0
votes
1answer
15 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
2
votes
0answers
37 views

Find the result of $ \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$

I would like to calculate the sum $$\displaystyle \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$$ where $k=\sqrt{2}+1$, $x$ is an odd integer and $\lfloor z \rfloor_o$ indicates ...
2
votes
0answers
47 views

Closed form for the sum $\sum_{a=1}^{b} a^3\cdot (b \bmod a)$

How can we simplify $\sum_{a=1}^{b} a^3\cdot (b \mod a)$? For $a \ge \frac{b+1}{2} $ to $a = b$ it reduces to $$\sum_{a\ge \frac{b+1}{2}}^{b}a^3\cdot (b-a)=b\cdot\sum_{a\ge ...
5
votes
1answer
88 views

Number of pairs $(i, j)$ where $1\leq i < j \leq N$ such that $i|j$

This is a programming question so I tried like this: for (int i=1; N/i != 1; i++) sum += N/i-1; I am getting the correct answer. But one solution I saw by ...
-1
votes
1answer
65 views

Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
3
votes
2answers
61 views

Which natural numbers can be represented as a sum of natural numbers raised to different powers?

Waring's problem asks about natural numbers that can be represented as a sum of natural numbers all raised to the same power $k$. I'm wondering which natural numbers can be represented as a sum of ...
3
votes
1answer
83 views

Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...
1
vote
0answers
31 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...
14
votes
3answers
443 views

Equality of sums with fractional parts of the form $\sum_{k=1}^{n}k\{\frac{mk}{n}\}$

I recently encountered the following equality ($\{\}$ denotes fractional part): $$\sum_{k=1}^{65}k\left\{\frac{8k}{65}\right\}=\sum_{k=1}^{65}k\left\{\frac{18k}{65}\right\}$$ and found it very ...
1
vote
1answer
64 views

Finding formulas for sums

I know that $\sum_{d \mid n} \mu(d) = 0$ whenever $n >1$, and I know that $\sum_{d \mid n} \phi(d) = n$. How can I use this in order to give a formula for $\sum_{d \mid n} \mu(d)\phi(d)$?
3
votes
0answers
164 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
4
votes
1answer
234 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
0
votes
1answer
51 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
26
votes
1answer
545 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
0
votes
0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
0
votes
2answers
66 views

Sum a series of series where each value increments by one

Can anyone suggest an elegant way to sum a series of numbers like this: (1, 2, 3, 4) (2, 3, 4, 5) (3, 4, 5, 6) (4, 5, 6, 7) That is for $n$ sets ...
3
votes
1answer
81 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
1
vote
3answers
265 views

Sum of digits of number from 1 to n

Is there any general formula for calculating the sum of digits of number from 1 to n? n < 10^9
1
vote
0answers
43 views

How find this sum $S=\sum_{i=1}^{m}(-1)^{a_{i}}\cdot 2^{m-i}$ and $2^i\equiv a_{i}\pmod n$

Question: let $n$ is give odd positive integer numbers,and $a_{i}\neq 1,0\le a_{i}\le n-1$, and $$2^i\equiv a_{i}\pmod n,i=1,2,\cdots,m-1$$ where $m(m\le n)$ such $2^m\equiv 1\pmod n$ ...
2
votes
0answers
48 views

Inverse logarithmic integral

If the expansion of the logarithmic interval is$$\text{li}(n) = \log \log n + \gamma + \sum_{k=1}^\infty \dfrac{(\log n)^k}{k! k}$$ what is the inverse of the function?
1
vote
0answers
35 views

Simplifying a sum- combinatorics

Hello fellow mathematicians, I have been working avidly as a high school project to prove Legendre's conjecture. The question below and the other questions I have posted are directly linked to a ...
0
votes
1answer
40 views

Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
2
votes
1answer
80 views

Integrality Conjectures

Here are some interesting conjectures I would like to prove. For all positive integers $a=bc,m,n$ the following expressions are integers: $$c\sum_{k=1}^{am}k\left\{\frac{kbn}{am}\right\}$$ ...
3
votes
2answers
140 views

Dedekind Sum Integrality Result

Can we prove the following is always an integer? $$6b\sum_{k=1}^bk\left\{\frac{ka}{b}\right\}$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the fractional part operator. UPDATE: Through the ...
7
votes
2answers
188 views

Evaluation of finite sum

How can one prove the following equality (for fixed positive integer $a$): $$12\sum_{k=1}^{an^2-1}k\left\{\frac{k(an-1)}{an^2}\right\}=3a^2n^4-a^2n^2-2$$ where $\{x\}=x-\lfloor x\rfloor$ denotes the ...
3
votes
3answers
277 views

Summation of Dedekind sums to zero

I'm trying to show that: $$\sum_{a=0}^b\text{GCD}(a,b)s(a,b)=0$$ More generally, can we also show: $$\sum_{a=0}^b\text{GCD}(a,b)^ls(a,b)=0$$ where $s$ is the Dedekind sum. Any ideas?
0
votes
0answers
63 views

A partial sum involving Euler's function

This is Exercise 2.1.17 of the book "H. Montgomery and R. Vaughan. Multiplicative Number Theory— I. Classical Theory". For $x\ge 2$, $\sum_{n\le x}\frac{\mu(n)^2}{\varphi(n)}=\log ...
1
vote
1answer
86 views

$\sum_{p\le x} \frac{1}{pq}$

I was given that $\sum_{p\le x} \frac{1}{p}$ = $\log\log x$+O(1). I need to show that $\sum_{pq\le x} \frac{1}{pq} = (\log \log x)^2 + O(\log \log x)$. Here we go: Break the sum into two sums: ...
11
votes
3answers
992 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
1
vote
0answers
28 views

Finding the summation of the series. [duplicate]

Is there any formula to find out the summation of the series. $$\sum_{i=1}^{n} \lfloor \frac{n}{i} \rfloor$$ Can someone help me with this.
11
votes
2answers
225 views

A different Harmonic series.

Let's call the following numbers than can be produced by playing with plus and minus: $$H_n'=\pm\frac{1}{1}\pm\frac{1}{2}\pm\frac{1}{3}\pm\cdots\pm\frac{1}{n}$$ "Harmonic kids" of $H_n$. We have a ...
7
votes
1answer
133 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
5
votes
1answer
72 views

Question about $ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) \forall s\in \Bbb N$

what I found from messing around was $$ \int_{-1}^{0}\sum_{n=1}^{x}n^sdx=\zeta (-s) $$ $$ s\in \mathbb{N} $$ when the partial sum is changed to an equivalent polynomial using Faulhaber's formula. ...
1
vote
1answer
39 views

What is the reasoning behind ways of splitting up this summation sign?

Some context: I've been studying Chebyshev's $\psi$ - function, which claims that $\psi(x) = \sum_{n \le x} \Lambda(n) = \sum_{p^k \le x} \log p$ where $p$ is prime and $\Lambda(n)$ is the von ...
1
vote
2answers
146 views

Summation of factorials modulo ten

I have read that$$\sum\limits_{i=1}^n i!\equiv3\;(\text{mod }10),\quad n> 3.$$ Why is the sum constant, and why is it $3$?
18
votes
4answers
311 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
4
votes
0answers
60 views

prime zeta function when $0<s<1$ [duplicate]

I would like to know if there is a good estimate for the sum which concerns all primes not exceeding $x$: $$\sum\limits_{p\leq x}\frac{1}{p^s}$$$$0<s<1$$. Only this. Thanks in advance!
1
vote
0answers
92 views

I am trying to prove this problem by induction, how can can i prove the following?

I am given $$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$ where, $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\psi = \frac{1 - \sqrt{5}}{2}$ The textbook states that it's equal to the n-th Fibonacci ...
15
votes
4answers
484 views

How find this sum $\sum_{ab+cd=2^m}ac=?$

Find this sum (where $m$ is a fixed positive integer) $$\sum_{\substack{ab+cd=2^m\\ a,b,c,d \text{ are odd}}}ac.$$ My idea: since $$ab+cd=2^m\Longrightarrow ab=2^m-cd$$ and $a,b,c,d$ is odd ...
0
votes
2answers
111 views

Does sum of all natural numbers contradict another rule?

I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to $-1/12$ and I am also aware that ...
0
votes
3answers
212 views

$1+2+3+4+5+… = -\frac{1}{12}$. Is there any intuition for this? [duplicate]

I was looking into a Numberphile video here. The guy says he was unable to find an intuition. Does there exist one? Is the premise, $1-1+1-1+...=\frac{1}{2}$, reasonable mathematically?
2
votes
0answers
70 views

Find Gcd summation fast?

Find the value of the summation: $$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$ Contraints $2\leq$number of summation terms$\leq 500$, $1\leq ...
16
votes
2answers
230 views

Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
0
votes
1answer
101 views

On Euler totient function sum

Let $q$ an arbitrary integer. Is there any chance of getting a bound like $$\underset{d\mid q}{\sum}\frac{1}{\phi\left(q/d\right)^{2}}\ll\frac{1}{\phi\left(q\right)^{2}}?$$
1
vote
0answers
67 views

Estimation of a logarithmic sum

I need to estimate the sum $$ \underset{r=2}{\overset{t}{\sum}}\left(\frac{\log\log r}{r}\right)^{2}. $$ I tried to use the Abel's partial summation, and I got $$ \frac{(\log\log ...
1
vote
1answer
82 views

Ramanujan Notebook Part 1 (1.16): $\sum q^{n^2} = (-q;q^2)_\infty^2(q^2;q^2)_\infty=\frac{(-q;-q)_\infty}{(q;-q)_\infty}$

I am having trouble with proving a statement in Ramanujan's Lost Notebook Part 1 (1.16). The statement is as follows: $\varphi(q)=f(q,q)=\sum_{n=-\infty}^\infty q^{n^2} = ...
2
votes
2answers
52 views

Is there another way to represent this summation?

I wish to calculate $\sum_{x=1}^{n}\sum_{y=1}^{n} f(x,y)$ where $x>2y$. I can do this by changing $y$'s upperbound to the floor of $(x-1)/2$ but this makes simplification of the summation harder ...
3
votes
0answers
44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
5
votes
4answers
247 views

factorial as difference of powers: $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$?

The successive difference of powers of integers leads to factorial of that power. I think this is the formula $\sum_{r=0}^{n}\binom{n}{r}(-1)^r(l-r)^n=n!$ But I found no proof on internet. Please ...
3
votes
3answers
512 views

Sum of all real number for any interval.

We know that sum of natural numbers over any interval always exists. For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ But what about real ...