Approaching The Euler-Mascheroni Constant











up vote
2
down vote

favorite
1












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 $$g(x) = lim_{mtoinfty} sum_{t=1}^m f(x,t)-int_1^m f(x,t)dt $$



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 $$g'(x) = lim_{mtoinfty} sum_{t=1}^m f_x(x,t)-int_1^m f_x(x,t)dt $$



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.










share|cite|improve this question

















This question has an open bounty worth +50
reputation from Mason ending in 5 days.


This question has not received enough attention.
















  • I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
    – Mason
    Nov 20 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
    2 days ago










  • 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
    2 days ago






  • 1




    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
    2 days ago

















up vote
2
down vote

favorite
1












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 $$g(x) = lim_{mtoinfty} sum_{t=1}^m f(x,t)-int_1^m f(x,t)dt $$



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 $$g'(x) = lim_{mtoinfty} sum_{t=1}^m f_x(x,t)-int_1^m f_x(x,t)dt $$



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.










share|cite|improve this question

















This question has an open bounty worth +50
reputation from Mason ending in 5 days.


This question has not received enough attention.
















  • I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
    – Mason
    Nov 20 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
    2 days ago










  • 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
    2 days ago






  • 1




    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
    2 days ago















up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





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 $$g(x) = lim_{mtoinfty} sum_{t=1}^m f(x,t)-int_1^m f(x,t)dt $$



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 $$g'(x) = lim_{mtoinfty} sum_{t=1}^m f_x(x,t)-int_1^m f_x(x,t)dt $$



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.










share|cite|improve this question















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 $$g(x) = lim_{mtoinfty} sum_{t=1}^m f(x,t)-int_1^m f(x,t)dt $$



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 $$g'(x) = lim_{mtoinfty} sum_{t=1}^m f_x(x,t)-int_1^m f_x(x,t)dt $$



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.







real-analysis sequences-and-series eulers-constant






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 20 at 17:03

























asked Nov 20 at 15:16









Mason

1,7001325




1,7001325






This question has an open bounty worth +50
reputation from Mason ending in 5 days.


This question has not received enough attention.








This question has an open bounty worth +50
reputation from Mason ending in 5 days.


This question has not received enough attention.














  • I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
    – Mason
    Nov 20 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
    2 days ago










  • 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
    2 days ago






  • 1




    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
    2 days ago




















  • I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
    – Mason
    Nov 20 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
    2 days ago










  • 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
    2 days ago






  • 1




    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
    2 days ago


















I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
– Mason
Nov 20 at 18:11




I suppose I could just ask more broadly about the class of functions $f$ such that $g(x)=g'(x)$
– Mason
Nov 20 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
2 days ago




Another way to think about this is that the $a$ value just changes the index of the summation and the integral.
– Mason
2 days ago












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
2 days ago




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
2 days ago




1




1




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
2 days ago






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
2 days ago

















active

oldest

votes











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














 

draft saved


draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006440%2fapproaching-the-euler-mascheroni-constant%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















 

draft saved


draft discarded



















































 


draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3006440%2fapproaching-the-euler-mascheroni-constant%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







Popular posts from this blog

Wiesbaden

Marschland

Dieringhausen