Are defined. The notation means the greatest common divisor of and the summation thus extends over the numbers with which to be prime. The summands in the sum are powers of a fixed complex root of unity .
For a clear presentation, abbreviations are used in number theory and the function is called the number-theoretical exponential function .
With the number theoretic exponential function, the Ramanujan sum can be expressed as
write.
For integers and one writes , read “a divides b”, if there is an integer with which there is no such number, one writes , read “a does not divide b”. The summation symbol means that the summation index runs through all positive factors of . For a prime power and an integer one writes (read “ divides b exactly”), but if , in other words, if .
Elementary properties
If you hold one of the variables or in Ramanujan's sum fixed, the result is a number theoretic function as a function of other variables, must as for that term variable to be limited. With fixed the function is - periodic , that is, it is true
if .
If one leaves out the condition of coprime numbers in the summation, one obtains
because then the left side is a geometric sum. If you sort the sum according to the greatest common divisor of and , you get a Dirichlet convolution of the number theoretic function with the constant function :
Ramanujan sums for representing number theoretic functions
Ramanujan already showed for some important special cases that one can obtain interesting representations for number theoretic functions with his sums. For this purpose, a special kind of discrete Fourier transformation is introduced for number theoretic functions of the greatest common divisor:
Be and a number theoretic function. Then is called
discrete Fourier transform of .
The following applies to this Fourier transform:
and
for the inverse transform .
In these transformations, the determining equations only have to take into account a finite number of coefficients with a positive index by forming the greatest common divisor.
Examples
Greatest common divisor:
. This representation allows an analytic continuation of the greatest common divisor in the first place on the entire function !
Euler's φ function:
. The trigonometric relations follow from this by dividing them into real and imaginary parts
and
The divisor function can be explicitly represented as a series using Ramanujan sums:
The calculation of the first values of shows the fluctuation around the "mean value" (the average order of magnitude ) :
A kind of orthogonality for Ramanujan sums: Let the number-theoretic one function, i.e. the neutral element of the convolution operation with
Then it follows by inverse Fourier transform for
That means: Exactly when the sum on the right does not vanish, the numbers and are prime. Then the right side of the equation has the value 1.
literature
Jörg Brüdern : Introduction to analytical number theory . Springer, Berlin, Heidelberg, New York 1995, ISBN 3-540-58821-3 .
John Knopfmacher: Abstract Analytic Number Theory . New edition. Dover Publications, 2000, ISBN 0-486-66344-2 .
Srinivasa Ramanujan: On Certain Trigonometric Sums and their Applications in the Theory of Numbers . In: Transactions of the Cambridge Philosophical Society . tape22 , no.15 , 1918, pp.259-276 .
Srinivasa Ramanujan: On Certain Arithmetical Functions . In: Transactions of the Cambridge Philosophical Society . tape22 , no.9 , 1916, pp.159-184 .
Srinivasa Ramanujan: Collected Papers . American Mathematical Society / Chelsea, Providence 2000, ISBN 978-0-8218-2076-6 .
Wolfgang Schramm: The Fourier Transform of Functions of the Greatest Common Divisor . In: Integers: Electronic Journal of Combinatorical Number Theory . tape8 , no.50 , 2008 ( emis.de [PDF]).
Ivan Matveevitch Vinogradov: The Method of Trigonometrical Sums in the Theory of Numbers . Translated from the Russian and annotated by Klaus Friedrich Roth and Anne Ashley Davenport. New York, Dover 2004.