A Math Contest P17 - Heatwaves - DMOJ: Modern Online Judge

A Math Contest P17 - Heatwaves

View as PDF

Submit solution

All submissions

Best submissions

Points: 45

Time limit: 7.0s

Memory limit: 512M

Author:

TechnobladeNeverDies

Problem type

You are given an integer $N > 2$ $N > 2$ and a function $\chi : \mathbb Z \to \mathbb C$ $χ : Z \to C$ such that

$\chi(n) = \chi(n+N)$ $χ (n) = χ (n + N)$ ;
$\chi(n) = 0 \iff \gcd(n,N) \ne 1$ $χ (n) = 0 ⟺ gcd (n, N) \neq 1$ ;
$\chi(n)\chi(m) = \chi(nm)$ $χ (n) χ (m) = χ (n m)$ ;
the nonzero values of $\chi$ $χ$ has period $\varphi(N)$ $φ (N)$ .

Consider the unique function $L_\chi(s) : \mathbb C \to \mathbb C$ $L_{χ} (s) : C \to C$ such that

when $\operatorname{Re}(s)>1$ $Re (s) > 1$ , then $L_\chi(s) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s}$ $L_{χ} (s) = \sum_{n = 1}^{\infty} \frac{χ (n)}{n^{s}}$ ;
$L_\chi(s)$ $L_{χ} (s)$ is differentiable everywhere.

Find

$L_\chi(s)$ $L_{χ} (s)$ for a given complex number $s$ $s$ ;
a complex number $s$ $s$ with a nonzero imaginary part such that $L_\chi(s) = 0$ $L_{χ} (s) = 0$ .

You are further informed that

The conditions of the function $\chi$ $χ$ are equivalent to the conditions of $\chi$ $χ$ being a primitive Dirichlet character.
$L_\chi(s)$ $L_{χ} (s)$ is the analytic continuation of the Dirichlet L-function onto the complex plane.
Zeta functions can be analytically continued to any portion of the complex plane given by $\operatorname{Re}(s) > \sigma$ $Re (s) > σ$ for real $\sigma$ $σ$ using the Euler-Maclaurin summation formula.
If we let $a = [\chi(-1)=-1]$ $a = [χ (- 1) = - 1]$ , and $\varepsilon_\chi = \frac{1}{i^a\sqrt{N}} \sum_{n=1}^N \chi(n) e^{2\pi in/N}$ $ε_{χ} = \frac{1}{i^{a} \sqrt{N}} \sum_{n = 1}^{N} χ (n) e^{2 π i n / N}$ , and $\xi_\chi(s) = \left(\frac{N}{\pi}\right)^{(s+a)/2} \Gamma\left(\frac{s+a}{2}\right) L_\chi(s)$ $ξ_{χ} (s) = {(\frac{N}{π})}^{(s + a) / 2} Γ (\frac{s + a}{2}) L_{χ} (s)$ , then $\xi_\chi(s)$ $ξ_{χ} (s)$ satisfies $\xi_\chi(s) = \varepsilon_\chi \xi_{\bar\chi}(1-s)$ $ξ_{χ} (s) = ε_{χ} ξ_{\bar{χ}} (1 - s)$ .

Constraints

$2 < N \le 1\,000$ $2 < N \leq 1 000$

$0 \le \operatorname{Re}(s) \le 1$ $0 \leq Re (s) \leq 1$ and $|\operatorname{Im}(s)| \le 5$ $| Im (s) | \leq 5$

$x \in \{0, 1\}$ $x \in {0, 1}$ and $0 \le y \le 2z \le 10^9$ $0 \leq y \leq 2 z \leq 10^{9}$ and $1 \le z$ $1 \leq z$

For the first question, the jury answer is precise to $10^{-50}$ $10^{- 50}$ , and the magnitude of the difference between your answer and the jury answer should be less than $10^{-12}$ $10^{- 12}$ .

For the second question, the checker is precise to $10^{-15}$ $10^{- 15}$ , and the magnitude of $L_\chi(s)$ $L_{χ} (s)$ of your answer should be less than $10^{-9}$ $10^{- 9}$ .

Input Specification

The first line contains an integer, $N$ $N$ .

The next $N$ $N$ lines contain three nonnegative integers $x$ $x$ , $y$ $y$ , and $z$ $z$ each, representing $\chi(0), \chi(1), \dots, \chi(N-1)$ $χ (0), χ (1), \dots, χ (N - 1)$ respectively. The three integers $x, y, z$ $x, y, z$ correspond to $xe^{yi\pi/z}$ $x e^{y i π / z}$ .

The next line contains two real numbers, with at most $5$ $5$ digits after the decimal point, representing the real and imaginary parts of $s$ $s$ .

Output Specification

On the first line, output the real and imaginary parts for the first answer.

On the second line, output the real and imaginary parts for the second answer.

Sample Input

Copy

5
0 0 1
1 0 2
1 1 2
1 3 2
1 2 2
0.69 4.2069

Sample Output

Copy

1.5186663729999338699 -0.8641952653252008201
0.5000000000000000000 6.1835781954508539144

Explanation for Sample

Comments

There are no comments at the moment.