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A298674
Square matrix read by antidiagonals up. Matrix of Dirichlet series associated with Sum_{n<=X} MangoldtLambda(n) * MangoldtLambda(n+2).
3
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.
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
KEYWORD
sign,tabl
AUTHOR
Mats Granvik, Jan 24 2018
STATUS
approved