Euler's Constant in the Proof of Theorem 3.2 (a)

Definition of Euler's Constant

$$C=\underset{n\to \mathrm{\infty }}{lim}(1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}-\log n)=\underset{x\to \mathrm{\infty }}{lim}(\sum _{n\le x}\frac{1}{n}-\log x).$$

Euler's Constant Emerging In the Proof of Theorem 3.2 (a)

$$\sum _{n\le x}\frac{1}{n}=\log x+\underset{\text{This is C as shown below}}{\underbrace{1-{\int }_1^{\mathrm{\infty }}\frac{t-[t]}{t^2}dt}}+O\left(\frac{1}{x}\right).$$

$$\sum _{n\le x}\frac{1}{n}-\log x=1-{\int }_1^{\mathrm{\infty }}\frac{t-[t]}{t^2}dt+O\left(\frac{1}{x}\right).$$

$$C=\underset{x\to \mathrm{\infty }}{lim}(\sum _{n\le x}\frac{1}{n}-\log x)=\underset{x\to \mathrm{\infty }}{lim}(1-{\int }_1^{\mathrm{\infty }}\frac{t-[t]}{t^2}dt+O\left(\frac{1}{x}\right))=1-{\int }_1^{\mathrm{\infty }}\frac{t-[t]}{t^2}dt.$$