Question

Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

score 253 · Answer 1 · 2014-09-13 12:44:03

253

Views

Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

Published on 13 Sep 2014 - 12:44

score 54 · Answer 2 · 2014-09-14 22:13:01

If an infinite product$\ P= 0$ times a constant and an unbounded function gives$\ 1$, will the same for$\ Q< \infty$ certainly be$\ >1$?

score 241 · Answer 3 · 2014-09-14 23:37:00

241

Views

Good lower bound on an infinite product

Published on 14 Sep 2014 - 23:37

score 70 · Answer 4 · 2014-09-16 05:52:35

70

Views

As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$?

Published on 16 Sep 2014 - 5:52

score 1.9k · Answer 5 · 2026-03-26 01:28:09

1.9k

Views

How can we apply the Borel-Cantelli lemma here?

Published on 26 Mar 2026 - 1:28

score 262 · Answer 6 · 2014-09-23 22:14:56

262

Views

Quotient of infinite products

Published on 23 Sep 2014 - 22:14

score 175 · Answer 7 · 2014-09-24 18:35:52

175

Views

$\prod\left(1-p_n\right)>0$

Published on 24 Sep 2014 - 18:35

score 94 · Answer 8 · 2014-09-28 13:10:43

94

Views

Calculate $\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$

Published on 28 Sep 2014 - 13:10

score 2k · Answer 9 · 2014-09-29 05:36:43

2k

Views

Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Published on 29 Sep 2014 - 5:36

score 206 · Answer 10 · 2026-03-25 11:16:18

206

Views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$

Published on 25 Mar 2026 - 11:16

score 13.3k · Answer 11 · 2026-03-25 21:22:59

13.3k

Views

Find the value of $\sqrt{10\sqrt{10\sqrt{10...}}}$

Published on 25 Mar 2026 - 21:22

score 110 · Answer 12 · 2014-10-04 21:22:17

110

Views

Infinite product: Problem I

Published on 04 Oct 2014 - 21:22

score 313 · Answer 13 · 2014-10-12 02:14:27

313

Views

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

Published on 12 Oct 2014 - 2:14

score 2k · Answer 14 · 2014-10-16 10:30:08

2k

Views

Convergence of infinite product of prime reciprocals?

Published on 16 Oct 2014 - 10:30

score 421 · Answer 15 · 2014-10-17 01:52:10

421

Views

Special values of the Dedekind eta function

Published on 17 Oct 2014 - 1:52

List Question

Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

If an infinite product$\ P= 0$ times a constant and an unbounded function gives$\ 1$, will the same for$\ Q< \infty$ certainly be$\ >1$?

Good lower bound on an infinite product

As$\ n \to \infty$, how does a product over the primes less than$\ p_n$ equal the same product over the primes less than$\ n$?

How can we apply the Borel-Cantelli lemma here?

Quotient of infinite products

$\prod\left(1-p_n\right)>0$

Calculate $\int_{-\pi}^{\pi}\prod_{n=1}^\infty \left(1-\frac{t^2}{n^2}\right) e^{-izt}dt$

Show that $\prod (1- P(A_n))=0$ iff $\sum P(A_n) = \infty$

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$

Find the value of $\sqrt{10\sqrt{10\sqrt{10...}}}$

Infinite product: Problem I

Evaluating a trigonometric product $\prod_{n=1}^{\infty}\cos^2\left(\frac{1}{n^2}\right)$

Convergence of infinite product of prime reciprocals?

Special values of the Dedekind eta function

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