Inequalities for the Euler–Mascheroni constant

Let γ=0.577215…γ=0.577215… be the Euler–Mascheroni constant, and let Rn=∑k=1n1k−log(n+12). We prove that for all integers n≥1n≥1, 124(n+a)2≤Rn−γ<124(n+b)2 with the best possible constants a=124[−γ+1−log(3/2)]−1=0.55106…andb=12. This refines the result of D. W. DeTemple, who proved that the double inequality holds with a=1a=1 and b=0b=0.