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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.
In mathematics, Ramanujan's Master Theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. Page from Ramanujan's notebook stating his Master theorem.
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
To get a continued fraction, let’s take our formula g(n+2) = xg(n+1) +(n +1)g(n) for n 0 and divide it by g(n+1): g(n+2) g (n+1) = x +(n+1) g(n) g for 0 This looks simpler in terms of the ratios rn = g(n+1)=g(n): rn+1 = x + (n +1) rn or solving for rn: rn = n +1 x+rn+1 for n 0
1 lip 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).
1 lip 2013 · Expanding the coefficient of $\,T^n\,$ on the left side and splitting the infinite sum into four infinite sums of geometric series we get the left side of equation $(2)$. The happy surprising result is how simple the numerator of the right side is.
13 gru 2010 · The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or what the motivation for it is?
