OFFSET
1,12
COMMENTS
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
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
REFERENCES
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011 (T(n,k) for n+k <= 142).
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 p. 410.
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_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]
From Mats Granvik, Oct 10 2016: (Start)
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)
T(n,k) = c_n(k) = Sum_{s | gcd(n,k)} s * Moebius(n/s). - Petros Hadjicostas, Jul 27 2019
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
EXAMPLE
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, ...
... [example edited by Petros Hadjicostas, Jul 27 2019]
MAPLE
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
# Petros Hadjicostas, Jul 27 2019
MATHEMATICA
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}]]
(* Petros Hadjicostas, Jul 27 2019 *)
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Apr 09 2000
EXTENSIONS
Name edited by Petros Hadjicostas, Jul 27 2019
STATUS
approved