I am writing my senior thesis (I am an undergrad math major at UCSB) on Dirichlet Series, which are, in the classical sense, series of the form(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\zeta (s)=\sum_{k=1}^{\infty}\frac{a_k}{k^s}[/tex]

where [tex]a_k,s\in\mathbb{C}[/tex] and are more generally given by (whence the name generalized Dirichlet series)

[tex]\xi (s)=\sum_{k=1}^{\infty}a_k e^{-\lambda_k s}[/tex]

where [tex]\{ \lambda_k\}[/tex] is a sequence of real numbers such that [tex]\lambda_k < \lambda_{k+1},\forall k\in\mathbb{Z} ^+[/tex] and such that [tex]\lambda_k\rightarrow\infty\mbox{ as }k\rightarrow\infty .[/tex] I note that the usual power series and classical Dirichlet series are both special cases of generalized Dirichlet series (the sufficiently curious will check this by putting [tex]\lambda_k = \log k[/tex] in [tex]\xi[/tex] to obtain the later, and put [tex] a_k = b_n[/tex] if [tex]k=2^{n}[/tex] and [tex]a_n = 0[/tex] otherwise in [tex]\zeta[/tex] to obtain the former.)

As is well known, the formula for multipling two absolutely convergent classical Dirichlet series, say f and g given by

[tex]f(s)=\sum_{k=1}^{\infty}\frac{a_k}{k^s}\mbox{ and }g(s)=\sum_{n=1}^{\infty}\frac{b_n}{n^s},[/tex]

is theso-calledDirichlet product or convolution given by

[tex]h(s):=f(s)g(s)=\left( \sum_{k=1}^{\infty}\frac{a_k}{k^s}\right) \left( \sum_{n=1}^{\infty}\frac{b_n}{n^s}\right) = \sum_{k=1}^{\infty} \left( \sum_{n|k}a_{n}b_{\frac{k}{n}}\right) \frac{1}{k^s} ,[/tex]

where the inner sum is over all positive divisors n of k (the symbol n|k is read "n divides k".) This is an analog of the Cauchy product of power series, namely

[tex]\left( \sum_{k=0}^{\infty}a_{k}z^{k}\right) \left( \sum_{n=0}^{\infty}b_{n}z^{n}\right) = \sum_{k=0}^{\infty} \left( \sum_{n=0}^{k}a_{n}b_{k-n}\right) z^{k} .[/tex]

I am attempting to formulate an analog of this multiplication formula for generalized Dirichlet series, this is what I've got:

Question: is the following correct?

The formula for multipling two absolutely convergent general Dirichlet series, say F and G given by

[tex]F(s)=\sum_{k=1}^{\infty}a_{k}e^{-\alpha_k s}\mbox{ and }G(s)=\sum_{n=1}^{\infty} b_{n}e^{-\beta_n s},[/tex]

is

[tex]H(s):=F(s)G(s)=\left( \sum_{k=1}^{\infty}a_{k}e^{-\alpha_k s}\right) \left( \sum_{n=1}^{\infty} b_{n}e^{-\beta_n s}\right) = \sum_{k=2}^{\infty} \left( \sum_{\substack{p+q=k\\p,q\geq 1}}a_{p}b_{q}e^{-\left(\alpha_p + \beta_q \right) s}\right) ,[/tex]

where the inner sum is over all integer partitions of k into exactly two summands p and q.

Note that this should reduce to the Dirichlet product and/or the Cauchy product under the respective special cases mentioned above.

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# I'm writing my senior thesis on Dirichlet Series

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