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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.
Learn the definition, properties and applications of Ramanujan's sum, a special sum involving the Mobius function and the principal Dirichlet character modulo q. See how Ramanujan's sum relates to Fourier transforms, Dirichlet series and the Riemann zeta-function.
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).
10 Οκτ 2016 · Can someone please explain the concept of a Ramanujan sum in easier language than Wikipedia and its relation to this question. Then, how to calculate the Ramanujan sum: $$\sum _{n\geq 1}^{\Re } n^{-1/s}$$ To show it equal to: $$\zeta(1/s)$$
17 Μαρ 2023 · Ramanujan sums. Trigonometric sums depending on two integer parameters $ k $ and $ n $: $$ c _ {k} ( n) = \sum _ { h } \mathop {\rm exp} \left ( \frac {2 \pi n h i } {k} \right ) = \ \sum _ { h } \cos \frac {2 \pi n h } {k} , $$. when $ h $ runs over all non-negative integers less than $ k $ and relatively prime to $ k $.
15 Ιουν 2018 · Ramanujan summation allows you to manipulate sums without worrying about operations on infinity that would be considered wrong. For example, you can use Ramanujan summation to assign a finite value to the infinite series 1-1+1-1+1-..., which we know diverges.
1 Ιουλ 2024 · Ramanujan's Sum Identity. Given the generating functions defined by. (OEIS A051028, A051029 , and A051030 ), then. Hirschhorn (1995) showed that. where. Hirschhorn (1996) showed that checking the first seven cases to 6 is sufficient to prove the result.
