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The Gelfond–Schnirelman method in prime number theory
 Canad. J. Math
, 2005
"... Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients 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 lowe ..."
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Abstract. The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients 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 ψfunction, 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 π(x) be the number of primes not exceeding x. The celebrated Prime Number Theorem (PNT), suggested by Legendre and Gauss, states that (1.1) π(x) ∼ x log x as x → ∞. We include a very brief sketch of its history, referring for details to many excellent books and surveys available on this subject (see, e.g., [8, 10, 17, 29]). Chebyshev [6] made the first important step towards the PNT in 1852, by proving the bounds (1.2) 0.921 x log x
DISTRIBUTION OF ALGEBRAIC NUMBERS
"... Abstract. Schur studied limits of the arithmetic means An of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that lim sup n→ ∞ An  ≤ 1 − √ e/2. We show that An → 0, and estimate the rate o ..."
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Abstract. Schur studied limits of the arithmetic means An of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that lim sup n→ ∞ An  ≤ 1 − √ e/2. We show that An → 0, and estimate the rate of convergence by generalizing the ErdősTurán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for lim infn→ ∞ An was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure. 1. Schur’s problems on means of algebraic numbers Let E be a subset of the complex plane C. Consider the set of polynomials Zn(E) of the exact degree n with integer coefficients and all zeros in E. We denote a subset of Zn(E) with simple zeros by Zs n(E). Given M> 0, we write Pn = anzn +... ∈ Zs n(E, M) if an  ≤ M and Pn ∈ Zs n(E) (respectively Pn ∈ Zn(E, M) if an  ≤ M and Pn ∈ Zn(E)). Schur [46, §48] studied the limit behavior of the arithmetic means of zeros for polynomials from Zs n(E, M) as n → ∞, where M> 0 is an arbitrary fixed number. His results may be summarized in the following statements. Let R+: = [0, ∞), where R is the real line. ∏ n k=1 Theorem A (Schur [46, p. 393], Satz IX) Given a polynomial Pn(z) = an (z − αk,n), define the arithmetic mean of squares of its zeros by Sn: = ∑n k=1 α2 k,n /n. If Pn ∈ Zs n(R, M) is any sequence of polynomials with degrees n → ∞, then
TRACE OF TOTALLY POSITIVE ALGEBRAIC INTEGERS AND INTEGER TRANSFINITE DIAMETER
"... Explicit auxiliary functions can be used in the “SchurSiegelSmyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm and we improve t ..."
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Explicit auxiliary functions can be used in the “SchurSiegelSmyth trace problem”. In the previous works, these functions were constructed only with polynomials having all their roots positive. Here, we use several polynomials with complex roots, which are found with Wu’s algorithm and we improve the known lower bounds of the trace of totally positive algebraic integers. This improvement has a consequence for the search of Salem numbers that have a negative trace. The same method also gives a small improvement of the upper bound for the integer transfinite diameter of [0,1]. 1
Minimal Mahler measures
, 2007
"... We determine the minimal Mahler measure of a primitive, irreducible, noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive, irreducible, noncyclotomic polynomials with measure less than 1.3 and degree at most 44. 1 ..."
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We determine the minimal Mahler measure of a primitive, irreducible, noncyclotomic polynomial with integer coefficients and fixed degree D, for each even degree D ≤ 54. We also compute all primitive, irreducible, noncyclotomic polynomials with measure less than 1.3 and degree at most 44. 1
DISTRIBUTION OF PRIMES AND A WEIGHTED ENERGY PROBLEM
"... Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Cheb ..."
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Abstract. We discuss a recent development connecting the asymptotic distribution of prime numbers with weighted potential theory. These ideas originated with the GelfondSchnirelman method (circa 1936), which used polynomials with integer coefficients and small sup norms on ¢ £¥¤§¦© ¨ to give a Chebyshevtype lower bound in prime number theory. A generalization of this method for polynomials in many variables was later studied by Nair and Chudnovsky, who produced tight bounds for the distribution of primes. Our main result is a lower bound for the integral of Chebyshev’s �function, expressed in terms of the weighted capacity for polynomialtype weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support. This new connection leads to some interesting open problems on weighted capacity.
Article electronically published on January 8, 2003 MONIC INTEGER CHEBYSHEV PROBLEM
"... Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈Mn(Z) satisfies ‖Mn‖E = inf Pn∈Mn(Z) ‖Pn‖E. and the monic integer Cheb ..."
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Abstract. We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let Mn(Z) denote the monic polynomials of degree n with integer coefficients. A monic integer Chebyshev polynomial Mn ∈Mn(Z) satisfies ‖Mn‖E = inf Pn∈Mn(Z) ‖Pn‖E. and the monic integer Chebyshev constant is then defined by tM (E): = lim n→ ∞ ‖Mn‖1/n E. This is the obvious analogue of the more usual integer Chebyshev constant that has been much studied. We compute tM (E) for various sets, including all finite sets of rationals, and make the following conjecture, which we prove in many cases. Conjecture. Suppose [a2/b2,a1/b1] is an interval whose endpoints are consecutive Farey fractions. This is characterized by a1b2 − a2b1 =1. Then tM [a2/b2,a1/b1] =max(1/b1, 1/b2). This should be contrasted with the nonmonic integer Chebyshev constant case, where the only intervals for which the constant is exactly computed are intervals of length 4 or greater. 1. Introduction and
THE MULTIVARIATE INTEGER CHEBYSHEV PROBLEM
"... Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivar ..."
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Abstract. The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in C d. We study this problem on general sets, but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the HilbertFekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single variable polynomials in the complex plane, our estimate coincides with the HilbertFekete result. 1. The integer Chebyshev problem and its multivariate counterpart The supremum norm on a compact set E ⊂ C d, d ∈ N, is defined by ‖f‖E: = sup f(z). z∈E We study polynomials with integer coefficients that minimize the sup norm on a set E, and investigate their asymptotic behavior. The univariate case (d = 1) has a long history, but the problem is virtually untouched for d ≥ 2. Let Pn(C) and Pn(Z) be the classes of algebraic polynomials in one variable, of degree at most n, respectively with complex and with integer coefficients. The problem of minimizing the uniform norm on E by monic polynomials from Pn(C) is the classical Chebyshev problem (see [5], [22], [26], etc.) For E = [−1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial of degree n: Tn(x): = 2 1−n cos(n arccos x), n ∈ N. By a linear change of variable, we immediately obtain that ( ) n () b − a 2x − a − b tn(x): = Tn 2 b − a is a monic polynomial with real coefficients and the smallest norm on [a, b] ⊂ R among all monic polynomials of degree n from Pn(C). In fact, ( ) n b − a (1.1) ‖tn ‖ [a,b] = 2, n ∈ N, 4 and the Chebyshev constant for [a, b] is given by 1/n b − a (1.2) tC([a, b]): = lim ‖tn‖ n→ ∞ [a,b]
Polynomials with integer coefficients and their zeros
"... Abstract. We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the ..."
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Abstract. We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.