A298674 Square matrix read by antidiagonals up. Matrix of Dirichlet series associated with Sum_{n<=X} MangoldtLambda(n) * MangoldtLambda(n+2).

1, 1, 2, 1, -2, -1, 1, 2, 2, 3, 1, -2, -1, -1, 2, 1, 2, -1, 3, 2, -2, 1, -2, 2, -1, -3, -4, 2, 1, 2, -1, 3, 2, -2, 2, 4, 1, -2, -1, -1, -3, 2, 2, 0, -3, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, -2, -1, -1, 2, 2, -5, 0, -3, -4, 2, 1, 2, -1, 3, -3, -2, 2, 4, -3, -6, 2, -3

Offset

1,3

Comments

For n > 1: Sum_{k>=1} T(n,k) = log(A014963(n))*log(A014963(n+2)).

Triangular submatrix of this matrix is A298824.

Row sums of A298824 are found in A298825. A298825(n)/n = A298826(n). A298826 appears to be relevant to the heuristic for the twin prime conjecture.

By varying the prime gap "h" in the program it appears that prime gaps that are powers of "h" have the same row sums of the triangular submatrix, which in turn seems to imply that prime gaps equal to powers of "h" have the same density.

Links

Table of n, a(n) for n=1..78.

Terence Tao, Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges. See second formula.

Formula

Let h = 2.

Let A(n,k) = 1 if n mod k = 0, otherwise 0.

Let B(n,k) = A008683(n)*n if k mod n = 0, otherwise 0.

Let T = A.B (where "." is matrix multiplication).

Take the Dirichlet convolution of a row in T(n,k) and a row in T(n+h,k) for n=1,2,3,4,5,... infinity, and form this matrix from the first columns of the convolutions. See Mathematica program for more precise description.

Example

The square matrix starts:
{
  {1,  2, -1,  3,  2, -2,  2,  4, -3,  4,  2, -3},
  {1, -2,  2, -1,  2, -4,  2,  0,  3, -4,  2, -2},
  {1,  2, -1,  3, -3, -2,  2,  4, -3, -6,  2, -3},
  {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},
  {1,  2,  2,  3, -3,  4, -5,  4,  3, -6,  2,  6},
  {1, -2, -1, -1,  2,  2,  2,  0, -3, -4,  2,  1},
  {1,  2, -1,  3,  2, -2, -5,  4, -3,  4,  2, -3},
  {1, -2,  2, -1, -3, -4,  2,  0,  3,  6,  2, -2},
  {1,  2, -1,  3,  2, -2,  2,  4, -3,  4, -9, -3},
  {1, -2, -1, -1, -3,  2,  2,  0, -3,  6,  2,  1}
}

Mathematica

h = 2; nn = 14;
A = Table[If[Mod[n, k] == 0, 1, 0], {n, nn}, {k, nn}];
B = Table[If[Mod[k, n] == 0, MoebiusMu[n]*n, 0], {n, nn}, {k, nn}];
T = (A.B);
TableForm[TwinMangoldt = Table[a = T[[All, kk]];
    F1 = Table[If[Mod[n, k] == 0, a[[n/k]], 0], {n, nn}, {k, nn}];
    b = T[[All, kk + h]];
    F2 = Table[If[Mod[n, k] == 0, b[[n/k]], 0], {n, nn}, {k, nn}];
    (F1.F2)[[All, 1]], {kk, nn - h}]];
Flatten[Table[TwinMangoldt[[n - k + 1, k]], {n, nn - h}, {k, n}]]

Crossrefs

Cf. A191898, A298824, A001694, A298825, A298826.

Sequence in context: A205107 A228667 A336005 * A211354 A211352 A087187

Adjacent sequences: A298671 A298672 A298673 * A298675 A298676 A298677

Keyword

sign,tabl

Author

Mats Granvik, Jan 24 2018

Status

approved