scholarly journals On the order of magnitude of Jacobsthal's function

1977 ◽  
Vol 20 (4) ◽  
pp. 329-331 ◽  
Author(s):  
R. C. Vaughan
Keyword(s):  
Prime Divisors ◽  
Order Of Magnitude ◽  
Consecutive Integers ◽  
Image Position

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, ifthenIt is probably true thatwhere ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that

scholarly journals The order of magnitude of the mth coefficients of cyclotomic polynomials

1985 ◽  
Vol 27 ◽  
pp. 143-159 ◽  
Author(s):  
H. L. Montgomery ◽  
R. C. Vaughan
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Order Of Magnitude ◽  
Euler’S Function ◽  
Image Position

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

scholarly journals The normalizer of Γ0(N) in PSL(2, ℝ)

1990 ◽  
Vol 32 (3) ◽  
pp. 317-327 ◽  
Author(s):  
M. Akbas ◽  
D. Singerman
Keyword(s):  
Positive Integer ◽  
Modular Group ◽  
Prime Divisors ◽  
Möbius Transformations ◽  
The Matrix ◽  
Image Position ◽  
Mobius Transformations

Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

scholarly journals ON THE ORDER OF MAGNITUDE OF SUMS OF NEGATIVE POWERS OF INTEGRATED PROCESSES

2012 ◽  
Vol 29 (3) ◽  
pp. 642-658 ◽  
Author(s):  
Benedikt M. Pötscher
Keyword(s):  
Lower Bounds ◽  
Random Variables ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Integrated Process ◽  
Order Of Magnitude ◽  
Image Position ◽  
Integrated Processes

Upper and lower bounds on the order of magnitude of $\sum\nolimits_{t = 1}^n {\lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } } $, where xt is an integrated process, are obtained. Furthermore, upper bounds for the order of magnitude of the related quantity $\sum\nolimits_{t = 1}^n {v_t } \lefttnq#x007C; {x_t } \righttnq#x007C;^{ - \alpha } $, where vt are random variables satisfying certain conditions, are also derived.

scholarly journals On the maximal length of two sequences of consecutive integers with the same prime divisors

1989 ◽  
Vol 54 (3-4) ◽  
pp. 225-236 ◽  
Author(s):  
R. Balasubramanian ◽  
T. N. Shorey ◽  
M. Waldschmidt
Keyword(s):  
Maximal Length ◽  
Prime Divisors ◽  
Consecutive Integers

scholarly journals Almost periodic generalized functions

1983 ◽  
Vol 94 (1) ◽  
pp. 149-166
Author(s):  
H. Burkill ◽  
B. C. Rennie
Keyword(s):  
Generalized Functions ◽  
Inner Product ◽  
Almost Periodic ◽  
Test Functions ◽  
Complex Functions ◽  
Entire Complex ◽  
Order Of Magnitude ◽  
The Individual ◽  
Image Position ◽  
Derivatives Of

In (4) a space C of generalized functions was defined which is rather larger than the simple space used to such effect by Lighthill in (3). At the core of C is the space C0 = T of test functions. These are entire (complex) functions f such that all derivatives of f and its Fourier transform F have order of magnitude not exceeding as x → ± ∞, where c is a positive number depending on the individual derivative concerned. If f, g∈ T, the inner product 〈f | g〉 is defined to be

A kaleidoscope of solutions for a Diophantine system

2010 ◽  
Vol 94 (529) ◽  
pp. 42-50
Author(s):  
Juan Pla
Keyword(s):  
Analogous Problem ◽  
Pell Equation ◽  
Pythagorean Triples ◽  
Diophantine System ◽  
Consecutive Integers ◽  
Image Position

A classical exercise in recreational mathematics is to find Pythagorean triples such that the legs are consecutive integers. It is equivalent to solve the Pell equation with k = 2. In this case it provides all the solutions (see [1] for details). But to obtain all the solutions of a Diophantine system in one stroke is rather exceptional. Actually this note will show that the analogous problem of finding four integers A, B, C and D such that

scholarly journals Water pressure and basal sliding on Storglaciären, northern Sweden

1995 ◽  
Vol 41 (138) ◽  
pp. 232-240 ◽  
Author(s):  
Peter Jansson
Keyword(s):  
Water Pressure ◽  
Surface Velocity ◽  
Northern Sweden ◽  
Overburden Pressure ◽  
Weakly Coupled ◽  
Basal Sliding ◽  
Order Of Magnitude ◽  
Image Position ◽  
Hydraulic Jacking ◽  
Small Valley

AbstractThe subglacial hydrology of the ablation area of Storglaciären, a small valley glacier in northern Sweden, is dramatically affected by a subglacial ridge, or riegel. Water pressures above this riegel are relatively constant, while down-glacier from it they vary significantly. The lower part of the glacier accelerates in response to peaks in basal water pressure. The upper part may be weakly coupled to the lower part during these peaks.A power-law fit of observed basal water pressures and measured surface velocities yieldswhereusis the surface velocity andPEis the effective water pressure (ice overburden pressure minus subglacial water pressure). Data from Findelengletscher, reported by Iken and Bindschadler (1986), yield an identical exponent and a coefficient one order of magnitude larger. The similar exponent implies that the process producing the velocity variations on both glaciers is similar. The variations in velocity are inferred to be due to hydraulic jacking on both glaciers.

Sign in / Sign up

Export Citation Format

Share Document