My discovery about mathematics

                             1

I show a formula giving a root of the algebraic equation. The formula is written by the coefficients of the
equation and some analytic symbols.

                             2

Set up that x_n=e_n (1≤n≤N), x_n=c₁x_n-1+…+c_Nx_n-N (n≥N+1).
Theorem 1. n≥N+1⇒ x_n=∑_1≤p≤Ne_p∑_{k1+…+ks=n-p, 1≤ki≤N, ks≥N+1-p} ∏_1≤i≤s c_ki
Proof. n≥N+1⇒ x_n=∑_1≤k1≤N c_k1x_n-k1=(∑_{1≤n-k1≤N,1≤k1≤N}c_k1x_n-k1)+(∑_{n-k1≥N+1,1≤k1≤N} c_k1x_n-k1)
∑_{1≤n-k1≤N,1≤k1≤N}c_k1e_n-k1=∑_{1≤p≤N, k1=n-p,1≤k1≤N}e_pc_k1, (k_s=k₁=n-p≥N+1-p)
By induction,you may assume that
n-k₁≥N+1⇒ x_n-k1=∑_1≤p≤Ne_p∑_{k2+…+ks=n-k1-p,1≤ki≤N,ks≥N+1-p} ∏_2≤i≤s c_ki.
Substituting these,you get the theorem.♦
Theorem 2. n≥N+1⇒
x_n=∑_1≤p≤N e_p∑_1≤s≤n-p∫_0≤t≤2π(∑_1≤k≤N c_ke^√-1kt)^s-1(∑_{N+1-p≤k≤N} c_ke^√-1kt)e^√-1(p-n)tdt/2π
Proof. ∑_{k1+…+ks=n-p,1≤ki≤N, ks≥N+1-p} ∏_1≤i≤s c_ki is the coefficient of x^n-p in
(∑_1≤k≤Nc_kx^k)^s-1(∑_{N+1-p≤k≤N} c_kx^k). etc..♦
Theorem 3. e₁=1, e_i=0 (2≤i≤N),x^N-c₁x^N-1-…-c_N=(x-v₁)…(x-v_N),
|v₁|-|v_i|≥0,≠0 (2≤i≤N), v_i≠v_k (i≠k),v_i≠0 (1≤i≤N) ⇒
x_n+1/x_n→v₁ (n→∞)
Proof. By theory of the difference equation,you can write x_n=b₁v₁^n-1+…+b_Nv_N^n-1.
(1,0,…0)=(b₁,…,b_N)A A=(v_i^k-1)_{1≤i≤N, 1≤k≤N} is a matrix. detA=∏_i-k≥0,≠0(v_i-v_k)≠0
(1,0,…0)A^-1=(b₁,…,b_N) b₁ is the (1,1) component of A^-1 which is v₁…v_N/∏_2≤i≤N(v_i-v₁)≠0.
i≥2⇒ (v_i/v₁)^m→0 (m→∞) x_n+2/x_n+1=(b₁v₁+b₂(v₂/v₁)ⁿv₂+…)/(b₁+b₂(v₂/v₁)ⁿ+…)→v₁ (n→∞)♦

ζ(2)=π²/6=1+∑_i∈r1/2ⁱ (r⊂N)
r is a countable set. Existence of a countable set leads contradiction.
(Refer to "Defect of the formal algebra 1",3"" in this blog.) The standard
mathematics treating ζ(s) isn't consistent. Riemann hypothesis is nonsense.
Theory of the function √x is nonsense too.