User:Virginia-American/Sandbox/Dirichlet convolutions

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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:

      [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

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

i.e.


Now consider

i.e.


Since A) times C) is one,

  1. ^ Hardy & Wright, § 17.2
  2. ^ Titchmarsh, § 1.1
  3. ^ Hardy & Wright, Thm. 287
  4. ^ Titchmarsh, Eq. 1.1.4
  5. ^ Hardy & Wright, Th. 291
  6. ^ Titchmarsh, Eq. 1.3.1
  7. ^ Hardy & Wright, Thm. 289
  8. ^ Titchmarsh, Eq. 1.2.1
  9. ^ Hardy & Wright, Thm. 290
  10. ^ Hardy & Wright, Thn. 288
  11. ^ Titchmarsh, Eq. 1.2.12
  12. ^ Hardy & Wright, Thm. 300
  13. ^ Titchmarsh, Eq. 1.2.11
  14. ^ Hardy & Wright, Th. 302
  15. ^ Titchmarsh, Eq. 1.2.7
  16. ^ Hardy & Wright, Thm. 301
  17. ^ Titchmarsh, Eq. 1.2.8
  18. ^ Titchmarsh, § 1.2.9
  19. ^ Hardy & Wright, Thm. 304
  20. ^ Titchmarsh, Eq. 1.2.10
  21. ^ Hardy & Wright, Thm. 294
  22. ^ Titchmarsh, Eq. 1.1.8
  23. ^ Hardy & Wright, Thm. 294