Euler–Mascheroni constant and the Meissel–Mertens constant

I've not written explicitly about either the Euler–Mascheroni constant or the Meissel–Mertens constant before, although the former is made mention of in a Numberphile video that I referenced in a post titled The Harmonic Series on October 12th 2016. 

Let's recount that the harmonic series is simply \(\zeta(1)\) and so:$$\zeta(1)=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ \dots =\sum_{n=1}^{\infty}\frac{1}{n}$$While this sequence does diverge it does so very slowly and that's what my post The Harmonic Series was all about. The Euler-Mascheroni constant can be defined as:$$\begin{align}

\gamma &= \lim_{n\to\infty}\left(-\log n + \sum_{k=1}^n \frac1{k}\right)\\

&=\int_1^\infty\left(-\frac1x+\frac1{\lfloor x\rfloor}\right)\,dx.

\end{align}$$Here, \(\lfloor x\rfloor\) represents the floor function. The numerical value of the Euler–Mascheroni constant, to 50 decimal places, is:

0.57721566490153286060651209008240243104215933593992... 

Below I've embedded the Numberphile video referred to earlier as it's really quite informative.

Like the harmonic series, the sum of the reciprocals of the prime numbers diverges also and even more slowly. The Meissel-Mertens constant is defined as:$$M = \lim_{n \rightarrow \infty } \left( \sum_{p \leq n} \frac{1}{p} - \ln(\ln n) \right)=\gamma + \sum_{p} \left[ \ln\! \left( 1 - \frac{1}{p} \right) + \frac{1}{p} \right]$$where \( \gamma \) is the Euler-Mascheroni constant. The value of M is approximately:

M ≈ 0.2614972128476427837554268386086958590516... 

Figure 1: source

The two constants are thus intimately linked. It's easy to generate approximations of these functions using SageMathCell. See Figure 2.

Figure 2: permalink

Looking at the results in Figure 2, it can be seen that:

Approximation of Euler-Mascheroni constant up to 100000 is 0.577220664893197

Approximation of Miessel-Mertens constant up to 100000 is 0.261801821365208

The light grey digits do not correspond to the known digits for these constants. It can be seen that the approximation to the Miessel-Mertens constant is less accurate than for the Euler-Mascheroni constant, reflecting the log(log) computation for the former versus the log computation for the latter.

For a post that shows how to determine the sum of the alternating harmonic series, see my post titled Alternating Series Test from April 23rd 2021. The alternating harmonic series converges thus:$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} \dots=\ln(2) \approx 0.693147180559945 \dots$$See also:

• Biography of Ernet Meissel
• Biography of Franz Mertens
• Biography of Lorenzo Mascheroni
• Biography of Leonhard Euler