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On the linear independence measure of logarithms of rational numbers
 Moscow Lomonosov State University Department of
, 2002
"... Abstract. In this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and, in particular, the linear independence measure of 1, log 2,log 3, log 5 and of 1, log 2, log 3,log 5, log 7. We also give a method to search for polynomials of smallest norm on ..."
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Cited by 10 (4 self)
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Abstract. In this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and, in particular, the linear independence measure of 1, log 2,log 3, log 5 and of 1, log 2, log 3,log 5, log 7. We also give a method to search for polynomials of smallest norm on a real interval [a, b] which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers. 1.
Small Polynomials With Integer Coefficients
"... this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively wi ..."
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Cited by 8 (6 self)
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this paper is the study of polynomials with integer coe#cients that minimize the sup norm on the set E. In particular, we consider the asymptotic behavior of these polynomials and of their zeros. Let P n (C) and P n (Z) be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coe#cients. The problem of minimizing the uniform norm on E by monic polynomials from P n (C) is well known as the Chebyshev problem (see [4], [31], [43], [16], etc.) In the classical case E = [1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: T n (x) := 2 1n cos(n arccos x), n # N. Using a change of variable, we can immediately extend this to an arbitrary interval [a, b] # R, so that t n (x) := # b  a 2 # n T n # 2x  a  b b  a # is a monic polynomial with real coe#cients and the smallest uniform norm on [a, b] among all monic polynomials from P n (C). In fact, (1.1) #t n # [a,b] = 2 # b  a 4 # n , n # N, and we find that the Chebyshev constant for [a, b] is given by (1.2) t C ([a, b]) := lim n## #t n # 1/n [a,b] = b  a 4 . The Chebyshev constant of an arbitrary compact set E # C is defined in a similar fashion: (1.3) t C (E) := lim n## #t n # 1/n E , where t n is the Chebyshev polynomial of degree n on E. It is known that t C (E) is equal to the transfinite diameter and the logarithmic capacity cap(E) of the set E (cf. [43, pp. 7175], [16] and [30] for the definitions and background material). 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15. Key words and phrases. Chebyshev polynomials, integer Chebyshev constant, integer transfinite diameter, zeros, multiple factors, asymptotic...
The GelfondSchnirelman Method In Prime Number Theory
 Canad. J. Math
"... The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for t ..."
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Cited by 4 (4 self)
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The original GelfondSchnirelman method, proposed in 1936, uses polynomials with integer coe#cients and small norms on [0, 1] to give a Chebyshevtype lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's #function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. 1. Lower bounds for arithmetic functions Let #(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that ##.
An Essay On Irrationality Measures Of Logarithms
"... ma 2.1). Let #, # # R be two irrational numbers. Suppose that sequences of linear forms b n x a n and b n x a # n , with integer coe#cients from the field of rationals or an imaginary quadratic field, satisfies 0 , lim n # # a # n # 0 for some positive real ..."
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ma 2.1). Let #, # # R be two irrational numbers. Suppose that sequences of linear forms b n x a n and b n x a # n , with integer coe#cients from the field of rationals or an imaginary quadratic field, satisfies 0 , lim n # # a # n # 0 for some positive real constants C 0 < C # 0 and C 1 . Then any element # Q# + Q# # is irrational with the bound (#) # 1 + C 1 /C 0 for the irrationality exponent. 1. Irrationality measure for log 2 (after E. Rukhadze) 1.1. Gauss hypergeometric function. It is worth performing a slightly general integral than (1), namely for nonnegative integers m,n 0 , n 1 , provided the condition max{m,n 0 # n 1 holds for further convenience. The integral in (2) is exactly Euler's integral for the Gauss hypergeometric series: #(n 0 + 1) #(n 1 + 1) 2 F 1 m+ 1, n 0 + 1 n 0 + n 1 + 2 # # #(n 1 + 1) #(m + 1) #(m + 1 + #) #(n 0 + 1 + #) #(1 + #) #(n 0 + n 1 + 2 + #) (3) (see, e.g., [
Let a ∈ Q ∩ (0, 2], a ̸ = 1. Then the sequence of quantities
, 2004
"... on the occasion of his 60th birthday xn (1 − x) n dx ∈ Q log a + Q, n = 0, 1, 2,..., (1) (1 − (1 − a)x) n+1 produces ‘good ’ rational approximations to loga. There are several ways of performing integration in (1) in order to show that the integral lies in Q log a + Q; we give an exposition of diffe ..."
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on the occasion of his 60th birthday xn (1 − x) n dx ∈ Q log a + Q, n = 0, 1, 2,..., (1) (1 − (1 − a)x) n+1 produces ‘good ’ rational approximations to loga. There are several ways of performing integration in (1) in order to show that the integral lies in Q log a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and log 3. Although methods presented below work in general situations (e.g., for certain Qlinear forms in logarithms) as well, the three numbers seem to be very nice and important models for our exposition. Bounds for irrationality measures are presented by means of upper estimates for irrationality exponents. Recall that the irrationality exponent of a real irrational number γ is defined by the relation µ = µ(γ) = inf{c ∈ R: the inequality γ − a/b  � b  −c has only finitely many solutions in a, b ∈ Z}. The estimates for µ(γ) are deduced by constructing sequences of linear forms involving γ and using standard tools of the following shapes.
Let a ∈ Q ∩ (0, 2], a ̸ = 1. Then the sequence of quantities ∫ 1
, 2004
"... n+1 produces ‘good ’ rational approximations to log a. There are several ways of perfoming integration in (1) in order to show that the integrals lies in Qlog a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integral ..."
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n+1 produces ‘good ’ rational approximations to log a. There are several ways of perfoming integration in (1) in order to show that the integrals lies in Qlog a + Q; we give an exposition of different methods below. The aim of this essay is to demonstrate how suitable generalizations of the integrals in (1) allow to prove the best known results on irrationality measures of the numbers log 2, π and log 3. Although methods presented below work in general situations (e.g., for certain Qlinear forms in logarithms) as well, the three numbers seem to be very nice and important models for our exposition. Bounds for irrationality measures are presented by means of upper estimates for irrationality exponents. Recall that the irrationality exponent of a real irrational number γ is defined by the relation µ = µ(γ) = inf{c ∈ R: the inequality γ − a/b  � b  −c has only finitely many solutions in a, b ∈ Z}. The estimates for µ(γ) are deduced by constructing sequences of linear forms involving γ and using standard tools of the following shapes. Proposition 1 ([Ha1], Lemma 3.1). Let γ ∈ R be irrational. Suppose that a sequence of linear forms bnx − an, with integer coefficients from the field of rationals or an imaginary quadratic field, satisfies lim sup n→∞ log bn n � C1, log bnγ − an lim n→ ∞ n = −C0 for some positive real C0 and C1. Then µ(γ) � 1 + C1/C0.