# Tagged Questions

Questions involving asymptotic analysis, including growth of functions, Big-O notation, Big-Omega and Big-Theta notations.

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### Asymptotics of the differences between successive zeta zeros

Does anyone know what the asymptotic of the differences between successive zeta zeros is? Update It appears that $\zeta(n)$ is not a bad asymptotic, when the data range is stretched: ...
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### Average of numbers converges, then what happens to the maximum

Let $\{a_n\}$ be a positive sequence of numbers such that $\displaystyle \frac1n\sum_{i=1}^n a_i \to a$ where $a>0$. Then can we say anything about the order of $\displaystyle b_n=\max_{i\in n} a_i$...
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### Extracting the divergent part of an integral

I want to evaluate the integral $$\int_0^1 \frac{2x(x-2)(1-x)}{(1-x)^2 + ax} \, \mathrm{d}x$$ in the limit of small $a$. For $a = 0$ this integral is divergent due to the $1/(1-x)$ pole. The exact ...
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### 10-Subset Sum: Given a set of integers K and an integer M, is there a subset of exactly 10 elements of K whose sum equals M?

I understand that the more general Subset Sum problem is NP-complete, but I am under the assumption that this more specific version of the problem can be solved in polynomial time. However, I can't ...
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### Approximation of a quotient that involves the Lambert function.

I would like to find an asymptotic upper bound for $$\frac{-\ln n}{W(- \ln^{-c}n)}$$ where $c$ is positive and $W$ is the Lambert function. More precisely, I want something which dominates this ...
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### More elegant derivation of the shift in median bin occupancy

In answering Median of a multinomial variable, I found to my own surprise through a somewhat tedious calculation that the expected value of the median of the ball counts in $3$ bins into which $n$ ...
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### Function $(2.2)^n$ — what is it?

The running time of an algorithms is $(2.2)^n$. I have to tell what is the maximum $n$ for reaching 1.000.000 steps. What type of a function is $(2.2)^n$? How its output depends on the input $n$? ...
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### Asymptotic behavior of integrals of Legendre polynomials

By definition $\int_{-1}^1 |P_n(x)|^2 dx = O(n^{-1})$. What about the other powers? Do we know how $\int_{-1}^1 |P_n(x)|^k dx$ behaves for any $k$? Maybe $O(n^{-k/2})$?
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### Determine numerical infinity for Schrodinger equation $−\psi''(z) − (iz)^ N \psi(z) = E\psi(z)$

Consider the following one dimensional Schrodinger equation within the complex plane of $z$ $$−ψ''(z) − (iz)^ N ψ(z) = Eψ(z).$$ where $N$ can be any real number, the boundary condition is $ψ(z) → 0$ ...
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### Which one is asymptotically larger?

The question is to find out which among $n^\sqrt{n}$ or $n^(log_2 n)$ is asymptotically larger? Now as a solution I read somewhere that if we take log on both sides and then compute which one is ...
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### Prove that the value of the constant $C$ must be $1$

After proving the prime number theorem in class, our professor directs us to a remark by Lagrange that for large values of $x$, $\pi(x)$ is approximately equal to $$\frac{x}{\log x - B}.$$ (This is ...
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### Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...