Inequalities for the Euler–Mascheroni constant

Let γ = 0.577215 … be the Euler–Mascheroni constant, and let R n = ∑ k = 1 n 1 k − log ( n + 1 2 ) . We prove that for all integers n ≥ 1 , 1 24 ( n + a ) 2 ≤ R n − γ < 1 24 ( n + b ) 2 with the best possible constants a = 1 24 [ − γ + 1 − log ( 3 / 2 ) ] − 1 = 0.55106 … and b = 1 2 . This refines the result of D. W. DeTemple, who proved that the double inequality holds with a = 1 and b = 0 .