An improvement of a theorem of Erdős and Sárközy

Horváth, Gábor (2007) An improvement of a theorem of Erdős and Sárközy. Pollack Periodica, 2 (Supple). pp. 155-161. ISSN 1788-1994

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Abstract

Let a ₁ < a ₂ < … be an infinite sequence of positive integers and denote by R ₂ ( n ) the number of solutions of n = a _i + a _j . P. Erdős and A. Sárközy proved that if g ( n ) is a monotonically increasing arithmetic function with g ( n ) → +∞ and g ( n ) = o ( n (log n ) ⁻² ) then | R ₂ ( n ) − g ( n )| = o (√ g ( n )) cannot hold. We will show that for any ɛ > 0, the inequality | R ₂ ( n ) − g ( n )| ≤ (1 − ɛ )√ g ( n ) cannot hold from a certain point on.

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