scholarly journals Asymptotic Formulas for Some Arithmetic Functions

1958 ◽  
Vol 1 (3) ◽  
pp. 149-153
Author(s):  
P. Erdős
Keyword(s):  
Arithmetic Functions ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
Increasing Function ◽  
General Conditions

Let f(x) be an increasing function. Recently there have been several papers which proved that under fairly general conditions on f(x) the density of integers n for which (n, f(n)) = 1 is 6/π2 and that (d(n) denotes the number of divisors of n)In particular both of these results hold if f(x) = xα, 0 < α < 1 and the first holds if f(x) = [α x], α irrational.

Asymptotic formulas for certain arithmetic functions

2008 ◽  
Vol 58 (3) ◽  
Author(s):  
M. Garaev ◽  
M. Kühleitner ◽  
F. Luca ◽  
W. Nowak
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Arithmetic Functions ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
Second Moments

AbstractThis is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere.

scholarly journals On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Randell Heyman ◽  
László Tóth
Keyword(s):  
Arithmetic Functions ◽  
Asymptotic Formulas ◽  
Common Generalization ◽  
Positive Integers

AbstractWe obtain asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{mn\le x} f((m,n))$$ ∑ m n ≤ x f ( ( m , n ) ) and $$\sum _{mn\le x} f([m,n])$$ ∑ m n ≤ x f ( [ m , n ] ) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions $$f(n)=\tau (n), \log n, \omega (n)$$ f ( n ) = τ ( n ) , log n , ω ( n ) and $$\Omega (n)$$ Ω ( n ) . We also define a common generalization of the latter three functions, and prove a corresponding result.

scholarly journals A note on some congruences involving arithmetic functions

2019 ◽  
Author(s):  
József Sándor
Keyword(s):  
Arithmetic Functions ◽  
Arithmetical Functions ◽  
Number Of Divisors ◽  
International Audience

International audience We consider some congruences involving arithmetical functions. For example, we study the congruences nψ(n) ≡ 2 (mod ϕ(n)), nϕ(n) ≡ 2 (mod ψ(n)), ψ(n)d(n) − 2 ≡ 0 (mod n), where ϕ(n), ψ(n), d(n) denote Euler's totient, Dedekind's function, and the number of divisors of n, respectively. Two duals of the Lehmer congruence n − 1 ≡ 0 (mod ϕ(n)) are also considered.

scholarly journals On the coprimality of some arithmetic functions

2010 ◽  
Vol 87 (101) ◽  
pp. 121-128
Author(s):  
Koninck De ◽  
Imre Kátai
Keyword(s):  
Positive Integer ◽  
Asymptotic Estimate ◽  
Counting Function ◽  
Arithmetic Functions ◽  
Euler Function ◽  
Number Of Divisors

Let ? stand for the Euler function. Given a positive integer n, let ?(n) stand for the sum of the positive divisors of n and let ?(n) be the number of divisors of n. We obtain an asymptotic estimate for the counting function of the set {n : gcd (?(n), ?(n)) = gcd(?(n), ?(n)) = 1}. Moreover, setting l(n) : = gcd(?(n), ?(n+1)), we provide an asymptotic estimate for the size of #{n ? x: l(n) = 1}.

scholarly journals On some upper bounds for the zeta-function and the Dirichlet divisor problem

2016 ◽  
Vol 12 (08) ◽  
pp. 2231-2239
Author(s):  
Aleksandar Ivić
Keyword(s):  
Zeta Function ◽  
Error Term ◽  
Upper Bounds ◽  
Mean Value ◽  
Divisor Problem ◽  
Asymptotic Formulas ◽  
Number Of Divisors ◽  
The Riemann Zeta Function ◽  
Dirichlet Divisor Problem ◽  
Riemann Zeta

Let [Formula: see text] be the number of divisors of [Formula: see text], let [Formula: see text] denote the error term in the classical Dirichlet divisor problem, and let [Formula: see text] denote the Riemann zeta-function. Several upper bounds for integrals of the type [Formula: see text] are given. This complements the results of [A. Ivić and W. Zhai, On some mean value results for [Formula: see text] and a divisor problem II, Indag. Math. 26(5) (2015) 842–866], where asymptotic formulas for [Formula: see text] were established for the above integral.

The Effect of Target Surround Density on Visual Search Performance

1975 ◽  
Vol 17 (4) ◽  
pp. 356-360 ◽  
Author(s):  
Timothy H. Monk ◽  
Brian Brown
Keyword(s):  
Visual Search ◽  
Geometric Mean ◽  
Search Time ◽  
Search Performance ◽  
Increasing Function ◽  
General Conditions

In an earlier study, Brown and Monk (1975) defined the area of display immediately adjacent to the target to be the “target surround”. Using highly specific configurations of nontargets in the target surround, they showed that congested target surrounds act to camouflage the target. The present study tests these results under more general conditions where no specific configurations are enforced. A linear increasing function is found between geometric mean search time and target surround density, using three measures of the latter. The implication of this result to studies of overall nontarget density is discussed.

Instrumentation for detecting channelling radiation in the high-voltage electron microscope

1983 ◽  
Vol 41 ◽  
pp. 318-319
Author(s):  
J. H. Butler ◽  
C. J. Humphreys
Keyword(s):  
Monochromatic Light ◽  
Incident Beam ◽  
Laboratory Frame ◽  
Relativistic Electrons ◽  
Voltage Electron ◽  
Dipole Emission ◽  
Sinusoidal Oscillations ◽  
Pass Through ◽  
Incident Beam Energy ◽  
Increasing Function

Electromagnetic radiation is emitted when fast (relativistic) electrons pass through crystal targets which are oriented in a preferential (channelling) direction with respect to the incident beam. In the classical sense, the electrons perform sinusoidal oscillations as they propagate through the crystal (as illustrated in Fig. 1 for the case of planar channelling). When viewed in the electron rest frame, this motion, a result of successive Bragg reflections, gives rise to familiar dipole emission. In the laboratory frame, the radiation is seen to be of a higher energy (because of the Doppler shift) and is also compressed into a narrower cone of emission (due to the relativistic “searchlight” effect). The energy and yield of this monochromatic light is a continuously increasing function of the incident beam energy and, for beam energies of 1 MeV and higher, it occurs in the x-ray and γ-ray regions of the spectrum. Consequently, much interest has been expressed in regard to the use of this phenomenon as the basis for fabricating a coherent, tunable radiation source.

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