Ramanujan’s sum

E355439

arithmetical function multiplicative function number-theoretic function

Ramanujan’s sum is a number-theoretic function introduced by Srinivasa Ramanujan, expressing certain periodic arithmetic functions as finite trigonometric sums over primitive roots of unity.

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All labels observed (2)

Label	Occurrences
Ramanujan expansions	1
Ramanujan’s sum canonical	1

Statements (49)

Predicate	Object
instanceOf	arithmetical function ⓘ multiplicative function ⓘ number-theoretic function ⓘ
alternativeForm	c_q(n) = sum_{a mod q, (a,q)=1} e^{2πi a n / q} ⓘ
appearsIn	papers of Srinivasa Ramanujan on highly composite numbers and related topics ⓘ
closedForm	c_q(n) = sum_{d \| gcd(n,q)} μ(q/d) d ⓘ c_q(n) = μ(q/(q,n)) φ((q,n)) / φ(q/(q,n)) ⓘ
definition	c_q(n) = sum_{1 ≤ a ≤ q, gcd(a,q)=1} exp(2πi a n / q) ⓘ
dependsOn	integer n ⓘ integer q ⓘ
domain	n ∈ ℤ ⓘ q ∈ ℕ, q ≥ 1 ⓘ
field	number theory ⓘ
generalizationOf	finite Fourier sums over primitive roots of unity ⓘ
hasSeriesUse	basis for expansions of arithmetic functions with period q ⓘ
hasVariable	argument n ⓘ modulus q ⓘ
introducedBy	Srinivasa Ramanujan ⓘ
isPeriodicIn	n modulo q ⓘ
isRealValued	true ⓘ
namedAfter	Srinivasa Ramanujan ⓘ
orthogonalityRelation	sum_{n mod q} c_q(n) = 0 for q > 1 ⓘ sum_{q ≥ 1} c_q(n) c_q(m) / φ(q) converges to a function of gcd(m,n) ⓘ
property	c_1(n) = 1 for all integers n ⓘ c_q(0) = φ(q) ⓘ c_q(n) depends only on gcd(n,q) ⓘ c_q(n) is an integer for all integers n and q ≥ 1 ⓘ c_q(n) is bounded in absolute value by φ(q) ⓘ multiplicative in q for fixed n ⓘ
relatedConcept	Fourier analysis on finite abelian groups ⓘ Ramanujan expansion of the divisor function ⓘ Ramanujan expansion of the von Mangoldt function ⓘ
relatedTo	Dirichlet characters ⓘ Euler’s totient function φ(n) ⓘ surface form: Euler’s totient function Fourier series on arithmetic progressions ⓘ Möbius function ⓘ primitive roots of unity ⓘ
specialCase	c_p(n) = -1 if p is prime and p ∤ n ⓘ c_p(n) = p-1 if p is prime and p \| n ⓘ c_q(1) = μ(q) ⓘ
symbol	c_q(n) ⓘ
takesValuesIn	integers ⓘ
usedFor	Ramanujan’s sum self-linksurface differs ⓘ surface form: Ramanujan expansions expansion of periodic arithmetic functions ⓘ expressing arithmetic functions as trigonometric sums ⓘ
usedIn	Waring’s problem and related additive problems ⓘ analytic number theory ⓘ circle method ⓘ study of multiplicative functions ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Srinivasa Ramanujan → notableWork → Ramanujan’s sum ⓘ

Ramanujan’s sum → usedFor → Ramanujan’s sum self-linksurface differs ⓘ

this entity surface form: Ramanujan expansions