Last time we revisited Robin’s theorem saying that 5040 being the largest counterexample to the bound
is equivalent to the Riemann hypothesis.
There’s an industry of similar results using other arithmetic functions. Today, we’ll focus on
Dedekind’s Psi function
where runs over the prime divisors of . It is series
A001615 in the online encyclopedia of integer sequences and it starts off with
1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, …
and here’s a plot of its first 1000 values

To understand this behaviour it is best to focus on the ‘slopes’ .
So, the red dots of minimal ‘slope’ correspond to the prime numbers, and the ‘outliers’ have a maximal number of distinct small prime divisors. Look at and its multiples and in the picture.
For this reason the
primorial numbers, which are the products of the fist prime numbers, play a special role. This is series
A002110 starting off with
1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870,…
In Patrick Solé and Michel Planat Extreme values of the Dedekind function, it is shown that the primorials play a similar role for Dedekind’s Psi as the superabundant numbers play for the sum-of-divisors function .
That is, if is the -th primorial, then for all we have that the 'slope' at is strictly below that of
which follows immediately from the fact that any can have at most distinct prime factors and is a strictly decreasing function.
Another easy, but nice, observation is that for all we have the inequalities
where is
Euler’s totient function
This follows as once from the definitions of and
But now it starts getting interesting.
In the proof of his theorem, Guy Robin used a result of his Ph.D. advisor Jean-Louis Nicolas

known as Nicolas’ criterion for the Riemann hypothesis: RH is true if and only if for all we have the inequality for the -th primorial number
From the above lower bound on we have for that
and combining this with Nicolas’ criterion we get
In fact, Patrick Solé and Michel Planat prove in their paper Extreme values of the Dedekind function that RH is equivalent to the lower bound
holding for all .
Dedekind’s Psi function pops up in lots of interesting mathematics.
In the theory of modular forms, Dedekind himself used it to describe the index of the congruence subgroup in the full modular group .
In other words, it gives us the number of tiles needed in the Dedekind tessellation to describe the fundamental domain of the action of on the upper half-plane by Moebius transformations.

When we have and we can view its fundamental domain via these Sage commands:
G=Gamma0(6)
FareySymbol(G).fundamental_domain()
giving us the 24 back or white tiles (note that these tiles are each fundamental domains of the extended modular group, so we have twice as many of them as for subgroups of the modular group)
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But, there are plenty of other, seemingly unrelated, topics where appears. To name just a few:
- The number of points on the projective line .
- The number of lattices at hyperdistance in Conway’s big picture.
- The number of admissible maximal commuting sets of operators in the Pauli group for the qudit.
and there are explicit natural one-to-one correspondences between all these manifestations of , tbc.