A054532 Ramanujan sum T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2*Pi*i*m*n / k), triangular array read by rows for n >= 1 and 1 <= k <= n.

1, 1, 1, 1, -1, 2, 1, 1, -1, 2, 1, -1, -1, 0, 4, 1, 1, 2, -2, -1, 2, 1, -1, -1, 0, -1, 1, 6, 1, 1, -1, 2, -1, -1, -1, 4, 1, -1, 2, 0, -1, -2, -1, 0, 6, 1, 1, -1, -2, 4, -1, -1, 0, 0, 4, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, 10, 1, 1, 2, 2, -1, 2, -1, -4, -3, -1, -1, 4, 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 12, 1

Offset

1,6

Comments

T(n, k) = c_k(n) = sum of the n-th powers of the k-th primitive roots of unity. - Petros Hadjicostas, Jul 27 2019

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 160.

Links

T. D. Noe, Rows n=1..50 of triangle, flattened

Tom M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math. 41 (1972), 281-293.

Eckford Cohen, A class of arithmetic functions, Proc. Natl. Acad. Sci. USA 41 (1955), 939-944.

A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), 173-188.

M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.

H. G. Gadiyar and R. Padma, Linking the circle and the sieve: Ramanujan-Fourier series, arXiv:math/0601574 [math.NT], 2006.

Emiliano Gagliardo, Le funzioni simmetriche semplici delle radici n-esime primitive dell'unità, Bollettino dell'Unione Matematica Italiana Serie 3, 8(3) (1953), 269-273.

Otto Hölder, Zur Theorie der Kreisteilungsgleichung K_m(x)=0, Prace mat.-fiz. 43 (1936), 13-23.

Peter H. van der Kamp, On the Fourier transform of the greatest common divisor, Integers 13 (2013), #A24. [See Section 3 for historical remarks.]

J. C. Kluyver, Some formulae concerning the integers less than n and prime to n, in: KNAW, Proceedings, 9 I, 1906, Amsterdam, 1906, pp. 408-414; see p. 410.

P. Moree and H. Hommerson, Value distribution of Ramanujan sums and of cyclotomic polynomial coefficients, arXiv:math/0307352 [math.NT], 2003.

K. Motose, Ramanujan's sums and cyclotomic polynomials, Math. J. Okayama U. 47, no 1, (2005), Article 5.

C. A. Nicol, On restricted partitions and a generalization of the Euler phi number and the Moebius function, Proc. Natl. Acad. Sci. USA 39(9) (1953), 963-968.

C. A. Nicol and H. S. Vandiver, A von Sterneck arithmetical function and restricted partitions with respect to a modulus, Proc. Natl. Acad. Sci. USA 40(9) (1954), 825-835.

K. G. Ramanathan, Some applications of Ramanujan's trigonometrical sum C_m(n), Proc. Indian Acad. Sci., Sect. A 20 (1944), 62-69.

Srinivasa Ramanujan, On certain trigonometric sums and their applications in the theory of numbers, Trans. Camb. Phil. Soc. 22 (1918), 259-276.

R. D. von Sterneck, Ein Analogon zur additiven Zahlentheorie, Sitzungsber. Akad. Wiss. Sapientiae Math.-Naturwiss. Kl. 111 (1902), 1567-1601 (Abt. IIa). [It may not be universally accessible.]

R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113. [Summary of the 1902 paper.]

Wikipedia, Ramanujan's sum.

Wikipedia, Robert Daublebsky von Sterneck der Jüngere.

Aurel Wintner, On a statistics of the Ramanujan sums, Amer. J. Math., 64(1) (1942), 106-114.

Formula

T(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} cos(2*Pi*m*n / k) = mu(k/gcd(k,n)) * phi(k) / phi(k/gcd(k,n)) = Sum_{d | gcd(k,n)} mu(k/d) * d. (All formulas were proved by Kluyver (1906, p. 410).) - Petros Hadjicostas, Aug 20 2019

Example

Triangle T(n,k) (with rows n >= 1 and columns k >= 1) begins as follows:
  1;
  1,  1;
  1, -1,  2;
  1,  1, -1,  2;
  1, -1, -1,  0,  4;
  1,  1,  2, -2, -1,  2;
  1, -1, -1,  0, -1,  1,  6;
  1,  1, -1,  2, -1, -1, -1, 4;
  1, -1,  2,  0, -1, -2, -1, 0, 6;
  ...

Mathematica

t[n_, k_] := Sum[ c = Exp[2*Pi*I*m*(n/k)]; If[ GCD[m, k] == 1, c, 0], {m, 1, k}] // FullSimplify; Flatten[ Table[ t[n, k], {n, 1, 15}, {k, 1, n}]] (* Jean-François Alcover, Mar 15 2012 *)
(* to get the triangle in the example *)
TableForm[Table[t[n, k], {n, 1, 9}, {k, 1, n}]]
(* Petros Hadjicostas, Jul 27 2019 *)

Crossrefs

Cf. A054533, A054534, A054535, A282634.

Sequence in context: A336709 A221362 A114116 * A260415 A370942 A214710

Adjacent sequences: A054529 A054530 A054531 * A054533 A054534 A054535

Keyword

sign,easy,nice,tabl

Author

N. J. A. Sloane, Apr 09 2000

Status

approved