Definition:Meissel-Mertens Constant
(Redirected from Definition:Kronecker's Constant)
Jump to navigation
Jump to search
Definition
Consider the expression:
- $\ds M = \map {\lim_{n \mathop \to \infty} } {\sum_{\substack {p \mathop \le n \\ \text {$p$ prime} } } \dfrac 1 p - \ln \ln n}$
Then:
- $M \approx 0 \cdotp 26149 \, 72128 \, 47642 \, 78375 \, 54268 \, 38608 \, 69585 \, 90516 \ldots$
This sequence is A077761 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
The constant $M$ is known as the Meissel-Mertens Constant.
Also known as
Also known as:
- the Mertens constant
- Kronecker's Constant, for Leopold Kronecker
- the Hadamard-de la Vallée-Poussin Constant, for Jacques Salomon Hadamard and Charles de la Vallée Poussin
- the Prime Reciprocal Constant
Also see
- Mertens' Second Theorem, where $M$ is proved to exist.
Source of Name
This entry was named for Daniel Friedrich Ernst Meissel and Franz Mertens.
Sources
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,2614972128$
- but beware -- there is a mistake in the formula they give
- 1994: Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.): Chapter $2$: Sums: $\S 2.1$: Notation
- Weisstein, Eric W. "Mertens Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MertensConstant.html