Definition: Number of Divisors

The number of divisors function $\tau:\mathbb N\to\mathbb N$ is an arithmetic function counting how many divisors a given number $n\in\mathbb N$ has. In the sum notation notation, the $\tau$ function can be written as $$\tau(n):=\sum_{d \mid n}1\quad\quad\forall n > 0.$$

Example.

The $\tau$ function can be visualized using SageMath. If you click on the evaluate button, you will see the values of $\tau(n)$ for $n=1,\ldots,100.$

sigmapoints= [(i, sigma(i,0)) for i in range(1,100)] list_plot(sigmapoints)

Examples: 1
Proofs: 2 3
Propositions: 4


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References

Bibliography

  1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
  2. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927