Question

Prove the equivalent form of the Selberg's Formula $\psi(x)\log x+\sum_{p\leq x}\psi\left(\frac xp\right)\log p=2x\log x+\mathcal O(x).$

score 284 · Answer 1 · 2019-01-29 02:46:23

Prove the equivalent form of the Selberg's Formula $\psi(x)\log x+\sum_{p\leq x}\psi\left(\frac xp\right)\log p=2x\log x+\mathcal O(x).$

score 71 · Answer 2 · 2019-01-29 04:04:18

71

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score 47 · Answer 3 · 2019-01-29 09:21:10

47

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score 170 · Answer 4 · 2019-01-29 13:23:21

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Asymptotic behavior of $\sum\limits_{n=0}^{\infty}x^{b^n}$ when $x\to1^-$

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score 2.3k · Answer 5 · 2026-03-27 04:17:47

2.3k

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score 29 · Answer 6 · 2019-01-30 07:32:51

29

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score 191 · Answer 7 · 2019-01-30 07:33:04

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51

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score 140 · Answer 13 · 2019-02-02 16:36:52

140

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score 255 · Answer 14 · 2019-02-02 23:21:20

255

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score 119 · Answer 15 · 2026-03-27 10:45:46

119

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List Question

Prove the equivalent form of the Selberg's Formula $\psi(x)\log x+\sum_{p\leq x}\psi\left(\frac xp\right)\log p=2x\log x+\mathcal O(x).$

Estimate about $\sum_{n\leq x}\mu(n)\log \frac xn$

Ordering w.r.t increase in growth rate

Asymptotic behavior of $\sum\limits_{n=0}^{\infty}x^{b^n}$ when $x\to1^-$

Asymptotics of Hypergeometric $_2F_1(a;b;c;z)$ for large $|z| \to \infty$?

Reciprocal of $x_{n-1}^2(1-\frac{x_{n-1}^2}{3}+o(x_{n-1}^2))$ where $x_n\searrow0$

Meaning of $ e^x = 1 + x + \Theta(x^2)$?

Sharp bounds for the principal branch of the Lambert W function?

Show $\frac{n+1}{2n^2+5} \in \Theta (\frac{1}{n})$ using definition

Combining little-o notation

sum of Time complexities

Alternating shifted central binomial sum with Cauchy weights

Asymptotic expansion of the sequence $u_n=\sum\limits_{k=1}^{n}e^{-b^{-k}}$

Solving the recurrence $T(n) \le 2T(n−1)+n$

Showing $7n^2 + n = \Omega(n)$

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