Multiplicative functions over short segments

Short interval results for certain prime-independent multiplicative functions

“…holds for y x 1 2r+1 +ε where r is given in (1). The purpose of this work is to establish an effective version of Zhai's result by giving a fully effective error term.…”

“…A prime-independent multiplicative function is a multiplicative arithmetic function f satisfying f (1) = 1 and such that there exists a map g : Z 0 −→ R such that g(0) = 1 and, for any prime powers p α f (p α ) = g(α). In this article, we only consider integer-valued prime-independent multiplicative functions f verifying f (p) = 1 for any prime p. This is equivalent to the fact that g(1) = 1 and we also assume that there exists r ∈ Z 2 such that (1) g(1) = · · · = g(r − 1) = 1 and α r ⇒ g(α) > 1.…”

“…Theorem 1. Let k ∈ Z 1 fixed and f be an integer-valued prime-independent multiplicative function such that f (p) = 1 for any prime p and let r ∈ Z 2 as in (1). Let r(3r−1) x ε .…”

“…In what follows, k ∈ Z 1 is fixed and f is an integer-valued prime-independent multiplicative function satisfying the hypothesis of Theorem 1, with r ∈ Z 2 given in (1).…”

Short interval results for certain arithmetic functions

“…For multiplicative functions F such that F (p) is close to 1 for every prime p, another method was developed in [2] using profound theorems from Filaseta-Trifonov and Huxley-Sargos results on integer points near certain smooth curves. This leads to very precise estimates of a large class of multiplicative functions.…”