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Ramanujan sum
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(Definition)
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For positive integers  and  , the complex number
is referred to as a Ramanujan sum, or a Ramanujan trigonometric sum. Since
 , an equivalent definition is
where  is some reduced residue system mod  , meaning any subset of
 containing exactly one element of each invertible residue class mod  .
Using a symmetry argument about roots of unity, one can show
Applying Möbius inversion, we get
which shows that  is a real number, and indeed an integer. In particular
 . More generally,
 if  Using the Chinese remainder theorem, it is not hard to show that for any fixed  , the function
 is multiplicative:
 if  If  is invertible mod  , then the mapping
 is a permutation of the invertible residue classes mod  . Therefore
 if   defines a periodic function on
 itself, and conversely. For another example, see Landsberg-Schaar relation.
Some writers use different notation from ours, reversing the roles of  and  in the expression  .
The name “Ramanujan sum” was introduced by Hardy.
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"Ramanujan sum" is owned by Mathprof. [ full author list (3) | owner history (2) ]
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(view preamble)
See Also: root of unity
| Also defines: |
Ramanujan trigonometric sum |
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Cross-references: expression, Landsberg-Schaar relation, periodic function, quotient ring, number theory, sums, permutation, mapping, multiplicative, function, Chinese remainder theorem, real number, Möbius inversion, roots of unity, argument, symmetry, residue class, invertible, subset, reduced residue system, complex number, integers, positive
There is 1 reference to this entry.
This is version 7 of Ramanujan sum, born on 2002-01-20, modified 2006-10-06.
Object id is 1504, canonical name is RamanujanSum.
Accessed 3777 times total.
Classification:
| AMS MSC: | 11L03 (Number theory :: Exponential sums and character sums :: Trigonometric and exponential sums, general) | | | 11T23 (Number theory :: Finite fields and commutative rings :: Exponential sums) |
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Pending Errata and Addenda
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