Approaching The Euler-Mascheroni Constant

I am looking for a value $$a \approx 14$$ with some nice property. So I am going to define some things with this value $$a$$ and then ask what $$a$$ does the trick I want (If there is some $$a$$ that does the trick at all).

Definitions and Intro

Let $$f(x,t) = \frac{\ln(t+a)^x}{t}$$ and note that $$f_x(x,t)=\frac{\ln(t+a)^{x}\ln(\ln(t+a))}{t}$$ where $$f_x$$ refers to $$\frac{d}{dx} f(x,t)$$

Now define $$g(x) = \lim_{m\to\infty} \sum_{t=1}^m f(x,t)-\int_1^m f(x,t)dt$$

Note that $$g(0) =\gamma$$ the Euler Mascheroni constant and the generalization above can be found under the generalization section of that wiki (So I am not conjuring this idea from thin air). In fact, when $$a=0$$ it seems that $$g(x)$$ is connected with what is referred to as Stieljes Constants.

It looks to me that there may exist some $$a$$ value that $$g(x)=g'(x)$$. Which would be kind of interesting. Because this would mean that $$g(x)= \gamma e^x$$.

Here's a graph which led me to these suspicions. I won't reproduce the image of the graph because it just looks like $$y=\gamma e^x$$. The interesting thing is that the numerical derivative nearly overlays the function.

The Question Does there exist some $$a$$ that does this? And what is it?

Some preliminary notes/ attempts to make progress

We should note that $$g'(x) = \lim_{m\to\infty} \sum_{t=1}^m f_x(x,t)-\int_1^m f_x(x,t)dt$$

Which allows for a little algebraic manipulations after we take the assumption $$g'(x) =g(x)$$. These manipulations haven't really helped me find out what $$a$$ is...

Motivations David Hilbert referred to the puzzle of proving the irrationality of $$\gamma$$ as "unapproachable." Which explains my title... I am just looking for some approaches to $$\gamma$$ which may communicate some information about this constant.

• I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$ – Mason Nov 20 '18 at 18:11
• Another way to think about this is that the $a$ value just changes the index of the summation and the integral. – Mason Nov 22 '18 at 14:32
• I would think that we can prove that there isn't one. Or there is one. I think there may be because of the graph which I've linked. – Mason Nov 22 '18 at 21:44
• Why do you think that for some $a$ then (for every $x$ in some interval) $g_a(x) = g_a'(x)$ ? The Stieltjes constant are the derivatives of $F_0(s) = (s-1) \zeta(s)$ at $s=1$. So try finding the analytic function $F_a(s)$ whose derivatives at $s=1$ are related to $g_a(n)$ – reuns Nov 22 '18 at 21:46
• A suggestion (rough method): Find out numerically, for which $a$ is $g’(0)=\gamma$ . Then choose $x_0\neq 0$ and test, if $g’(x_0)=g(x_0)$ (e.g. up to 8 digits behind the decimal point). Then you know, whether it makes sense to expect $g'(x)=g(x)$ or not. – user90369 Nov 29 '18 at 13:34