Leonard Euler discovered one the classical formula of analytical number theory;

where is the Euler–Mascheroni constant. The sum on the RHS is called the harmonic number and is denoted by . In this article, I shall represent this classical asymptotic formula of Euler in terms of the Riemann zeta function.

**Entry 1**. If then

**Entry 2**. Taking we obtain a new formula for the Euler–Mascheroni constant

**Entry 3**. If denotes the fractional part of then

**Entry 4**. If then

**Entry 5**. If denotes the fractional part of then

The RHS of the above asymptotic formula is exactly equal to that of Euler’s formula and thus we have

**Entry 6**.

### Like this:

Like Loading...

*Related*