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I was inspired by thisthis topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHelms for $n \approx 10^{100}$ (see answer below). enter image description here

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHelms for $n \approx 10^{100}$ (see answer below). enter image description here

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHelms for $n \approx 10^{100}$ (see answer below). enter image description here

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LRDPRDX
LRDPRDX
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I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHalms@GottfriedHelms for $n \approx 10^{100}$ (see answer below). enter image description here

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHalms for $n \approx 10^{100}$ (see answer below). enter image description here

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHelms for $n \approx 10^{100}$ (see answer below). enter image description here

added 100 characters in body
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LRDPRDX
LRDPRDX
  • 251
  • 1
  • 6

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

Please, let me know if it is obviouslyThe following picture based on data provided by @GottfriedHalms for $n \approx 10^{100}$ (or not obviouslysee answer below) false. enter image description here

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

Please, let me know if it is obviously (or not obviously) false.

I was inspired by this topic on Math.SE.
Suppose that $H_n = \sum\limits_{k=1}^n \frac{1}{k}$ - $n$th harmonic number. Then

Conjecture

Let $M$ be a set of all $n$ such that $$H_n - \lfloor{H_n\rfloor} < \frac{1}{n^{1+\epsilon}}.$$ Then $$\forall\epsilon>0 : |M| = \bar\eta(\epsilon) < \infty.$$

Picture below illustrates my conjecture, where I have checked this conjecture for $n < 10^6$ for each $\epsilon\in(0,1.1)$ with step 0.01 enter image description here

The following picture based on data provided by @GottfriedHalms for $n \approx 10^{100}$ (see answer below). enter image description here

added 228 characters in body
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LRDPRDX
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added 82 characters in body
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deleted 68 characters in body
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edited body
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