A191898 Symmetric square array read by antidiagonals: T(n,1)=1, T(1,k)=1, T(n,k) = -Sum_{i=1..k-1} T(n-i,k) for n >= k, -Sum_{i=1..n-1} T(k-i,n) for n < k.

1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, -2, 1, 1, -2, 1, 1, 1, -1, 1, -1, -4, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, -1, -6, -1, 1, -1, 1, -1, 1

Offset: 1

Author

Mats Granvik, Jun 19 2011

Keywords

• sign
• tabl

Comments

Rows equal columns and are periodic. No zero elements are found (conjecture). The recurrence is related to the recurrence for the Mahonian numbers. The main diagonal is the Dirichlet inverse of the Euler totient function A023900 (conjecture). [The 2nd and 3rd formulas state that the conjecture is correct. R. J. Mathar, Sep 16 2017]

The sums from n=1 to infinity of T(n,k)/n converge to the Mangoldt function for column k (conjecture).

If gcd(n,k)=1 then T(n,k)=1 and if T(n,k)=1 then gcd(n,k)=1 (conjecture).

The Dirichlet generating functions for s > 1 for the columns appear to be (see A054535):

Zeta(s)*(1 + (Sum over all possible combinations of products of negative distinct prime factors of k, up to rearrangement, 1/((-1* first distinct prime factor)*(-1*second distinct prime factor)*(-1*third distinct prime factor * ...))^(s-1))).

Examples:

k=1: Zeta(s)

k=2: Zeta(s)*(1 - 1/2^(s-1))

k=3: Zeta(s)*(1 - 1/3^(s-1))

k=4: Zeta(s)*(1 - 1/2^(s-1))

k=5: Zeta(s)*(1 - 1/5^(s-1))

k=6: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) + 1/6^(s-1))

k=7: Zeta(s)*(1 - 1/7^(s-1))

...

k=30: Zeta(s)*(1 - 1/2^(s-1) - 1/3^(s-1) - 1/5^(s-1) + 1/6^(s-1) + 1/10^(s-1) + 1/15^(s-1) - 1/30^(s-1))

...

(conjecture)

See triangle A142971 for negative distinct prime factors.

This could probably be checked by matrix multiplication.

The signs of the eigenvalues of this matrix are a rearrangement of the Mobius function A008683 (conjecture). The first few eigenvalues are:

{1.0000}

{-1.4142, 1.4142}

{-2.6554, 1.8662, -1.2108}

{-3.4393, 2.1004, -1.6611, 0}

{-4.7711, -3.3867, 2.5910, -1.4332, 0}

{-5.2439, -3.4641, 3.4641, 2.5169, -2.2730, 0}

The relation to Dirichlet characters for the entries in this matrix appears appears to be formulated in terms of the sequence A089026 which is equal to n if n is a prime, otherwise equal to 1. See Mathematica program below. [Mats Granvik, Nov 23 2013]

From Mats Granvik, Jun 19 2016: (Start)

Remark about the Dirichlet generating function for the whole matrix: Subtracting the first column (in the form of zeta(c)) of the matrix gives us the limit: lim_{c->1} zeta(s)*zeta(c)/zeta(c+s-1)-zeta(c) = -zeta'(s)/zeta(s) which is the classical Dirichlet generating function for the von Mangoldt function.

For n >= k, see A231425, this matrix has row sums equal to zero except for the first row:

1=1

1-1=0

1+1-2=0

1-1+1-1=0

1+1+1+1-4=0

...

log(A014963(n)) = Sum_{k>=1} A191898(n,k)/k, for n>1.

log(A014963(k)) = Sum_{n>=1} A191898(n,k)/n, for k>1.

log(A014963(n)) = limit of zeta(s)*(Sum_{d divides n} A008683(d)/d^(s-1)) as s->1, for n>1.

A008683(n) = Sum_{k=1..n} A191898(n,k)*exp(-i*2*Pi*k/n)/n.

A008683(n) = Sum_{n=1..k} A191898(n,k)*exp(-i*2*Pi*n/k)/k.

(End)

From Mats Granvik, Aug 09 2016: (Start)

The Dirichlet generating function for the matrix is zero for any pair c and s = 2 - c and any pair s and c = 2 - s except at the pole c = 1 and s = 1 where it is indeterminate.

In the Mathematica program section, in the expression for the matrix as Dirichlets characters, the variables s and c can apparently be any pair of positive integers.

Limits related to the Dirichlet generating function for the matrix: Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s-1) = ZetaZero(n). Let s = ZetaZero(n), then lim_{c->1} zeta(s*c)/zeta(c+s*c-1) = ZetaZero(n)/(1+ZetaZero(n)).

(End)

Examples

			Array starts:
n\k | 1    2    3    4    5    6    7    8    9   10
----+-----------------------------------------------------
1   | 1,   1,   1,   1,   1,   1,   1,   1,   1,   1, ...
2   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
3   | 1,   1,  -2,   1,   1,  -2,   1,   1,  -2,   1, ...
4   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
5   | 1,   1,   1,   1,  -4,   1,   1,   1,   1,  -4, ...
6   | 1,  -1,  -2,  -1,   1,   2,   1,  -1,  -2,  -1, ...
7   | 1,   1,   1,   1,   1,   1,  -6,   1,   1,   1, ...
8   | 1,  -1,   1,  -1,   1,  -1,   1,  -1,   1,  -1, ...
9   | 1,   1,  -2,   1,   1,  -2,   1,   1,  -2,   1, ...
10  | 1,  -1,   1,  -1,  -4,  -1,   1,  -1,   1,   4, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 antidiagonals of the array
• Mats Granvik, Is this similarity to the Fourier transform of the von Mangoldt function real?
• Mats Granvik, Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
• Mats Granvik, Primes approximated by eigenvalues?
• Mats Granvik, Are the primes found as a subset in this sequence a(n)?
• Mats Granvik, Will every eigenvalue in this type of matrix eventually be a common eigenvalue to infinitely many subsequent larger matrices of the same form?
• Mats Granvik, How write Dirichlet character sums for the terms of the von Mangoldt function?
• Mats Granvik, Do these series converge to the von Mangoldt function?
• Mats Granvik, Is this sum equal to the Möbius function?
• Mats Granvik, Can the Riemann hypothesis be relaxed to say that this matrix A consists of square roots?
• Mats Granvik, Elementary proof of the prime number theorem?
• Mats Granvik, Is this Dirichlet series generating function of the von Mangoldt function matrix correct?
• Mats Granvik, Question about ratios of polynomials evaluated at x=1

Crossrefs

Cf. A023900, A014963, A008302, A033999, A061347, A142971.

Cf. A007917, A199514, A260237. - Mats Granvik, Jun 19 2016

Cf. A343555, A343556.

Programs

Mathematica

T[ n_, k_] := T[ n, k] = Which[ n < 1 || k < 1, 0, n == 1 || k == 1, 1, k > n, T[k, n], n > k,T[k, Mod[n, k, 1]], True, -Sum[ T[n, i], {i, n - 1}]]; (* Michael Somos, Jul 18 2011 *)
(* Conjectured expression for the matrix as Dirichlet characters *) s = RandomInteger[{1, 3}]; c = RandomInteger[{1, 3}]; nn = 12; b = Table[Exp[MangoldtLambda[Divisors[n]]]^-MoebiusMu[Divisors[n]], {n, 1, nn^Max[s, c]}]; j = 1; MatrixForm[Table[Table[Product[(b[[n^s]][[m]]*DirichletCharacter[b[[n^s]][[m]], j, k^c] - (b[[n^s]][[m]] - 1)), {m, 1, Length[Divisors[n]]}], {n, 1, nn}], {k, 1, nn}]] (* Mats Granvik, Nov 23 2013 and Aug 09 2016 *)

PARI

{T(n, k) = if( n<1 || k<1, 0, n==1 || k==1, 1, k>n, T(k, n), kMichael Somos, Jul 18 2011 */

Python

from sympy.core.cache import cacheit
@cacheit
def T(n, k): return 0 if n<1 or k<1 else 1 if n==1 or k==1 else T(k, n) if k>n else T(k, (n - 1)%k + 1) if n>k else -sum([T(n, i) for i in range(1, n)])
for n in range(1, 21): print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Oct 23 2017

Formula

T(n,1)=1, T(1,k)=1, n>=k: -Sum_{i=1..k-1} T(n-i,k), n

T(n, n) = A023900(n). - Michael Somos, Jul 18 2011

T(n, k) = A023900(gcd(n,k)). - Mats Granvik, Jun 18 2012

Dirichlet generating function for sequence in the n-th row: zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1). - Mats Granvik, Jun 18 2012 & Jun 19 2016

From Mats Granvik, Jun 19 2016: (Start)

Dirichlet generating function for the whole matrix: Sum_{k>=1} (Sum_{n>=1} T(n,k)/(n^c*k^s)) = Sum_{n>=1} (zeta(s)*Sum_{ d divides n } mu(d)/d^(s-1))/n^c = zeta(s)*zeta(c)/zeta( c + s - 1 ).

T(n,k) = A127093(n,k)^(1/2-i*a(k))*transpose(A008683(k)*(A127093(n,k)^(1/2+i*a(n)))) where a(x) is some real number. An example would be T(n,k) = A127093(n,k)^(zetazero(k))*transpose(A008683(k)*(A127093(n,k)^(zetazero(-k)))) but this is of course not special for only the zeta zeros.

Recurrence for a subset of A191898 that is a cross-directional variant of the recurrence in A051731: T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..k-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..n-1} T(k-i,n) - T(k-i,n-1). Notice that the identity matrix in linear algebra satisfies a similar recurrence:

T(1,1)=1, T(1,2..k)=0, T(2..n,1)=0, n >= k: -Sum_{i=1..n-1} T(n-i,k) - T(n-i,k-1), n < k: -Sum_{i=1..k-1} T(k-i,n) - T(k-i,n-1).

(End)

This array equals A051731*transpose(A143256). - Mats Granvik, Jul 22 2016

T(n,k) = sqrt(A143256(n,k))*transpose(sqrt(A143256(n,k))). - Mats Granvik, Aug 10 2018

Dirichlet generating function for absolute values: Sum_{k>=1} (Sum_{n>=1} abs(T(n,k))/(n^c*k^s)) = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(2*(s + c - 1))*Product_{k>=1} (1 - 2/(prime(k) + prime(k)^(s + c))). After Vaclav Kotesovec in A173557. - Mats Granvik, Apr 25 2021

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