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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.
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
11 Ιαν 2019 · After converting the double factorials to Gamma functions, Maple evaluates your first sum as − 2EllipticE(i) π and your second as − Γ(3 / 4)2 √2π3 / 2. The first appears to agree numerically with your conjecture, but the second doesn't. Yours is off by about 7.69 × 10 − 6. 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).
the denominator of Bn contains each of the factors 2 and 3 once and only once, 2n(2n − 1)Bn. /. n is an integer and 2 (2n − 1)Bn consequently is an odd integer. In his 17-page paper "Some Properties of Bernoulli's Numbers" (1911), Ramanujan gave three proofs, two corollaries and three conjectures. [61]
9 Φεβ 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.
Sums involving \( c_q(n) \) are known as Ramanujan sums; these were also used in applications including the proof of Vinogradov's theorem that every sufficiently large odd positive integer is the sum of three primes.
