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Not to be confused with Ramanujan summation. In number theory, Ramanujan's sum, usually denoted cq (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. Srinivasa Ramanujan mentioned the sums in a 1918 paper. [ 1 ]
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.
The Ramanujan sum cq (n) has been used by mathematicians to derive many important infinite series expansions for arithmetic-functions in number theory. Interestingly, this sum has many properties which are attractive from the point of view of digital signal processing.
- Palghat P. Vaidyanathan, Srikanth Tenneti
- 2020
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
- 163KB
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In number theory, a branch of mathematics, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula \( c_q(n)= \sum_{a=1\atop (a,q)=1}^q e^{2 \pi i \tfrac{a}{q} n}, \) where (a, q) = 1 means that a only takes on values coprime to q.
17 de mar. de 2023 · Ramanujan sums. Trigonometric sums depending on two integer parameters $ k $ and $ n $: $$ c _ {k} ( n) = \sum _ { h } \mathop {\rm exp} \left ( \frac {2 \pi n h i } {k} \right ) = \ \sum _ { h } \cos \frac {2 \pi n h } {k} , $$.
22 de ago. de 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).
