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Questions tagged [harmonic-numbers]

For questions regarding harmonic numbers, which are partial sums of the harmonic series. The $N$-th harmonic number is the sum of reciprocals of the first $N$ natural numbers.

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186 views

Consecutive prime numerators of harmonic numbers?

Let $$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\frac{a}{b}$$ and let $a$ and $b$ are coprime, $h_{n}=a$. $h_{n}$ is prime for $$n=2,3,5,8,9,21,26,41,56,62,69,79,89,91,122,127,143,...
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697 views

stirling numbers and harmonic number identities

Permit me a brief introduction before I state the question, three questions in fact. Inspired by this MSE link I computed the following harmonic sum identities: $$1/6\, \left( {H_{{n}}}^{(1)} \right) ...
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128 views

On binomial sums $\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$ and log sine integrals

Seven years ago, I asked about closed-forms for the binomial sum $$\sum_{n=1}^\infty \frac{1}{n^k\,\binom {2n}n}$$ Some alternative results have been made. Up to a certain $k$, it seems it can be ...
6
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177 views

A cubic nonlinear Euler sum

Any idea how to solve the following Euler sum $$\sum_{n=1}^\infty \left( \frac{H_n}{n+1}\right)^3 = -\frac{33}{16}\zeta(6)+2\zeta(3)^2$$ I think It can be solved it using contour integration but ...
6
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174 views

An integral to prove that $\log(2n+1) \ge H_n$

Dalzell integral The equation $$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ proves that $\frac{22}{7}-\pi>0$ because the integrand is positive. Some Dalzell-type integrals for $\log(...
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97 views

Formulas involving the polynomials $\frac{1}{n!}x^n(a_n-b_nx)^n,$ where the coprime integers $a_n,b_n$ satisfy $H_n=\frac{a_n}{b_n}$ for $n\geq 1$

For integers $n\geq 1$ let $$H_n=1+\frac{1}{2}+\ldots+\frac{1}{n}$$ the $nth$ harmonic number. After I've seen the form of the polynomials used by Niven in [1] I wanted to create a puzzle with a new ...
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238 views

Is there a closed-form approximation to a band-limited sawtooth?

A partial Fourier Series with no coefficients is equal to the closed form expression: $${A \over n} \sum_{k=1}^n \cos(k\theta) = {A \over 2n} \left\{{\sin([2n + 1]\theta/2) \over \sin(\theta/2)} - 1\...
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89 views

Closed form of finite Euler sum $\sum_{k=1}^n \frac{ H_{k}}{(2k+1)}$

Recently (Finite sums of products of harmonic numbers like $\sum_{k=1}^n H_{k} H_{2k}$) I came across this finite Euler sum $$p_{2}(n) = \sum_{k=1}^n \frac{H_{k}}{2k+1}\tag{1}$$ and I wonder if it ...
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155 views

On the generating function $\sum_{n=1}^\infty\left(e^{H_n}\log\left(H_n\right)\right)x^n$, for $0<x<1$

You can read Lagarias equivalence to the Riemann's Hypothesis from this MathWorld, I am saying (5), or well from the reference [1]. Let for real numbers $0<x<1$ the factor $x^n$, where $n\geq 1$...
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2k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
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104 views

Is there an infinite hierarchy of sequences whose reciprocals diverge, starting with the natural numbers?

It is well known that the sum of the reciprocals of the function $f_0(n)=n$ (the harmonic series) diverges: $$\sum_{n=1}^\infty\,\frac{1}{n}=\infty$$ Similarly, the sum of the reciprocals of the ...
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50 views

How to evaluate $\sum_{n=1}^{\infty}\frac{H_{kn}^2-[\gamma+\ln(kn)]^2}{n}?$

From this post @Olivier Oloa gives the closed form for this sum $(1)$ $$\sum_{n=1}^{\infty}\frac{H_n^2-(\gamma+\ln n)^2}{n}=\frac{5}{3}\zeta(3)-\frac{2}{3}\gamma^3-2\gamma\gamma_1-\gamma_2\tag1$$ I ...
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88 views

Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

I am trying to evaluate the following integral $$\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_{n}, \hspace{0.5cm} 0<...
3
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71 views

Closed expression for almost-elliptic theta $\sum _{k=1}^{\infty } \frac{1}{2^{k^2} k}$?

Motivation: A more complicated generalization of Closed expressions for harmonic-like multiple sums are sums of a type for which the follwing is the simplest case $$s=\sum _{m=1}^{\infty } \sum _{n=...
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34 views

Asymptotics of $(n+1)^{-(1+a)} H_n^{(-a)}$

Interested by this post, running cases, I have been able to observe that, for any value of $a$ (positive integer, rational, irrational, complex) $$\frac{\sum_{i=1}^n i^a}{n^{a+1}}= \frac{H_n^{(-a)}}{(...
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124 views

Relation between Stirling numbers of first kind and harmonic numbers

We have the following nice relations for Striling numbers of the first kind $$\left[n\atop 2\right] = \Gamma(n) H_{n-1}$$ $$\left[n\atop 3\right] = \frac{\Gamma(n)}{2} ((H_{n-1})^2-H_{n-1}^{(2)})$$ ...
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87 views

An Identity with Binomials and Harmonic Numbers

Let $m,n,p$ positive integers with $m\geq n$ and $H_m=1+1/2+1/3+\cdots+1/m$ the $m-$ith Harmonic Number with $H_0:=0$. Show that for the values of $m,p,n$ for which the denominators do not vanish, ...
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191 views

Question concerning the divergence of a kind of “hyperharmonic” series different than the definition of Conway and Guy

Conway and Guy defined $$H_k^0=\sum_{n=1}^k\dfrac1n$$ and $$H_k^r=\sum_{n=1}^kH_n^{r-1}$$ for $k,r\in\Bbb Z^+$. I would prefer a definition of an $r$-hyperharmonic number to have some chance of ...
2
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35 views

sum of powers of unit fractions

I wonder if it is possible to evaluate explicitly the sum $$S(N):=\sum_{j=1}^{\left\lfloor\frac{N-1}{2}\right\rfloor}\left(1-\frac{2j}{N}\right)^{N+1},\quad N\in\mathbb{N}.$$ In the large $N$ limit ...
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32 views

A “binomial” generalization of harmonic numbers

For positive integers $s$ and $n$ (let's limit the generality), define $$H_s(n)=\sum_{k=1}^{n}\frac{1}{k^s},\qquad G_s(n)=\sum_{k=1}^{n}\binom{n}{k}\frac{(-1)^{k-1}}{k^s}.$$ The former is well-known; ...
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97 views

A sort of Wolstenholme's $p^2$-congruence.

It is well-known (Wolstenholme Theorem) that for any prime $p$ such that $p>3$ the Harmonic number $H_{p-1}$ satisfies the congruence $$ H_{p-1}:= \sum_{i=1}^{p-1}\frac{1}{i}\equiv 0 \pmod {p^2}$$ ...
2
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51 views

Sum with harmonic number of squared argument as e.g. $\sum_{k=1}^\infty \frac{H(k^2)}{k^2}$

I wonder if closed expression can be found for sums of harmonic numbers with a squared argument. Examples are $$s_{1}=\sum_{k=1}^\infty \frac{ H(k^2)}{k^2} \simeq 3.28709\tag{1}$$ $$s_{2}=\sum_{k=1}^...
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54 views

A very strict inequality for $\ln n$ and $H_{n-1}$, with $n>2$, searching for proof

I have been fooling around with logarithms and experimentally found a very strict inequality from above (which I haven't been able to improve further): $$\ln n < H_{n-1}+\frac{1}{8(n-1)}+\frac{3}...
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88 views

Asymptotic behaviour of Schatten $p$-norm

Consider the following matrix $$\Gamma(4) = \begin{pmatrix} 0 & 1 & 0 & 0\\ -1 & 0 & \sqrt{2} & 0\\ 0 & -\sqrt{2} & 0 & \sqrt{3}\\ 0 & 0 & -\sqrt{3} &0 ...
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69 views

Mellin Transform

If $$ H(x)=\sum_{n \leq x} \frac{1}{n} $$ what is its Mellin transform? I was able to find the Mellin transform of $\log(x+1)$ and of $\frac{1}{x+1}$, but I'm quite a bit inexperienced so I haven't ...
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64 views

Justify an approximation of $-\sum_{n=2}^\infty H_n\left(\frac{1}{\zeta(n)}-1\right)$, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, let $\mu(n)$ the Möbius function, see this MathWorld, and let $H_n$ the $n$th harmonic number as $H_n=1+1/2+\ldots+1/n$. Also we denote the Riemann zeta function with $\zeta(s)$...
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67 views

Partial sum involving harmonic numbers

QUESTION: I need to know how to compute the partial sum $$\sum_{k=1}^n \frac{H_{k+1}^2-H_{k+1}^{(2)}}{k+2}$$ in terms of the generalized harmonic numbers $H_n^{(m)}$. CONTEXT: This problem arose ...
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210 views

What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ...
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183 views

Finding a solution to $\sum _{n=1}^{n=k} \frac{1}{n^x}+\sum _{n=1}^{n=k} \frac{1}{n^y}=0$

Scroll down to the update to see what I am meaning. The Mathematica program below finds a solution to the equation: $$\sum _{n=1}^5 \frac{1}{n^x}+\sum _{n=1}^5 \frac{1}{n^y}=0$$ My question is if you ...
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38 views

Prove that the eigenvalues of a random matrix of this form, are invariant regardless of the value of the exponent $s$.

Consider the sequence: $$\frac{\sum _{k=1}^1 \frac{1^s}{k^s}}{1^s},\frac{\sum _{k=1}^2 \frac{2^s}{k^s}}{2^s},\frac{\sum _{k=1}^3 \frac{3^s}{k^s}}{3^s},...,\frac{\sum _{k=1}^n \frac{n^s}{k^s}}{n^s} \;\...
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101 views

Does the following limit exist? (involving harmonic numbers)

Let $H_m$ denote the $m$-th harmonic number (with the convention $H_0:=0$). Fix an integer $n$. Define for $k=0,1,\dots,n-1$ $$ d_{n,k}:={1\over{n^2}}\biggl\{\sum_{j=0}^k \bigl(H_n-H_k+H_{n-1}-H_{n-k+...
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111 views

Similarity of two limits related to the sum of divisors $\sigma(n)$ and the harmonic numbers $H_n$

Given that the sum of divisors has the form: $$\large \sigma(n) = \sum _{k=1}^n \lim_{s\to 0} \, \left(\frac{(s+1) (-1)^{\frac{2 n}{k}}+s-1}{k \cdot s \cdot 2}\right)^{-1}$$ $$1, 3, 4, 7, 6, 12, 8, ...
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199 views

Prime Zeta Function

Does $$\sum_{p \text{ prime}} \frac{1}{p^s} \sim \log \zeta(s) \quad \text{as} \quad s \to 1^+$$ imply $$\sum_{p \leq n} \frac{1}{p} \sim \log H_n \quad \text{as} \quad n \to \infty,$$ where $H_n$ is ...
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30 views

Leave-one-out Harmonic number

I am trying to find a number sequence $x_k\geq 0$ such that $$\text{for all integers }n\geq k\geq 1,\quad H_{n-1 - x_k(n-k)}\geq H_n-\frac{1}{k}.$$ I recall that the harmonic numbers $s\mapsto H_s$ ...
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65 views

Why can't the divergence of the harmonic series be resolved?

Calculus 1 students are taught that some infinite series converge while others do not. However, there are other tricks one can do whereby some of those "diverging" series do in fact converge but in ...
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78 views

Series involving product of Harmonic Numbers

Are there any identities involving this product of Harmonic Numbers? $$u_{n,m}=\prod_{k=1}^{n-1} H_k \prod_{k=1}^m H_{k+n}$$ My motivation is an attempt to study a series of the form $$\sum_{n=1}^\...
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38 views

Simple formula for $H_n = m + \alpha $?

Let $H_i$ be the $i$ th harmonic number. For a given positive integer $m$ we want to find the smallest possible positive integer value $n$ such that $H_n = m + \alpha $, where $\alpha > 0$. Let us ...
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44 views

On integer sequences of the form $\sum_{n=1}^N (a(n))^2H_n^2,$ where $H_n$ is the $n$th harmonic number: refute my conjecture and add yourself example

After I've read a problem posed by Édouard Lucas from the first paragraph of the biography from this Wikipedia's article I am trying to get different variations of this problem using number theoretic ...
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0answers
37 views

A non obvious example of a sequence $a(k)\cdot H_{b(k)}$ whose general term is integer many times, where $H_n$ denotes the $n$th harmonic number

For integers $n\geq 1$, $H_n$ denotes the $n$th harmonic number. I am looking examples of arithmetic function $a(k)$ and $b(k)$, whose terms are integers $\geq 1$ and such that the terms of this ...
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0answers
61 views

Generating function for harmonic number times log ($H_k log(k)$)

During my studies of generating functions for expressions composed of harmonic numbers and its asyptotic approximations I found a simple sum which seems to have been neglected so far $$s171219a=\sum ...
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0answers
31 views

Set Consisting of Harmonics of Integer Powers of 2?

If one defines "harmonics" as natural number fractions of a number, is there a name for the set consisting of all harmonics of all integer powers of two?
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58 views

Series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges

I was wondering if there are series whose $a_n \rightarrow 0$ as $n \rightarrow \infty$ and diverges. I know one example, that is $a_n = \frac{1}{n}$. Slight modifications don't count, such as $a_n = \...
1
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0answers
117 views

Infinite sum of harmonic number over general polynomial

As an extension to my question Infinite sum of harmonic number over polynomial of 2nd degreee I found a general formula for a polynomial of arbitrary degree which I had presented as a self answer but ...
1
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0answers
100 views

The Harmonic Logarithm and its relation to the Prime Number Theorem

For the purposes of this question, the Harmonic Number, $H_n$ is defined by the finite series sum $H_n=\sum_{k=1}^n\frac{1}{k}$ (n and k being positive integers) and the Harmonic Logarithm, $hlog(n)$ ...
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0answers
107 views

Sum Identity without Parseval's Theorem

Show that $\int_0^\pi \log^2\left(\tan\frac{ x}{4}\right)dx=\frac{\pi^3}{4}.$ The following Sum Identity was derived by applying Parseval's Theorem to the integral in the question linked above. $\...
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0answers
49 views

Asymptotic (divergent) behavior of modified alternating harmonic series?

I was fiddling around with some numerical sums in Mathematica, and I noticed that the partial sums of the following series seemed to follow a nice pattern: $$\sum_{n=1}^{10^k}\left( \frac{1}{n} \...
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0answers
73 views

Evaluation of two Euler type sums

We know that the harmonic number sum (also called Euler type sum) enter link description here $$\sum\limits_{n = 1}^\infty {\frac{{H_n^{\left( 2 \right)}}}{{{n^2}{2^n}}}} = {\rm{L}}{{\rm{i}}_4}\...
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0answers
89 views

On identities from derivatives of high order of the Gamma function and harmonic numbers, likes $\Gamma'(m+1)=m!\left(-\gamma+H_m\right)$

I know that the Gamma function is related with harmonic numbers since in the sum $$1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$ for $n>1$, one can get particular values of the Gamma function as ...
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0answers
69 views

How would look a harmonic and p-harmonic series vector in a multidimensional space?

Learning about the harmonic series divergence and p-harmonics series convergence, I tried to manipulate them globally as follows. $$\lim_{n \to \infty}A \cdot I = \lim_{n \to \infty} \begin{pmatrix} \...
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72 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ https://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...