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%I #130 Aug 22 2019 09:20:03
%S 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,
%T 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,
%U -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
%N 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).
%C 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
%C We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and
%C A054534(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k). That is, the current array is the transpose of array A054534. Dirichlet g.f.'s for these two arrays are given below by _R. J. Mathar_ and _Mats Granvik_. - _Petros Hadjicostas_, Jul 27 2019
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003.
%D E. C. Titchmarsh and D. R. Heath-Brown, The theory of the Riemann zeta-function, 2nd ed., 1986.
%H Robert Israel, <a href="/A054535/b054535.txt">Table of n, a(n) for n = 1..10011</a> (T(n,k) for n+k <= 142).
%H Tom M. Apostol, <a href="http://dx.doi.org/10.2140/pjm.1972.41.281">Arithmetical properties of generalized Ramanujan sums</a>, Pacific J. Math. 41 (1972), 281-293.
%H Eckford Cohen, <a href="https://dx.doi.org/10.1073/pnas.41.11.939">A class of arithmetic functions</a>, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.
%H A. Elashvili, M. Jibladze, and D. Pataraia, <a href="http://dx.doi.org/10.1023/A:1018727630642">Combinatorics of necklaces and "Hermite reciprocity"</a>, J. Algebraic Combin. 10 (1999), 173-188.
%H M. L. Fredman, <a href="https://doi.org/10.1016/0097-3165(75)90008-4">A symmetry relationship for a class of partitions</a>, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
%H Emiliano Gagliardo, <a href="http://www.bdim.eu/item?id=BUMI_1953_3_8_3_269_0">Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità</a>, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.
%H Otto Hölder, <a href="http://matwbn.icm.edu.pl/ksiazki/pmf/pmf43/pmf4312.pdf">Zur Theorie der Kreisteilungsgleichung K_m(x)=0</a>, Prace mat.-fiz. 43 (1936), 13-23.
%H Peter H. van der Kamp, <a href="http://emis.impa.br/EMIS/journals/INTEGERS/papers/n24/n24.Abstract.html">On the Fourier transform of the greatest common divisor</a>, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]
%H J. C. Kluyver, <a href="https://www.dwc.knaw.nl/DL/publications/PU00013765.pdf">Some formulae concerning the integers less than n and prime to n</a>, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.
%H C. A. Nicol, <a href="https://dx.doi.org/10.1073/pnas.39.9.963">On restricted partitions and a generalization of the Euler phi number and the Moebius function</a>, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.
%H C. A. Nicol and H. S. Vandiver, <a href="https://dx.doi.org/10.1073/pnas.40.9.825 ">A von Sterneck arithmetical function and restricted partitions with respect to a modulus</a>, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.
%H K. G. Ramanathan, <a href="https://www.ias.ac.in/article/fulltext/seca/020/01/0062-0069">Some applications of Ramanujan's trigonometrical sum C_m(n)</a>, Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.
%H Srinivasa Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram21.pdf">On certain trigonometric sums and their applications in the theory of numbers</a>, Trans. Camb. Phil. Soc. 22 (1918), 259-276.
%H R. D. von Sterneck, <a href="https://play.google.com/books/reader?id=V1I-AQAAMAAJ&hl=de&printsec=frontcover&pg=GBS.PA1567">Ein Analogon zur additiven Zahlentheorie</a>, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]
%H R. D. von Sterneck, <a href="https://eudml.org/doc/144877">Über ein Analogon zur additiven Zahlentheorie</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ramanujan%27s_sum">Ramanujan's sum</a>.
%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Robert_Daublebsky_von_Sterneck_der_J%C3%BCngere">Robert Daublebsky von Sterneck der Jüngere</a>.
%H Aurel Wintner, <a href="https://www.jstor.org/stable/2371672">On a statistics of the Ramanujan sums</a>, Amer. J. Math., 64(1) (1942), 106-114.
%F 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]
%F 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]
%F From _Mats Granvik_, Oct 10 2016: (Start)
%F For n >= 1 and k >= 1 let
%F A(n,k) := if n mod k = 0 then k^r, otherwise 0;
%F B(n,k) := if n mod k = 0 then k/n^s, otherwise 0.
%F Then the Ramanujan's sum matrix equals
%F inverse(A).transpose(B) evaluated at s=0 and r=0.
%F Equals inverse(A051731).transpose(A127093).
%F 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)
%F T(n,k) = c_n(k) = Sum_{s | gcd(n,k)} s * Moebius(n/s). - _Petros Hadjicostas_, Jul 27 2019
%F 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
%e Square array T(n,k) = c_n(k) (with rows n >= 1 and columns k >= 1) starts as follows:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, ...
%e -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, ...
%e 0, -2, 0, 2, 0, -2, 0, 2, 0, -2, 0, 2, 0, ...
%e -1, -1, -1, -1, 4, -1, -1, -1, -1, 4, -1, -1, -1, ...
%e 1, -1, -2, -1, 1, 2, 1, -1, -2, -1, 1, 2, 1, ...
%e -1, -1, -1, -1, -1, -1, 6, -1, -1, -1, -1, -1, -1, ...
%e 0, 0, 0, -4, 0, 0, 0, 4, 0, 0, 0, -4, 0, ...
%e ... [example edited by _Petros Hadjicostas_, Jul 27 2019]
%p 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_
%p # to get the example above
%p for n to 8 do
%p seq(c(n, k), k = 1 .. 13);
%p end do
%p # _Petros Hadjicostas_, Jul 27 2019
%t 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_ *)
%t (* To get the example above in table format *)
%t TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 13}]]
%t (* _Petros Hadjicostas_, Jul 27 2019 *)
%Y Transpose of array in A054534. Cf. A054532, A054533, A282634.
%Y Cf. A086831=c_n(2) (2nd column), A085097=c_n(3) (3rd column), A085384=c_n(4) (4th column), A085639=c_n(5) (fifth column), A085906=c_n(6) (sixth column), A099837=c_3(n) (third row), A176742=c_4(n) (fourth row), A100051=c_6(n) (sixth row).
%K sign,tabl,nice
%O 1,12
%A _N. J. A. Sloane_, Apr 09 2000
%E Name edited by _Petros Hadjicostas_, Jul 27 2019
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