Consider two arithmetic functions *f*(*n*) and *g*(*n*), with Dirichlet series generating functions *F*(*s*) and *G*(*s*), respectively. What, then, can we say about the product *F*(*s*)*G*(*s*)?

Using different summation indicies *a* and *b*, we have

and . The product, then, is

.

Now, suppose we reorder the sum so as to group terms with the same value of *a**b*. We let *n*=*a**b*; then we have , and there is one term with given *n* for each *a* which divides that *n*, so within each set of terms with the same *n*, we sum over *a*|*n*, and we get:,

and so we see the product *F*(*s*)*G*(*s*) is the Dirichlet series generating function of the Dirichlet convolution of *f*(*n*) and *g*(*n*) (compare to the relationship between the convolution and the Fourier transform).

Now, this implies that the Dirichlet series generating function of the unit function *ε*(*n*) is 1, which should be obvious, as

.

And since , we see that if *f*(*n*) has Dirichlet series generating function *F*(*s*), then its Dirichlet inverse *f*^{-1}(*n*) has Dirichlet series generating function 1/*F*(*s*). For example, consider the Dirichlet inverses 1(*n*) and *μ*(*n*), which have Dirichlet series generating functions and , respectively. You may also recall that here I showed that the Dirichlet series generating function for |*μ*| is . I also demonstrated here that |*μ*| and the Liouville function *λ*(*n*) are Dirichlet inverses, so we can see that *λ*(*n*) has Dirichlet series generating function ; we can confirm this via the Euler product for Dirichlet series generating functions of completely multiplicative functions.

One should also note that if a function *f*(*n*) has Dirichlet series generating function *F*(*s*), then the function has Dirichlet series generating function *F*(*s*–*a*); since

,

then

.

For example, the Dirichlet series generating function of is thus .

Combining this with our above relationship between Dirichlet convolution and Dirichlet series generating functions, we can develop a number of proofs. For example, the proof that the Dirichlet series generating function of the divisor function σ* _{x}*(

*n*) is becomes much more simple than the proof seen here:

Since , we see ; since the Dirichlet series generating function of 1(

*n*) is , and the Dirichlet series generating function of is , the Dirichlet series generating function of σ

*(*

_{x}*n*) is their product, .

Similarly, we can use the Möbius inversion formula relationship that , found here, to confirm that

*φ*(

*n*) has Dirichlet series generating function , as we found via Euler product here.

Tags: Dirichlet Convolution, DIrichlet Inverse, Dirichlet Series Generating Function, Divisor Functions, Euler Product, Math, Möbius Function, Möbius Inversion Formula, Monday Math, Riemann Zeta Function, Totient

November 2, 2009 at 1:51 am |

[…] that is nonzero only for , p prime and k a positive integer; in that case, . Thus . Now, using the relationship between Dirichlet convolution and Dirichlet series, we have that the Dirichlet series generating function for is thus the product of those for f-1(n) […]