数论函数
数论函数指定义域为正整数集，值域是整数集的函数
积性函数和完全积性函数
对于一个数论函数(f)，当(gcd(a,b)=1)时，(f(ab)=f(a)f(b))，则称其为积性函数
对于一个数论函数(f)，若 (f(ab)=f(a)f(b))，则称其为完全积性函数
莫比乌斯函数
[mu(n) = egin{cases} 1 qquad &n=1 \ (-1)^k qquad &n=prodlimits_{i=1}^{k} p_i \ 0 qquad & ext{otherwise} end{cases}
]
莫比乌斯函数是积性函数
狄利克雷卷积
[h(n)=sum_{d mid n}f(d)g(frac{n}{d})
]
满足结合律，交换律
存在单位元(f imes epsilon= f)，(epsilon(n)=[n=1]=egin{cases}1&n=1\0&n e1end{cases})
存在逆元，对于每个(f(1) e 0)的函数(f)，都存在函数(g)，使得(f imes g= epsilon)
一些卷积的结论
[varphi imes 1=id
]
[μ imes id=varphi
]
[μ imes 1=epsilon
]
[epsilon imes1 =1
]
莫比乌斯反演
(f=g imes 1iff g=f imes μ)
证明：
[f=f imes epsilon=f imes mu imes 1=g imes 1 iff g=f imes μ
]
数论分块
若要求(sumlimits_{i=1}^{n}lfloorfrac{n}{i} floor)，可以用数论分块实现(O(sqrt n))计算
(code:)
for(int l=1,r;l<=n;l=r+1)
r=n/(n/l),ans+=(r-l+1)*(n/l);
ZAP-Queries
求(sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}[gcd(i,j)=k] (n leqslant m))
[egin{aligned} &sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}[gcd(i,j)=k] \ =&sum_{i=1}^{lfloor frac{n}{k}
floor}sum_{j=1}^{lfloor frac{m}{k}
floor}[gcd(i,j)=1] \ =&sum_{i=1}^{lfloor frac{n}{k}
floor}sum_{j=1}^{lfloor frac{m}{k}
floor}epsilon(gcd(i,j)) \ =&sum_{i=1}^{lfloor frac{n}{k}
floor}sum_{j=1}^{lfloor frac{m}{k}
floor}sum_{d mid i wedge d mid j}mu(d)\ =&sum_{d=1}^{lfloor frac{n}{k}
floor}mu(d) sum_{i=1 wedge d mid i}^{lfloor frac{n}{k}
floor}sum_{j=1 wedge d mid j}^{lfloor frac{m}{k}
floor}1\ =&sum_{d=1}^{lfloor frac{n}{k}
floor}mu(d) (sum_{i=1 wedge d mid i}^{lfloor frac{n}{k}
floor}1)(sum_{j=1 wedge d mid j}^{lfloor frac{m}{k}
floor}1)\=&sum_{d=1}^{lfloor frac{n}{k}
floor}mu(d) lfloor frac{n}{kd}
floor lfloor frac{m}{kd}
floorend{aligned}
]
约数个数和
求(sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}d(ij) (n leqslant m))
[ egin{aligned}
&sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}d(ij) \
=&sumlimits_{i=1}^nsumlimits_{j=1}^msumlimits_{xmid i}sumlimits_{ymid j} [gcd(x,y)=1] \
=&sumlimits_{i=1}^nsumlimits_{j=1}^msumlimits_{xmid i}sumlimits_{ymid j}epsilon(gcd(x,y)) \
=&sumlimits_{i=1}^nsumlimits_{j=1}^msumlimits_{xmid i}sumlimits_{ymid j}sum_{d mid x wedge d mid y}mu(d) \
=&sum_{d=1}^nmu(d)sumlimits_{i=1}^nsumlimits_{j=1}^msumlimits_{xmid i}sumlimits_{ymid j}[d mid x wedge d mid y]\
=&sum_{d=1}^nmu(d)sum_{x=1}^{lfloor frac{n}{d}
floor}sum_{y=1}^{lfloor frac{m}{d}
floor}lfloorfrac{n}{dx}
floorlfloorfrac{m}{dy}
floor\
=&sum_{d=1}^nmu(d)sum_{x=1}^{lfloor frac{n}{d}
floor}lfloorfrac{n}{dx}
floorsum_{y=1}^{lfloor frac{m}{d}
floor}lfloorfrac{m}{dy}
floor\
=&sum_{d=1}^nmu(d)f(lfloorfrac{n}{d}
floor)f(lfloorfrac{m}{d}
floor)\
&f(n)=sum_{i=1}^n lfloorfrac{n}{i}
floor
end{aligned}
]
Crash的数字表格 / JZPTAB
求(sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}lcm(i,j) (n leqslant m))
[ egin{aligned} &sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}lcm(i,j) \ =&sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}frac{ij}{gcd(i,j)} \ =&sumlimits_{d=1}^{n}sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}frac{ij}{d}[gcd(i,j)=d] \ =&sumlimits_{d=1}^{n}sum_{i=1}^{lfloor frac{n}{d}
floor}sum_{j=1}^{lfloor frac{m}{d}
floor}sum_{k mid i wedge k mid j}mu(k)ijd \ =&sumlimits_{d=1}^{n}dsum_{i=1}^{lfloor frac{n}{d}
floor}sum_{j=1}^{lfloor frac{m}{d}
floor}sum_{k=1}^{lfloor frac{n}{d}
floor}mu(k)ij[k mid i wedge k mid j] \ =&sumlimits_{d=1}^{n}dsum_{k=1}^{lfloor frac{n}{d}
floor}mu(k)sum_{i=1}^{lfloor frac{n}{d}
floor}i[k mid i]sum_{j=1}^{lfloor frac{m}{d}
floor}j[k mid j] \ =&sumlimits_{d=1}^{n}dsum_{k=1}^{lfloor frac{n}{d}
floor}mu(k)k^2sum_{i=1}^{lfloor frac{n}{kd}
floor}isum_{j=1}^{lfloor frac{m}{kd}
floor}j \ =&sumlimits_{d=1}^{n}df(lfloor frac{n}{d}
floor,lfloor frac{m}{d}
floor) \ &f(n,m)=sum_{k=1}^{n}mu(k)k^2sum_{i=1}^{lfloor frac{n}{k}
floor}isum_{j=1}^{lfloor frac{m}{k}
floor}j \ end{aligned}
]
数字表格
求(prodlimits_{i=1}^{n}prodlimits_{j=1}^{m}f(gcd(i,j)) (n leqslant m))，其中(f)为斐波那契数列
[ egin{aligned} &prodlimits_{i=1}^{n}prodlimits_{j=1}^{m}f(gcd(i,j)) \ =&prodlimits_{d=1}^{n}f(d)^{sumlimits_{i=1}^{n}sumlimits_{j=1}^{m}[gcd(i,j)=d]} \=&prodlimits_{d=1}^{n}f(d)^{sumlimits_{i=1}^{lfloor frac{n}{d}
floor}sumlimits_{j=1}^{lfloor frac{m}{d}
floor}[gcd(i,j)=1]} \=&prodlimits_{d=1}^{n}f(d)^{sumlimits_{i=1}^{lfloor frac{n}{d}
floor}sumlimits_{j=1}^{lfloor frac{m}{d}
floor}sumlimits_{k mid i wedge k mid j}mu(k)} \=&prodlimits_{d=1}^{n}f(d)^{sumlimits_{k=1}^{lfloor frac{n}{d}
floor}mu(k)sumlimits_{i=1}^{lfloor frac{n}{d}
floor}sumlimits_{j=1}^{lfloor frac{m}{d}
floor}[k mid i wedge k mid j]} \=&prodlimits_{d=1}^{n}f(d)^{sumlimits_{k=1}^{lfloor frac{n}{d}
floor}mu(k)lfloor frac{n}{kd}
floor lfloor frac{m}{kd}
floor} \=&prodlimits_{d=1}^{n}prodlimits_{k=1}^{lfloor frac{n}{d}
floor}f(d)^{mu(k)lfloor frac{n}{kd}
floor lfloor frac{m}{kd}
floor} \=&prodlimits_{T=1}^{n}left(prodlimits_{d mid T}f(d)^{mu(frac{T}{d})}
ight)^{lfloor frac{n}{T}
floor lfloor frac{m}{T}
floor} \end{aligned}
]