Tagged Questions

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

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: ...
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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. ...
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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.
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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 ...
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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 ...
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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. ...
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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 ...
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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 $3$?
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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: $$n^k > 1^k + 2^k ... 0answers 60 views prime zeta function when 0<s<1 [duplicate] I would like to know if there is a good estimate for the sum which concerns all primes not exceeding x:$$\sum\limits_{p\leq x}\frac{1}{p^s}0<s<1$$. Only this. Thanks in advance! 0answers 92 views I am trying to prove this problem by induction, how can can i prove the following? I am given$$ F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}} $$where, \varphi = \frac{1 + \sqrt{5}}{2} and \psi = \frac{1 - \sqrt{5}}{2} The textbook states that it's equal to the n-th Fibonacci ... 4answers 484 views How find this sum \sum_{ab+cd=2^m}ac=? Find this sum (where m is a fixed positive integer)$$\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$$and a,b,c,d is odd ... 2answers 111 views Does sum of all natural numbers contradict another rule? I must say that I am not a mathematician, just a enthusiast who likes to read all the "weird" results in mathematics. I read that sum of all natural number equals to -1/12 and I am also aware that ... 3answers 212 views 1+2+3+4+5+… = -\frac{1}{12}. Is there any intuition for this? [duplicate] I was looking into a Numberphile video here. The guy says he was unable to find an intuition. Does there exist one? Is the premise, 1-1+1-1+...=\frac{1}{2}, reasonable mathematically? 0answers 70 views Find Gcd summation fast? Find the value of the summation:$$ val=\left( \sum_{i=1}^a \sum_{j=1}^b\sum_{k=1}^c....\sum_{x=1}^p GCD(i,j,k,..x) \right)$$Contraints 2\leqnumber of summation terms\leq 500, 1\leq ... 2answers 230 views Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes Starting from 2, the sequence of sums of all consecutive primes is:$$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
Let $q$ an arbitrary integer. Is there any chance of getting a bound like $$\underset{d\mid q}{\sum}\frac{1}{\phi\left(q/d\right)^{2}}\ll\frac{1}{\phi\left(q\right)^{2}}?$$