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Let $a,b,c$ denote positive integers greater than $0$ and $q_n$ a sequence of real numbers. Now assume $a,b$ and $c$ are parameters of the Dirichlet series $\displaystyle \sum_{n\geq a}\frac{q_n}{(bn+c)^s}$, valid for $s>1$.

I would like to find a notation for this series so that the three parameters $a,b$ and $c$ are clearly shown, something like $\zeta_{a,b,c}(s)$. But as you can see, having three parameters in the subscript is quite inconvenient. On the other hand, notation like $\zeta^a_{b,c}(s)$ looks confusing.

How can I notate the series $\displaystyle \sum_{n\geq a}\frac{q_n}{(bn+c)^s}$ with the three parameters $a,b,c$ in the most convenient way?

EDIT: I am aware that this might be opinion-based. In that case, I also welcome examples in published literature that manage to notate series with 3 or more parameters in a convenient way.

Xander Henderson
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Klangen
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    This seems to be a matter of opinion. That said, my PhD thesis uses the notation $\zeta_{E,E_\delta}(s)$. I also toyed with the notation $\zeta^{\text{loc}}{\Omega}(s;x)$ before settling on $\zeta^{\text{loc}}{x,\Omega}(s)$. The notation is ugly, but seems to get the job done. – Xander Henderson May 12 '20 at 16:13

2 Answers2

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One way I've sometimes seen (including several times on this site, although I don't recall which posts offhand & am not sure how to easily find any) is to specify the parameters after a semi-colon after the variables in the function definition. Your specific example would then be something like

$$\zeta(s; a,b,c) \tag{1}\label{eq1A}$$

In What does the semicolon ; mean in a function definition, this answer says

A semicolon is used to separate variables from parameters. Quite often, the terms variables and parameters are used interchangeably, but with a semicolon the meaning is that we are defining a function of the parameters that returns a function of the variables.

A comment to that answer gives a link to Definition of Parameter which says basically the same thing.

However, note this notation of using a semicolon separator is not necessarily always interpreted this way, with it possibly meaning different things depending on the context. For example, that linked post's accepted answer states:

The semicolon is used sometimes to optically separate some variable group. So the semicolon is not more than a reading aid.

As for the methods used to specify function parameters in published literature, I don't know of any examples offhand which I can point you to.

John Omielan
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You may try this: for each triple $t = (a, b, c)$ of positive integers, let $$\zeta_t(s) = \sum_{n\geq a}\frac{q_n}{(bn+c)^s}$$

J.-E. Pin
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