Question

Prove that $\sum_{p\leq x}\frac{1}{p}=\log \log x+O(1)$ using PNT

score 1.1k · Answer 1 · 2016-12-12 13:35:41

Prove that $\sum_{p\leq x}\frac{1}{p}=\log \log x+O(1)$ using PNT

Published on 12 Dec 2016 - 13:35

score 114 · Answer 2 · 2016-12-13 16:07:42

Rewriting the expression $\sum_{p}\sum_{k=1}^{\infty}\frac{1}{k}\frac{1}{p^{ks}}=\sum_{n=2}^{\infty}\frac{\Lambda (n)}{\log n}\frac{1}{n^s}$

score 367 · Answer 3 · 2016-12-14 13:43:54

367

Views

Number of primes in $]x,2x]$ compared to in $[0,x]$ for $x > 0$

Published on 14 Dec 2016 - 13:43

score 58 · Answer 4 · 2026-03-26 10:59:50

58

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Are these two sequences known to be infinite : A120628 and A079149 ? Reference request

Published on 26 Mar 2026 - 10:59

score 86 · Answer 5 · 2016-12-14 23:23:59

86

Views

Artin's primitive roots conjecture

Published on 14 Dec 2016 - 23:23

score 170 · Answer 6 · 2026-03-28 15:26:16

170

Views

Zeta function of number field and linear factors

Published on 28 Mar 2026 - 15:26

score 150 · Answer 7 · 2026-03-26 06:29:56

150

Views

Relation between $\zeta(s), \ Re(s) < 1$ and the summation $\sum_{k=1}^\infty k^{-s}$

Published on 26 Mar 2026 - 6:29

score 67 · Answer 8 · 2026-03-26 06:21:45

67

Views

$\sum_{n=1}^7\frac{\mu(n)}{n^s}$ has a zero with $\Re s>1$, where $\mu(n)$ is the Möbius function

Published on 26 Mar 2026 - 6:21

score 282 · Answer 9 · 2016-12-16 10:41:38

282

Views

Maximal order of a group algebra over $\mathbb{Q}$

Published on 16 Dec 2016 - 10:41

score 703 · Answer 10 · 2026-03-25 15:21:41

703

Views

Dirichlet series associated to squared Möbius

Published on 25 Mar 2026 - 15:21

score 214 · Answer 11 · 2026-03-26 06:22:31

214

Views

On the convergence of a series involving the Riemann Zeta function

Published on 26 Mar 2026 - 6:22

score 175 · Answer 12 · 2016-12-18 14:52:57

175

Views

Estimation of $\pi_2(x)=\#\{n\leq x:n=p_1p_2\}$, where $p_1$ and $p_2$ are two distinct primes

Published on 18 Dec 2016 - 14:52

score 125 · Answer 13 · 2026-03-25 13:13:28

125

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On the behaviour of $\sum_{\substack{n\leq x\\(n,k)=1}}\frac{1}{\sqrt[\alpha]{n}}$

Published on 25 Mar 2026 - 13:13

score 73 · Answer 14 · 2016-12-19 11:48:39

73

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What is the value of $\sum_{n \geq 1} \epsilon^n/n$, where $\epsilon$ is a $k^\text{th}$ root of $1$

Published on 19 Dec 2016 - 11:48

score 245 · Answer 15 · 2016-12-19 15:37:54

245

Views

Rewrite $\sum_{p\leq \sqrt x}\frac{x}{p\log(x/p)}$ using Abel's summation formula

Published on 19 Dec 2016 - 15:37

List Question

Prove that $\sum_{p\leq x}\frac{1}{p}=\log \log x+O(1)$ using PNT

Rewriting the expression $\sum_{p}\sum_{k=1}^{\infty}\frac{1}{k}\frac{1}{p^{ks}}=\sum_{n=2}^{\infty}\frac{\Lambda (n)}{\log n}\frac{1}{n^s}$

Number of primes in $]x,2x]$ compared to in $[0,x]$ for $x > 0$

Are these two sequences known to be infinite : A120628 and A079149 ? Reference request

Artin's primitive roots conjecture

Zeta function of number field and linear factors

Relation between $\zeta(s), \ Re(s) < 1$ and the summation $\sum_{k=1}^\infty k^{-s}$

$\sum_{n=1}^7\frac{\mu(n)}{n^s}$ has a zero with $\Re s>1$, where $\mu(n)$ is the Möbius function

Maximal order of a group algebra over $\mathbb{Q}$

Dirichlet series associated to squared Möbius

On the convergence of a series involving the Riemann Zeta function

Estimation of $\pi_2(x)=\#\{n\leq x:n=p_1p_2\}$, where $p_1$ and $p_2$ are two distinct primes

On the behaviour of $\sum_{\substack{n\leq x\\(n,k)=1}}\frac{1}{\sqrt[\alpha]{n}}$

What is the value of $\sum_{n \geq 1} \epsilon^n/n$, where $\epsilon$ is a $k^\text{th}$ root of $1$

Rewrite $\sum_{p\leq \sqrt x}\frac{x}{p\log(x/p)}$ using Abel's summation formula

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