一些結果的收集（不定期更新）

05-18

一些結果的收集（不定期更新）

來自專欄我的數學綱領

這裡會收集一些在看東西當中遇到的感覺有趣的結論

1 (serre) Let $f =sum limits_{n=0}^{infty} a_n q^n$ be a holomorphic modular form of integer weight $k geq 0$ and some level $Gamma_1(N)$ such that the cofficients $a_n$ are integers. p is a prime number, then we have $| { n < x , a_n eq 0 : ( mod : p ) } | ll dfrac{ x} {(logx)^{eta}}$ .

2. Wiener Ikehara theorem :

$F(s) =sum dfrac{a_n}{n^s}$ be a Dirichlet series with complex coefficients. Suppose there exists a Dirichlet series $F^+(s) = sum dfrac{a_n^+}{n^s}$ with positive real coefficients such that

(a) $|a_n| leq a_n^+$

(b) The series $F^+$ converges for $Re(s) >1$

(c)The function $F^+$ ( respective $F$ ) can be extended to a meromorphic function on $Re(s)>1$ having no pole except for a simple pole at $s=1$ with residue $C_+> 0$ ( respective $C>0$ ) .

Then $sum limits_ { m leq n } a_n =Cn +O(n)$