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If you take logs of both sides, and expand the left hand side in a power series around infinity (to first order), you get: $$\frac{\frac{j}{2}-\frac{j^2}{2}}{n}+j \left(-2 \log (j)-\log \left(\fra …

By the prime number theorem, the number of primes between $N/2$ and $N$ is on the order of $N/\log N.$ So, the sum of just those primes squared is of order $N^3/ \log N.$

Mathematica tells us that the principal value can be evaluated in closed form thus: $$e^{-A} 2^{-A N+B-1} \left((-1)^{A N-B+1} \Gamma (-B+A N+1) \Gamma (B-A N,-A)-i \pi \right).$$ (OK, the $i \pi …

I don't really understand the significance of your example, but the univariate version of the question is covered very thoroughly in Flajolet-Sedgewick's Analytic Combinatorics, while the (much harder …

See Kemkes, Graeme; Sato, Cristiane M.; Wormald, Nicholas, Asymptotic enumeration of sparse $2$-connected graphs, Random Struct. Algorithms 43, No. 3, 354–376 (2013). Zbl 1273.05127. PDF and referen …

I am not sure I understand the question. Any matrix $A$ (integer or not, positive or not) has a Jordan canonical form $A = MJM^{-1},$ whereupon $A^n = M J^n M^{-1}.$ If $A$ is integer and nonsingular, …

This is a cosine transform of $t^{-a} \mathbb{1}[0, 1],$ which can be evaluated explicitly using the trusty mathematica, which gives: $ \frac{t^{a+1} \, _1F_2\left(\frac{a}{2}+\frac{1}{2};\frac{1}{ …

If you look at: Microlocal WKB expansions by A. Martinez (available through cite seer), you will see that this is a fairly popular subject, where many results have been obtained by Sjostrand et al.

(the authors were unaware of Newman's work), with full asymptotics, and in a companion paper, a "softer" result was derived by Eskin and McMullen by ergodic-theoretic methods in the very well-known paper …

I am not sure about the asymptotic expansion, but approximating $1/\Gamma(x-1)$ by $x$ near $0$ (and $1/(1-x)$ near $1$) does show that the decrease is faster than $1/\log n$ - the integral of $x n^{x …

That the probability is at least $O(1/n^2)$ is immediate, since the probability that both permutations are $n$-cycles is $1/n^2.$ For the upper bound, there is a convergence speed estimate by Zacharov …

.$ I leave the final computation of the asymptotics to the interested reader. EDIT Actually, this is not quite Watson's lemma. …

Following Fedor Petrov's comment: the arguments of Gaussian primes are known to be uniformly distributed, see Example 7.20 in Number theory: an introduction to class field theory by Kato et al (2000). …

{y}\right)}{y} $$ Since the middle binomial coefficient is approximately $4^n/\sqrt{n},$ that cancels the $(xy)^n$ term (under your assumption) leaving the $1/\sqrt{n}$, so you just need to check the asymptotics …

Not an answer. First, Mathematica says that $$a_n = \frac{-4 n \, _2F_1\left(\frac{1}{2},-n-1;-4 n;2\right)+n \, _2F_1\left(\frac{3}{2},-n;1-4 n;2\right)+\, _2F_1\left(\frac{3}{2},-n;1-4 n;2\ri …