The original Gelfond-Schnirelman 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 ##.

Explicit auxiliary functions can be used in the “Schur-Siegel-Smyth 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

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 Erdős-Turán 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, §4-8] 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

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 Hilbert-Fekete 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 Hilbert-Fekete 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], [23], [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):=