Abstract. Define the Liouville function for A, a subset of the primes P , by λ A (n) = (−1) Ω A (n) , where Ω A (n) is the number of prime factors of n coming from A counting multiplicity. For the traditional Liouville function, A is the set of all primes. Denoten .
It is known that for eachGiven certain restrictions on the sifting density of A, asymptotic estimates for n≤x λ A (n) can be given. With further restrictions, more can be said. For an odd prime p, define the character-like function λ p as λ p (pk + i) = (i/p) for i = 1, . . . , p − 1 and k ≥ 0, and λ p (p) = 1, where (i/p) is the Legendre symbol (for example, λ 3 is defined by λ 3 (3k + 1) = 1, λ 3 (3k + 2) = −1 (k ≥ 0) and λ 3 (3) = 1). For the partial sums of character-like functions we give exact values and asymptotics; in particular, we prove the following theorem.
Theorem. If p is an odd prime, thenThis result is related to a question of Erdős concerning the existence of bounds for number-theoretic functions. Within the course of discussion, the ratio φ(n)/σ(n) is considered.
We investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials
with coefficients {−1, 1}) inside or on the unit circle |z| = 1, denoted by N(p) and U(p), respectively. Two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in
the sequence of coefficients and Littlewood polynomials with one negative coefficient. We obtain
explicit formulas for N(p), U(p) for polynomials p(z) of these types. We show that if n + 1 is a prime
number, then for each integer k, 0 ≤ k ≤ n − 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U(p) = 0. Furthermore, we describe some cases where the ratios N(p)/n and
U(p)/n have limits as n → ∞ and find the corresponding limit values.
Abstract. There is a 1941 conjecture of Erdős and Turán on what is now called additive basis that we restate:Conjecture 0.1 (Erdős and Turán). Suppose that 0 = δ 0 < δ 1 < δ 2 < δ 3 · · · is an increasing sequence of integers andSuppose thatOur main purpose is to show that the sequence {b n } cannot be bounded by 7. There is a surprisingly simple, though computationally very intensive, algorithm that establishes this.
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