Generating functions[edit]
The following formulas are known:
[1][2]
[3][4]
[5][6]
[7][8]
[9]
[10][11]
[12][13]
[14][15]
[16][17]
[18]
[19][20]
Let χ(n) be the characteristic function of the squares:
![{\displaystyle \chi (n)={\begin{cases}&1{\mbox{ if }}n{\mbox{ is a square }}\\&0{\mbox{ if }}n{\mbox{ is not square.}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfe21f942abbf22903fef5c32735ac84604e20e6)
[21][22]
[23]
Convolutions[edit]
A) × C) = F) × G) = 1.
A) × E) = B1.
E) × D0) = D1.
A) × G) = H.
A) × H) = I.
A) × I) = J.
D0) × H) = J.
A) × F) = K.
A) × L) = M.
Derivations[edit]
We have the series and Euler product formulas for ζ(s):
and thus
Let
From the Euler product
![{\displaystyle {\begin{aligned}{\frac {1}{\zeta (s)}}&=\prod _{p}\left(1-{\frac {1}{p^{s}}}\right)\\&=\left(1-{\frac {1}{2^{s}}}\right)\left(1-{\frac {1}{3^{s}}}\right)\left(1-{\frac {1}{5^{s}}}\right)\dots \\&=1-{\frac {1}{2^{s}}}-{\frac {1}{3^{s}}}-{\frac {1}{5^{s}}}+{\frac {1}{6^{s}}}-{\frac {1}{7^{s}}}+{\frac {1}{10^{s}}}\dots \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8813aab9b32e55fbf67eb452cee02a5351b0c2c)
i.e. a(n) = μ(n), or
Let A) times B) be
Then
giving
Setting k to 0 and 1 gives the special cases
Let B1) times C) be
Then
Consider
![{\displaystyle {\begin{aligned}{\frac {\zeta (s)}{\zeta (2s)}}&=\prod _{p}{\frac {1-p^{2s}}{1-p^{s}}}\\&=\prod _{p}\left(1+p^{s}\right)\\&=\left(1+{\frac {1}{2^{s}}}\right)\left(1+{\frac {1}{3^{s}}}\right)\left(1+{\frac {1}{5^{s}}}\right)\dots \\&=\sum _{n=1}^{\infty }{\frac {|\mu (n)|}{n^{s}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17a6264ae8b05fe510210af38eb61ecd788c87ce)
i.e.
Now consider
![{\displaystyle {\begin{aligned}{\frac {\zeta (2s)}{\zeta (s)}}&=\prod _{p}{\frac {1-p^{s}}{1-p^{2s}}}\\&=\prod _{p}\left(1+p^{s}\right)^{-1}\\&=\left(1-{\frac {1}{2^{s}}}+{\frac {1}{4^{s}}}-{\frac {1}{8^{s}}}+\dots \right)\left(1-{\frac {1}{3^{s}}}+{\frac {1}{9^{s}}}-{\frac {1}{27^{s}}}+\dots \right)\left(1-{\frac {1}{5^{s}}}+\dots \right)\dots \\&=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b327aa6f14d0b5010174ef9b5368d6d2741b71ae)
i.e.
![{\displaystyle {\frac {\zeta ^{2}(s)}{\zeta (2s)}}=\sum _{n=1}^{\infty }{\frac {2^{\omega (n)}}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ee041360ff7ba99c243115a8b45a2afad45cc7)
![{\displaystyle {\frac {\zeta ^{3}(s)}{\zeta (2s)}}=\sum _{n=1}^{\infty }{\frac {d(n^{2})}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0edae629d080670dd429fbad7cfada42eff8ea)
![{\displaystyle {\frac {\zeta ^{4}(s)}{\zeta (2s)}}=\sum _{n=1}^{\infty }{\frac {d^{2}(n)}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e93a5477df45b37cb7e6c37893e454f67e81c4e)
![{\displaystyle {\frac {\zeta (2s)}{\zeta (s)}}=\sum _{n=1}^{\infty }{\frac {\lambda (n)}{n^{s}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7455bd682be1dedd5a43cb220eb07f26f4b83198)
Since A) times C) is one,
- ^ Hardy & Wright, § 17.2
- ^ Titchmarsh, § 1.1
- ^ Hardy & Wright, Thm. 287
- ^ Titchmarsh, Eq. 1.1.4
- ^ Hardy & Wright, Th. 291
- ^ Titchmarsh, Eq. 1.3.1
- ^ Hardy & Wright, Thm. 289
- ^ Titchmarsh, Eq. 1.2.1
- ^ Hardy & Wright, Thm. 290
- ^ Hardy & Wright, Thn. 288
- ^ Titchmarsh, Eq. 1.2.12
- ^ Hardy & Wright, Thm. 300
- ^ Titchmarsh, Eq. 1.2.11
- ^ Hardy & Wright, Th. 302
- ^ Titchmarsh, Eq. 1.2.7
- ^ Hardy & Wright, Thm. 301
- ^ Titchmarsh, Eq. 1.2.8
- ^ Titchmarsh, § 1.2.9
- ^ Hardy & Wright, Thm. 304
- ^ Titchmarsh, Eq. 1.2.10
- ^ Hardy & Wright, Thm. 294
- ^ Titchmarsh, Eq. 1.1.8
- ^ Hardy & Wright, Thm. 294