A054533 Triangular array giving Ramanujan sum T(n,k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) for n >= 1 and 1 <= k <= n.

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

Offset: 1

Author

N. J. A. Sloane, Apr 09 2000

Keywords

• sign
• easy
• nice
• tabl

Comments

From Wolfdieter Lang, Jan 06 2017: (Start)

Periodicity: c_n(k+n) = c_n(k). See the Apostol reference p. 161.

Multiplicativity: c_n(k)*c_m(k) = c_{n*m}(k), if gcd(n,m) = 1. For the proof see the Hardy reference, p. 138.

Dirichlet g.f. for fixed k: D(n,s) := Sum_{n>=1} c_n(k)/n^s = sigma_{1-s}(k)/zeta(s) = sigma_{s-1}(k)/(k^(s-1)*zeta(s)) for s > 1, with sigma_m(k) the sum of the m-th power of the divisors of k. See the Hardy reference, eqs. (9.6.1) and (9.6.2), pp. 139-140, or Hardy-Wright, Theorem 292, p. 250.

Sum_{n>=1} c_n(k)/n = 0. See the Hardy reference, p. 141. (End)

Right border gives A000010. - Omar E. Pol, May 08 2018

Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} T(d, v) * binomial((n + k)/d, k/d) = S(k, n, v). This was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 1) = A051168(n + k, k). - Petros Hadjicostas, Jul 09 2019

We have T(n, k) = c_n(k) = Sum_{m=1..n, (m,n)=1} exp(2 Pi i m k / n) and A054532(n, k) = c_k(n) = Sum_{m=1..k, (m,k)=1} exp(2 Pi i m n / k) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Jul 27 2019

Examples

			Triangle begins
   1;
  -1,  1;
  -1, -1,  2;
   0, -2,  0,  2;
  -1, -1, -1, -1,  4;
   1, -1, -2, -1,  1,  2;
  -1, -1, -1, -1, -1, -1,  6;
   0,  0,  0, -4,  0,  0,  0,  4;
   0,  0, -3,  0,  0, -3,  0,  0,  6;
   1, -1,  1, -1, -4, -1,  1, -1,  1,  4;
  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10;
   0,  2,  0, -2,  0, -4,  0, -2,  0,  2,  0,  4;
  -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12;
   ...
[Edited by _Jon E. Schoenfield_, Jan 03 2017]
Periodicity and multiplicativity: c_6(k) = c_2(k)*c_3(k), e.g.: 2 = c_6(6) = c_2(6)*c_3(6) = c_2(2)*c_3(3) = 1*2 = 2. - _Wolfdieter Lang_, Jan 05 2017
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp. 160-161.
• G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 137-139.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Oxford Science Publications, Clarendon Press, Oxford, 2003, pp. 237-238.

Links

• Seiichi Manyama, Rows n=1..140 of triangle, flattened (Rows 1..50 from T. D. Noe)
• 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.
• 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 bottom of p. 410, where the author proves that Sum cos(2*Pi*q*v/n) = mu(n/D) * phi(n) /phi(n/D), where D is the gcd of n and q. The summation is over integers v "less than n and prime to n" (top of p. 408).]
• 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).
• R. D. von Sterneck, Über ein Analogon zur additiven Zahlentheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung 12 (1903), 110-113.
• M. V. Subbarao, The Brauer-Rademacher identity, Amer. Math. Monthly 72 (1965), 135-138.
• 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.

Crossrefs

Cf. A008683, A054532, A054534, A054535, A282634, A000010.

Programs

Mathematica

c[k_, n_] := Sum[ If[GCD[m, k] == 1, Exp[2 Pi*I*m*n/k], 0], {m, 1, k}]; A054533 = Flatten[ Table[c[n, k] // FullSimplify, {n, 1, 14}, {k, 1, n}] ] (* Jean-François Alcover, Jun 27 2012 *)
(* to get the triangle in the example above *)
FormTable[Table[c[n, k] // FullSimplify, {n, 1, 13}, {k, 1, n}]]
(* Petros Hadjicostas, Jul 28 2019 *)

PARI

T(n,k) = sumdiv(gcd(n,k), d, d*moebius(n/d));
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(T(n,k), ", "); ); print(); ); }; \\ Michel Marcus, Jun 14 2018

Formula

T(n, k) = Sum_{m=1..n, gcd(m,n) = 1} exp(2*Pi*i*m*k / n), n >= 1, 1 <= k <= n, where i is the imaginary unit.

T(n, k) = Sum_{d | gcd(n,k)} d*Moebius(n/d), n >= 1, 1 <= k <= n.

Extensions

Name edited by Petros Hadjicostas, Jul 27 2019