Ramanujan's sum

As Ramanujansumme is in number theory, a branch of mathematics, a certain finite sum whose value depends on the natural numbers and the integer, respectively. It is carried

Defined. The notation represents the greatest common divisor of and, therefore, the summation extends over the numbers are prime. The summands in the sum are powers of a fixed complex root of unity.

S. Ramanujan led these totals 1916. They play an important role in the circle method of Hardy, Littlewood and Vinogradov. → See also trigonometric polynomial.

By Ramanujansummen can win interesting representations for number-theoretic functions that allow an analytic continuation of these functions.

Spellings

For a clear presentation is written abbreviation in number theory and the function is called number theoretical exponential function.

With the number-theoretic exponential function allows the Ramanujansumme as

For integers and to write, read " a divides b", if an integer exists applies to the, there is no such number, to write, read " a divides b is not ." The summation symbol means that the summation index runs through all positive divisors of. For a prime power and an integer to write ( read " divides b if "), but if, in other words, if.

Elementary Properties

If you hold one of the variables or fixed in the Ramanujansumme, we obtain a number-theoretic function as a function of other variables, must be limited to as a variable for this term. For fixed, the function - periodic, which means that it applies

If you leave the condition of coprimality in the summation continues, obtained

Because then the left side is a geometric sum. You sorted in the sum by the greatest common divisor of and, then there is a Dirichlet convolution of the number-theoretic function with the constant function:

It follows from the Möbius inversion formula:

It follows then:

  • The Ramanujansumme always assumes real and even integer values ​​,
  • It is important,
  • It is at a fixed multiplicative number-theoretic function, ie
  • And it is always.
  • One can represent the Ramanujansumme by the Euler φ - function and the Möbius function:
  • Their values ​​are by limited in amount with a fixed,
  • Is the natural number for a prime number of shared, then.

Ramanujansummen for the representation of number-theoretic functions

Already Ramanujan showed for some important special cases that one can gain interesting representations for number-theoretic functions with its sums. This requires a special kind of discrete Fourier transform for number-theoretic functions, the greatest common divisor is introduced:

For this Fourier transform applies:

In these transformations, the governing equations by the formation of the greatest common divisor must consider many coefficients with positive index only finite.

Examples

  • Greatest common divisor:
  • Euler's φ - function:
  • A kind of orthogonality for Ramanujansummen: Be the number theoretic one function, so the neutral element of the convolution operation with
G. H. Hardy Dudley E. Littlewood Entire function American Mathematical Society
671765
de