Dekoninck 2004 sur la proximite des nombres

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De Koninck, Jean-Marie and Luca, Florian, Sur la proximité des nombres puissants. Acta Arith. 114 (2004), 149--157.

Written in French, the paper studies powerful (squarefull) numbers between consecutive squares. Theoreme 1 states that there are infinitely many n for which the open interval (n^2,(n+1)^2) contains more than (9/20)(log n / log log n)^{1/3} powerful numbers, so arbitrarily many k powerful numbers occur in such an interval for every fixed k. Theoreme 2 shows the counting function V(N) of n <= N whose interval (n^2,(n+1)^2) contains no powerful number satisfies V(N)=C_2 N + O(N/sqrt(log log N)) with C_2 = prod_{m>=2}(1-mu^2(m)m^{-3/2}) approx 0.275, so that set has positive density. The proof of Theoreme 1 (Section 3) uses Dirichlet's simultaneous rational approximation theorem applied to the irrationals d_j^{-1/2}, where d_j = m_j^2+1 and m_1 < ... < m_{2k} are the first 2k positive integers m with m^2+1 squarefree; with D = d_1...d_{2k} and n = Dq, the 2k distinct powerful numbers d_j D^2 p_j^2 lie within less than 2n-1 of n^2, so at least k of them fall in (n^2,(n+1)^2) or in ((n-1)^2,n^2). The observation that the powerful-number count S(N)=C_1 sqrt(N)+O(N^{1/3}) has C_1 = zeta(3/2)/zeta(3) approx 2.1732 > 2 already gives infinitely many intervals with at least two powerful numbers. These results bear on problem 942, which asks how many powerful numbers can lie between consecutive squares.

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Bears on. #942

Results to transcribe.

• Theoreme 1: There are infinitely many n such that (n^2,(n+1)^2) contains more than (9/20)(log n/log log n)^{1/3} powerful numbers.
• Theoreme 2: The number V(N) of n <= N with no powerful number in (n^2,(n+1)^2) is C_2 N + O(N/sqrt(log log N)), C_2 = prod_{m>=2}(1-mu^2(m)/m^{3/2}) approx 0.275.
• Equation (1): The count of powerful numbers up to N is C_1 sqrt(N)+O(N^{1/3}) with C_1 = zeta(3/2)/zeta(3) approx 2.1732, exceeding 2.