Alladi–Grinstead constant

The Alladi–Grinstead constant is a mathematical constant that arises in the study of the factorial function, and in particular the decomposition of factorials into prime numbers. It is named for the mathematicians Krishnaswami Alladi and Charles Grinstead. The decimal representation of the Alladi–Grinstead constant begins,

0.80939402054063913071793188059409131721595399242500030424202871504... (sequence A085291 in the OEIS).

The exact value of the constant can be written as the exponential of a certain infinite series expressed using the Riemann zeta function. The sequence to which the Alladi–Grinstead constant is the limit arises in the investigation of how to write n factorial as the product of n large factors, a topic studied by Paul Erdős and others.

Definition

The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers. For example, the factorial of 3 is

$3!=3\cdot 2\cdot 1$

$\;\;\;=3^{1}\cdot 2^{1}$

As a less trivial example:

$10!=10\cdot 9!$

$\;\;\;=2\cdot 5\cdot 9!$

$\;\;\;=2\cdot 5\cdot 3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 5\cdot 7\cdot 2^{3}$

$\;\;\;=3^{2}\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{2}\cdot 2\cdot 5\cdot 5\cdot 7\cdot 2^{3}$

$\;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 2^{3}\cdot 5\cdot 5\cdot 7\cdot 2^{3}$

$\;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 2^{2}\cdot 5\cdot 5\cdot 7\cdot 2^{3}\cdot 2^{3}$

$\;\;\;=3\cdot 3\cdot 3\cdot 3\cdot 4\cdot 5\cdot 5\cdot 7\cdot 8\cdot 8$

Consider all decompositions of ${\textstyle n!}$ that have length ${\textstyle n}$, and let the index $i$ label those decompositions. Then for each $i$,

$$n!=\prod _{j=1}^{n}p_{i,j}^{b_{i,j}},$$

where the ${\textstyle p_{i,j}}$ are prime numbers and the $b_{i,j}$ are integers greater than or equal to 1. Let

$$\alpha (n)={\frac {\ln \max _{i}\left(p_{i,1}^{b_{i,1}}\right)}{\ln n}}.$$

As ${\textstyle n}$ tends to infinity, $\alpha (n)$ approaches a limiting value, the Alladi–Grinstead constant. This constant can be written as an exponential:

$$\lim _{n\to \infty }\alpha (n)=e^{c-1}\approx 0.80939402,$$

where ${\textstyle c}$ is given by

$$c=\sum _{k=2}^{\infty }{\frac {1}{k}}\ln {\frac {k}{k-1}}\approx 0.78853057.$$

This constant can alternatively be expressed in terms of the Riemann zeta function, as follows:

$$c=\sum _{n=1}^{\infty }{\frac {\zeta (n+1)-1}{n}}.$$

This series for the constant ${\textstyle c}$ converges more rapidly than the one before.

See also

• Mathematical constants
• Golomb–Dickman constant

References

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