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