scholarly journals The Difference between Consecutive Prime Numbers V

1963 ◽  
Vol 13 (4) ◽  
pp. 331-332 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Prime Numbers ◽  
Euler’S Constant ◽  
Prime Factors ◽  
The Difference ◽  
Image Position ◽  
Positive Integers ◽  
Efficient Selection ◽  
Euler's Constant ◽  
Selection Of

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, whereHerer x is large and α and δ are positive constants to be chosen suitably.

scholarly journals Some Finite Analogues of the Poisson Summation Formula

1961 ◽  
Vol 12 (3) ◽  
pp. 133-138 ◽  
Author(s):  
L. Carlitz
Keyword(s):  
Summation Formula ◽  
Poisson Summation Formula ◽  
Euler’S Constant ◽  
Image Position ◽  
Positive Integers ◽  
Euler's Constant ◽  
Poisson Summation

1. Guinand (2) has obtained finite identities of the typewhere m, n, N are positive integers and eitherorwhere γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

A note on the diophantine equation (xm − l) / (x − 1) = yn + l

1994 ◽  
Vol 116 (3) ◽  
pp. 385-389 ◽  
Author(s):  
Le Maohua
Keyword(s):  
Diophantine Equation ◽  
Rational Numbers ◽  
Prime Factors ◽  
Image Position ◽  
Positive Integers

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

scholarly journals A variance method in combinatorial number theory

1969 ◽  
Vol 10 (2) ◽  
pp. 126-129 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Number Theory ◽  
Variance Method ◽  
Combinatorial Number Theory ◽  
Prime Factors ◽  
Integer Solutions ◽  
Image Position ◽  
Positive Integers

Let s = s(a1, a2,...., ar) denote the number of integer solutions of the equationsubject to the conditionsthe ai being given positive integers, and square brackets denoting the integral part. Clearly s (a1,..., ar) is also the number s = s(m) of divisors of which contain exactly λ prime factors counted according to multiplicity, and is therefore, as is proved in [1], the cardinality of the largest possible set of divisors of m, no one of which divides another.

scholarly journals An Elementary Derivation of the Exponential Limit and of Euler's Constant

1935 ◽  
Vol 29 ◽  
pp. xxi-xxiv ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Area Under The Curve ◽  
Euler’S Constant ◽  
Elementary Derivation ◽  
Fundamental Properties ◽  
Image Position ◽  
The Right ◽  
Euler's Constant

By defining a logarithm aswe may visualise the function as the area under the curve, measured to the right from the zero value at the ordinate AB. The fundamental properties follow at once from (1): for if u = av, then

scholarly journals On Series for calculating Euler's Constant, and the Constant in Stirling's Theorem

1909 ◽  
Vol 28 ◽  
pp. 48-59
Author(s):  
K. J. Sanjana
Keyword(s):  
Euler’S Constant ◽  
Elementary Methods ◽  
Image Position ◽  
Euler's Constant ◽  
Made In

Let γn denote the value ofwhere n is a definite integer; and let γ denote the limit ofwhen the integer k is indefinitely increased. It is known that the expansion of γn – γ in ascending powers of 1/n iswhere B1, B3, B5… are the numbers of Bernoulli. The series (3) is, however, divergent, as B2r+1 not only increases indefinitely with r, but bears† an infinite ratio to B2r–1 in this case. It is proposed to find by elementary methods the expansion of γn – γ up to the term in nr and to estimate the error (of order l/nr+1) made in omitting further terms of series (3). I shall take the case of r = 9, but the process is quite general.

scholarly journals RATIONAL APPROXIMATIONS TO VALUES OF BELL POLYNOMIALS AT POINTS INVOLVING EULER’S CONSTANT AND ZETA VALUES

2012 ◽  
Vol 92 (1) ◽  
pp. 71-98
Author(s):  
KH. HESSAMI PILEHROOD ◽  
T. HESSAMI PILEHROOD
Keyword(s):  
Zeta Function ◽  
Riemann Zeta Function ◽  
Bell Polynomials ◽  
Rational Approximations ◽  
Euler’S Constant ◽  
The Riemann Zeta Function ◽  
Zeta Values ◽  
Riemann Zeta ◽  
Image Position ◽  
Euler's Constant

AbstractIn this paper we present new explicit simultaneous rational approximations which converge subexponentially to the values of the Bell polynomials at the points where m=1,2,…,a, a∈ℕ, γ is Euler’s constant and ζ is the Riemann zeta function.

scholarly journals On Dirichlet's divisor problem

1931 ◽  
Vol 134 (823) ◽  
pp. 192-202 ◽  
Keyword(s):  
Positive Integer ◽  
Divisor Problem ◽  
Euler’S Constant ◽  
Number Of Divisors ◽  
Prime Factors ◽  
Euler's Constant

1. Let d ( n ) denote the number of divisors of the positive integer n , so that, if n = p 1 a 1 . . . p r ar is the canonical expression of n in prime factors, d ( n ) = (1 + a 1 ) . . . (1 + a r ), and let d ( x ) = 0 if x is not an iteger; then if (1. 1) D ( x ) = Σ' n ≤ x d ( n ) = Σ n ≤ x d ( n ) ─ ½ d ( x ), and (1. 2) Δ ( x ) = D ( x ) ─ x log x ─ (2C ─ 1) x ─ ¼, where C is Euler's constant, it was proved by Dirichlet in 1849 that (1. 21) Δ ( x ) = O (√ x ),

On the function

1963 ◽  
Vol 59 (4) ◽  
pp. 735-737
Author(s):  
A. S. Meligy ◽  
E. M. EL Gazzy
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Bessel Functions ◽  
Exponential Integral ◽  
Euler’S Constant ◽  
Image Position ◽  
Euler's Constant

In a previous paper (3) one of us reported an expansion for the exponential integralin terms of Bessel functions. In this note, we shall obtain the more general formulawhere n is any positive integer, γ is Euler's constant andIt reduces to that in (3) when n = 1.

scholarly journals On Measures of Polynomials in Several Variables: Corrigendum

1982 ◽  
Vol 26 (2) ◽  
pp. 317-319 ◽  
Author(s):  
Gerald Myerson ◽  
C.J. Smyth
Keyword(s):  
Error Term ◽  
Euler’S Constant ◽  
Several Variables ◽  
Explicit Constant ◽  
Image Position ◽  
Euler's Constant

In [2], it was asserted in Theorem 3 that the measure M(x0 + … + xx) is asymptotically , where c was an explicit constant. The value of c given was incorrect, and should be e-½γ where γ is Euler's constant. This was pointed out by the first author. In factwhere we have tried to make amends by improving the error term.

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