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In number theory, Ramanujan's sum, usually denoted c q (n), is a function of two positive integer variables q and n defined by the formula = (,) =, where (a, q) = 1 means that a only takes on values coprime to q.
10 Οκτ 2016 · Then, how to calculate the Ramanujan sum: ∑n≥1R n−1/s ∑ n ≥ 1 ℜ n − 1 / s. To show it equal to: ζ(1/s) ζ ( 1 / s) summation. ramanujan-summation. Share.
1 Ιουλ 2024 · The sum c_q(m)=sum_(h^*(q))e^(2piihm/q), (1) where h runs through the residues relatively prime to q, which is important in the representation of numbers by the sums of squares. If (q,q^')=1 (i.e., q and q' are relatively prime), then c_(qq^')(m)=c_q(m)c_(q^')(m).
A function tau(n) related to the divisor function sigma_k(n), also sometimes called Ramanujan's tau function. It is defined via the Fourier series of the modular discriminant Delta(tau) for tau in H, where H is the upper half-plane, by Delta(tau)=(2pi)^(12)sum_(n=1)^inftytau(n)e^(2piintau) (1) (Apostol 1997, p. 20).
Therefore we can write Ramanujan’s sum c q(l) as c q(l) = q˜b(l), thus c q= q˜b. The above gives us an expression for c q(l) as a multiple of the Fourier trans-form of the principal Dirichlet character modulo q. c q: Z=q!C, and we can write the Fourier transform of c q as cb q(k) = 1 q X j2Z=q c q(j)e 2ˇijk=q = X j2Z=q ˜b(j)e 2ˇijk=q: 1
1 Ιουλ 2024 · Given the generating functions defined by (1+53x+9x^2)/(1-82x-82x^2+x^3) = sum_(n=1)^(infty)a_nx^n (1) (2-26x-12x^2)/(1-82x-82x^2+x^3) = sum_(n=0)^(infty)b_nx^n (2) (2+8x-10x^2)/(1-82x-82x^2+x^3) = sum_(n=0)^(infty)c_nx^n (3) (OEIS A051028, A051029, and A051030), then a_n^3+b_n^3=c_n^3+(-1)^n.
