Theory of numbers: Dirichlet series

This lecture is part of an online undergraduate course on the theory of numbers.
We describe the correspondence between Dirichlet series and arithmetic functions, and work out the Dirichlet series of the arithmetic functions in the previous lecture.
Correction: Dave Neary pointed out that on the first sheet (bottom right) 13x39 should be 13x139
For the other lectures in the course see • Theory of numbers

[@James-kg8fp]

I think you are tacitly using that no two distinct arithmetic functions have the same Dirichlet series. For those wondering how to prove this, consider the limit as s goes to infinity of k^s(F(s)-G(s)) where k is the first index that f and g differ.

[@omargaber3122]

Excuse me
I dont understand what you say
Do you mean there is an error in the explanation in the video?
Or is this just an additional clarification?

[@James-kg8fp]

@@omargaber3122 Not an error! More of an omission I was filling in. He uses relationships amongst Dirichlet series F,G and then concludes things about the underlying arithmetic functions f,g. This is a correct and useful thing to do, but it is only possible because if two arithmetic functions have the same Dirichlet series then they are the same function, a fact which was not mentioned in the video.

[@omargaber3122]

1-Do you mean that this only applies to functions that have the same arithmetic functions?
2-Could you PLEASE cite an example to the contrary?
3-Do you understand British Dyer's cojecture?

[@richarde.borcherds7998]

Alternatively you can just work with formal Dirichlet series, where convergence does not matter, and the Dirichlet series is just a suggestive way of writing an infinite sequence of integers.

[@James-kg8fp]

@@richarde.borcherds7998 I like the idea of using formal series to avoid issues of convergence, which also avoids the problem of two different functions having the same Dirichlet series, since obsiously their formal series would be different. However, for formal series it seems one would have to justify why operations like F(s)/G(s) * G(s) = F(s) make sense. After all, what does division or multiplication of formal series mean without appealing to evaluations as series of reals? In that case one would have to do the bookkeeping of showing that the calculus of formal series is consistent.

[@martinepstein9826]

This was extremely interesting. For some more examples, in the previous lecture you mentioned some additive functions like omega(n) = # of distinct prime divisors of n, and you pointed out we could exponentiate them to get multiplicative functions. I took the Dirichlet series of b^omega(n) and ended up with an interesting looking identity: b^omega(n) = sum over square-free d|n of (b-1)^omega(d). Once you see it it's not hard to confirm by elementary means. I think I'll take a break then look at b^Omega(n).

[@dneary]

I wonder if the Dirichlet series plus Mobius inversion formula gives a way to calculate a closed form for the sum of floor(n/i) for i=1..n - it seems like there is a close relationship between the floor and divisors of n - I have figured out that \sum_{i=1}^n [n/i] = \sum_{i=1}^n \sigma_0(i) - is there a nice way to get a closed form for this?

[@martinepstein9826]

In analytic number theory does one take analytic continuations of all these Dirichlet series and subject them to complex analysis?

[@martinepstein9826]

Now that you've covered the Moebius function a video on Mertens conjecture could be interesting.

[@omargaber3122]

You are wonderfull professor.👏👏

[@cboniefbr]

I love the you explain things, it all seems very fluid! Thank you so much for your content!

[@omargaber3122]

Thank you very much doctor , please explain brich-dyre conjecture.

[@migarsormrapophis2755]

yeeeee
bsa ain sxg eg

[@downinthehole]

Dear Prof. Borcherds, thank you very much for all your content, it's a pleasure to listen to you lecture! Allow me to make one tiny "stylistic" notice though: in the thumbnail for this and the previous video (on multiplicative functions) you have an extra "Type something" prompt above the title that you may have forgotten to erase (thought you might like to correct this).

[@richarde.borcherds7998]

Thanks; it was so small I overlooked it.

[@downinthehole]

@@richarde.borcherds7998 I would also like to note that the thumbnail for this video has changed to an incorrect one for whatever reason and is now the thumbnail for the video on congruences that comes after this one.