11 Iwaniec-Kowalski
11.1 Blueprint for Iwaniec-Kowalski Chapter 1
Here we collect facts from Chapter 1 that are not already in Mathlib. We will try to upstream as much as possible.
Additive function.
Completely additive function.
If ‘g‘ is a multiplicative arithmetic function, then for any \(n \neq 0\), \(\sum _{d | n} \mu (d) \cdot g(d) = \prod _{p \in n.\text{primeFactors}} (1 - g(p))\).
Multiply out and collect terms.
The Dirichlet convolution of \(\zeta \) with itself is \(\tau \) (the divisor count function).
By definition of \(\zeta \), we have \(\zeta (n) = 1\) for all \(n \geq 1\). Thus, the Dirichlet convolution \((\zeta * \zeta )(n)\) counts the number of ways to write \(n\) as a product of two positive integers, which is exactly the number of divisors of \(n\), i.e., \(\tau (n)\).
The L-series of \(\tau \) equals the square of the Riemann zeta function for \(\Re (s) {\gt} 1\).
From the previous theorem, we have that the Dirichlet convolution of \(\zeta \) with itself is \(\tau \). Taking L-series on both sides, we get \(L(\tau , s) = L(\zeta , s) \cdot L(\zeta , s)\). Since \(L(\zeta , s)\) is the Riemann zeta function \(\zeta (s)\), we conclude that \(L(\tau , s) = \zeta (s)^2\) for \(\Re (s) {\gt} 1\).