On primary pseudo-polynomials (around Ruzsa’s conjecture)

Every polynomial Formula: see text satisfies the congruences Formula: see text for all integers Formula: see text. An integer valued sequence Formula: see text is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long-standing conjecture of Ruzsa says that a pseudo-polynomial Formula: see text is a polynomial as soon as Formula: see text. Under this growth assumption, Perelli and Zannier proved that the generating series Formula: see text is a Formula: see text-function. A primary pseudo-polynomial is an integer valued sequence Formula: see text such that Formula: see text for all integers Formula: see text and all prime numbers Formula: see text. The same conjecture has been formulated for them, which implies Ruzsa’s, and this paper revolves around this conjecture. We obtain a Hall type characterization of primary pseudo-polynomials. We give a new proof and generalize a result due to Zannier that any primary pseudo-polynomial with an algebraic generating series is a polynomial. We make the Perelli–Zannier Theorem effective and we prove a Pólya type result.