# What properties characterize the function $L(x) = x+\exp(x) \log(x)$?

As might be known, the function $$L(x) = x+\exp(x)\log(x)$$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis:

$$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$

My question is, what "abstract" properties does this function have (like for example the $$\exp$$-function has the property $$\exp(x+y) = \exp(x)\exp(y)$$) and do these properties characterize this function?

Thanks for your help! (Also I am not sure how to properly tag this question)

Edit: Another related question: Dou you know of any other situation where the function $$L(x)$$ occurs? That would also be quite interesting!

Second Edit: I found a very exciting connection to "Logarithmic numbers" as defined by J. M. Gandhi:

http://oeis.org/A002741

The numbers $$\frac{d^n}{dx^n} L(x)$$ at $$x=1$$ are related to the Logarithmic numbers as defined by Gandhi.

There are two papers by Gandhi, on this topic:

http://oeis.org/A002741/a002741.pdf which is a little bit hard to read because it is scanned, and https://www.tandfonline.com/doi/abs/10.1080/00029890.1966.11970871 where there is made a connection to $$\sigma(n)$$ and the logarithmic numbers.

Third edit (18.05.2019): I think I found a very interesting property which seems to always hold:

If $$A$$ is a normal ($$A^TA=AA^T$$) and non-singular matrix and such that $$\frac{1}{|A|}\cdot A$$ is a doubly stochastic, positive matrix, then we have:

$$L(|A|) = |L(A)|$$ and $$L(A)$$ is a normal, non-singular matrix such that $$\frac{1}{|L(A)|}\cdot L(A)$$is doubly stochastic, positive matrix. where $$|.|$$ denotes the spectral norm.

To be more concrete I will tell how I construct the matrix for a given finite group $$G$$:

Let $$\rho$$ be the regular representation of $$G$$. $$S \subset G$$ a generating set, $$|g| := |g|_S=$$ word length with respect to $$S$$. Then I construct such a matrix, where we have some ordering $$g_i$$ of the group $$G$$:

$$a_{i,j} = \frac{1}{1+|g_i g_j^{-1}|}$$

This is a group matrix as defined by Dedekind and Frobenius. Let $$H_G:= \sum_{g \in G} \frac{1}{|g|+1}$$ be the harmonic number associated to $$S$$ and $$G$$.

Here are my conjectures concerning this matrix some of which I can prove:

1. $$H_G = |A|$$ [proved by Perron-Frobenius theorem]

2. (If 1. is true, then by definition of $$A$$ we must have that $$1/H_G A = 1/|A| A$$ is a doubly stochastic matrix [thats clear by 1. and definition of $$A_G$$.]

3. $$A = \sum_{g \in G} \frac{1}{1+|g|} \rho(g)$$ is the Birkhoff-Neunmann decomposition induced by the doubly stochastic matrix [proved by definition of $$A_G$$ and $$\rho$$]

4. Using 2. I can prove that $$A$$ is a normal matrix

5. $$A$$ is non-singular. [that remains mysterious]

My updated question is, if (any of) this can be proven (or if it is known, then any reference would also be nice)?

• One way to characterize it is $x L'' - (2 x - 1) L' + (x - 1) L =x^2 - 3 x + 1$, $L(1)=1$, $L''(1)=e$ – Matt F. Apr 30 at 11:14
• @MattF. I think you forgot the condition $L'(1) = e+1$. Thanks for your help. – orgesleka Apr 30 at 11:28
• WolframAlpha confirms the uniqueness of the characterization I proposed: wolframalpha.com/input/…. The alternative suggestion has $L(1)=e$. – Matt F. Apr 30 at 12:42
• Can you clarify your third edit? What is the Euclidean norm of a matrix? Frobenius/Hilbert-Schmidt norm? Or operator/spectral norm? Or something else? I've tried a few guesses, but numerics quickly disprove $L(|A|) = |L(A)|$ for every guess that I've made for what the norm should be. – Nathaniel Johnston May 18 at 15:42