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A054535
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Square array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n), read by antidiagonals upwards (n >= 1, k >= 1).
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17
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1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -2, 2, 1, 1, 1, -1, 0, -1, -1, 1, -1, -1, -1, 2, -1, 1, 1, 0, -1, -2, -1, 0, 2, -1, 1, 0, 0, -1, -1, 4, -2, -1, 1, 1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, -1, -3, -4, -1, 2, -1, 2, 2, 1, 1, 0, -1, 1, 0, 0, -1, 1, -1, 0, -1, -1, 1, -1, 2, -1, -1, 0, 0, 6, -1, -1, -2, -1, 1, 1, 1, -1
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OFFSET
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1,12
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COMMENTS
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Replace the first column in A077049 with any k-th column in A177121 to get a new array. Then the matrix inverse of the new array will have the k-th column of A054535 (this array) as its first column. - Mats Granvik, May 03 2010
We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003.
E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
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LINKS
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FORMULA
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T(n,k) = c_n(k) = phi(n) * Moebius(n/gcd(n, k))/phi(n/gcd(n, k)). - Emeric Deutsch, Dec 23 2004 [The r.h.s. of this formula is known as the von Sterneck function, and it was introduced by him around 1900. - Petros Hadjicostas, Jul 20 2019]
Dirichlet series: Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) where sigma is the sum-of-divisors function. Sum_{n>=1} c_k(n)/n^s = zeta(s)*Sum_{d|k} mu(k/d)*d^(1-s). [Hardy & Wright, Titchmarsh] - R. J. Mathar, Apr 01 2012 [We have sigma_{1-s}(k) = Sum_{d|k} d^{1-s} = Sum_{d|k} (k/d)^{1-s} = sigma_{s-1}(k) / k^{s-1}. - Petros Hadjicostas, Jul 27 2019]
For n >= 1 and k >= 1 let
A(n,k) := if n mod k = 0 then k^r, otherwise 0;
B(n,k) := if n mod k = 0 then k/n^s, otherwise 0.
Then the Ramanujan's sum matrix equals
inverse(A).transpose(B) evaluated at s=0 and r=0.
Dirichlet g.f.: Sum_{n>=1} Sum_{k>=1} T(n,k)/(n^r*k^s) = zeta(s)*zeta(s + r - 1)/zeta(r) as in Wikipedia. (End)
Lambert series and a consequence: Sum_{n >= 1} c_n(k) * z^n / (1 - z^n) = Sum_{s|k} s * z^s and -Sum_{n >= 1} (c_n(k) / n) * log(1 - z^n) = Sum_{s|k} z^s for |z| < 1 (using the principal value of the logarithm). - Petros Hadjicostas, Aug 15 2019
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EXAMPLE
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Square array T(n,k) = c_n(k) (with rows n >= 1 and columns k >= 1) starts as follows:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 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, 2, -1, ...
0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, ...
-1, -1, -1, -1, 4, -1, -1, -1, -1, 4, -1, -1, -1, ...
1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, ...
-1, -1, -1, -1, -1, -1, 6, -1, -1, -1, -1, -1, -1, ...
0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, -4, 0, ...
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MAPLE
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with(numtheory): c:=(n, k)->phi(n)*mobius(n/gcd(n, k))/phi(n/gcd(n, k)): for n from 1 to 13 do seq(c(n+1-j, j), j=1..n) od; # gives the sequence in triangular form # Emeric Deutsch
# to get the example above
for n to 8 do
seq(c(n, k), k = 1 .. 13);
end do
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MATHEMATICA
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nmax = 14; t[n_, k_] := EulerPhi[n]*(MoebiusMu[n / GCD[n, k]] / EulerPhi[n / GCD[n, k]]); Flatten[ Table[t[n - k + 1, k], {n, 1, nmax}, {k, 1, n}]] (* Jean-François Alcover, Nov 10 2011, after Emeric Deutsch *)
(* To get the example above in table format *)
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 13}]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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