euclidean plane R2 the inequalityis soluble in integers q1, q2, p with 0 < max (|q1|, |q2|) ≤ N.

IV.—On Least Squares and Linear Combination of Observations

1936 ◽  
Vol 55 ◽  
pp. 42-48 ◽  
Author(s):  
A. C. Aitken
Keyword(s):  
Least Squares ◽  
Linear Combination ◽  
Mean Square Error ◽  
Mean Square ◽  
Approximate Representation ◽  
Linear Combinations ◽  
The Mean ◽  
Image Position

In a series of papers W. F. Sheppard (1912, 1914) has considered the approximate representation of equidistant, equally weighted, and uncorrelated observations under the following assumptions:–(i) The data beingu1, u2, …, un, the representation is to be given by linear combinations(ii) The linear combinations are to be such as would reproduce any set of values that were already values of a polynomial of degree not higher than thekth.(iii) The sum of squared coefficientswhich measures the mean square error ofyi, is to be a minimum for each value ofi.

The mean value theorem in second order arithmetic

2001 ◽  
Vol 66 (3) ◽  
pp. 1353-1358 ◽  
Author(s):  
Christopher S. Hardin ◽  
Daniel J. Velleman
Keyword(s):  
Mean Value ◽  
Second Order ◽  
Mean Value Theorem ◽  
Natural Numbers ◽  
Second Order Arithmetic ◽  
Elementary Analysis ◽  
Logical Connectives ◽  
The Mean ◽  
Order Arithmetic ◽  
Image Position

This paper is a contribution to the project of determining which set existence axioms are needed to prove various theorems of analysis. For more on this project and its history we refer the reader to [1] and [2].We work in a weak subsystem of second order arithmetic. The language of second order arithmetic includes the symbols 0, 1, =, <, +, ·, and ∈, together with number variables x, y, z, … (which are intended to stand for natural numbers), set variables X, Y, Z, … (which are intended to stand for sets of natural numbers), and the usual quantifiers (which can be applied to both kinds of variables) and logical connectives. We write ∀x < t φ and ∃x < t φ as abbreviations for ∀x(x < t → φ) and ∃x{x < t ∧ φ) respectively; these are called bounded quantifiers. A formula is said to be if it has no quantifiers applied to set variables, and all quantifiers applied to number variables are bounded. It is if it has the form ∃xθ and it is if it has the form ∀xθ, where in both cases θ is .The theory RCA0 has as axioms the usual Peano axioms, with the induction scheme restricted to formulas, and in addition the comprehension scheme, which consists of all formulas of the formwhere φ is , ψ is , and X does not occur free in φ(n). (“RCA” stands for “Recursive Comprehension Axiom.” The reason for the name is that the comprehension scheme is only strong enough to prove the existence of recursive sets.) It is known that this theory is strong enough to allow the development of many of the basic properties of the real numbers, but that certain theorems of elementary analysis are not provable in this theory. Most relevant for our purposes is the fact that it is impossible to prove in RCA0 that every continuous function on the closed interval [0, 1] attains maximum and minimum values (see [1]).Since the most common proof of the Mean Value Theorem makes use of this theorem, it might be thought that the Mean Value Theorem would also not be provable in RCA0. However, we show in this paper that the Mean Value Theorem can be proven in RCA0. All theorems stated in this paper are theorems of RCA0, and all of our reasoning will take place in RCA0.

scholarly journals One Kind Special Gauss Sums and their Mean Square Values

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jianhong Zhao ◽  
Jiejie Gao
Keyword(s):  
Mean Value ◽  
Gauss Sums ◽  
Mean Square ◽  
Analytic Methods ◽  
The Mean

In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.

Sign in / Sign up

Export Citation Format

Share Document