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Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series.
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 paź 2016 · Does Ramanujan summation evaluate the series $\sum \frac{1}{n^s}$ to $\zeta(s)$ or $\zeta(s)-\frac{1}{s-1}$?
Since we multplied the sum by a number that is not 1, the sum must be equal to 0.) Thus we can express the M obius function using Ramanujan’s sum as (q) = c q(1). Because d(k) = dif kjdand d(k) = 0 if k6jd, we have c q(k) = X djq;djk q d d= X dr=q;djk (r)d: So c q(k) qs = X dr=q;djk 1 qs (r)d= X dr=q;djk 1 dsrs (r)d= X dr=q;djk 1 rs (r)d1 s ...
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).
17 mar 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 $.
9 lut 2018 · Ramanujan sum. For positive integers s s and n n, the complex number. is referred to as a Ramanujan sum, or a Ramanujan trigonometric sum . Since e2πi = 1 e 2 π i = 1, an equivalent definition is.
