Gennady Bachman

On the coefficients of ternary cyclotomic polynomials

Abstract We study coefficients of ternary cyclotomic polynomials Φpqr(z)=∏ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.

Ternary cyclotomic polynomials with an optimally large set of coefficients

Ternary cyclotomic polynomials are polynomials of the form Φ pqr (z) = Π ρ (z - p), where p < q < r are odd primes and the product is taken over all primitive pqr-th roots of unity p. We show that for every p there exists an infinite family of polynomials Φ pqr such that the set of coefficients of each of these polynomials coincides with the set of integers in the interval [-(p - 1)/2, (p + 1)/2]. It is known that no larger range is possible even if gaps in the range are permitted.

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum ∑n=a+1a+Nf(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.

Exponential sums with multiplicative coefficients

We provide estimates for the exponential sum