LetK/Qbe an extension of degreemandζK(s)be the corresponding zeta function defined in (I.22). Similar to Montgomery [68], we define the pair correlation function for the zeros ofζK(s)by
FK(α):=FK(α,T):=
mT 2π logT
−1
∑
0<γ,γ0≤T
Timα(γ−γ0)w(γ−γ0),
where the sum is over all non-trivial zerosρ,ρ0ofζK(s)with imaginary partsγ,γ0respectively, whereα,T∈ RwithT ≥2, andw(u) = 4
4+u2 as before. We again can think of this as a twisted average of the differences of imaginary parts for the non-trivial zeros. Using similar techniques to Montgomery, we are able to prove the following.
Theorem VI.2.1(Alsharif–Gibson–de Laat–Milinovich–Rolen–T.–Wagner, 2020). Assume GRH. If K/Qis an abelian extension, then uniformly for|α|≤m1 as T→∞, we have
FK(α) = (m+oK(1))T−2m|α|logT+m|α|+oK(1).
For general extensions K/Q, we have for|α|≤m1 as T →∞,
FK(α)≤(m+oK(1))T−2m|α|logT+m|α|+oK(1).
In addition to finding the exact asymptotics of the pair correlation function when the Galois group of the extension is abelian, there are select other cases that we were able to compute using the same methods. For example, if the Galois group is isomorphic toDm(the dihedral group of orderm), then we find that
FK(α) = (m+oK(1))T−2m|α|logT+3m+2
4 |α|+oK(1).
Although we are unable to find the exact asymptotics for all values ofK, the inequality of Theorem VI.2.1 is sufficient in order to determine a proportion of zeros that are distinct. To state our result, we first begin by defining
NKs(T):=
∑
0<γ≤T mρ=1
1, NKd(T):=
∑
0<γ≤T
1
mρ, NK∗(T):=
∑
0<γ≤T
mρ,
where each sum is over the zerosρ=β+iγofζK(s)forβ,γ∈Rand wheremρis the multiplicity of the zero ρ. Notice that the counting functions from the previous section are the same in the case thatK=Q. Using Theorem VI.2.1 and the techniques of [27], we were able to prove the following.
Theorem VI.2.2(Alsharif–Gibson–de Laat–Milinovich–Rolen–T.–Wagner, 2020). Assume GRH. Then for any extension K/Qof degree m, we have
NK∗(T)≤(cm+oK(1))NK(T),
where
cm=
2.3226 for m=2,
3.3232 for m=3,
4.3235 for m=4,
(1+10−10)m+0.3243 for m≥1,
m+1/3 for m≥1.
While this is a bound on the multiplicities of the zeros ofζK(s), the meaning of whatNK∗(T)is counting is likely less intuitive that it is forNKd(T)orNKs(T). Luckily, we are able to use simple inequalities to prove the following bound for the number of distinct zeros.
Corollary VI.2.3(Alsharif–Gibson–de Laat–Milinovich–Rolen–T.–Wagner, 2020). Assuming GRH and let- ting cmbe as defined in Theorem VI.2.2, we have that
NKd(T)≥ 1
cm+oK(1)
NK(T).
For quadratic extensions, we have the additional bound
NKd(T)≥(0.4585+oK(1))NK(T).
Before our work, there does not appear to have been any explicit results given for the proportion of distinct zeros for Dedekind zeta functions. The only bound that we were able to easily prove from the literature came from the works of [11, 27, 30]. Combining the bounds in these papers, one can find that for extensionsK/Q of degree 2, at least 39.76% of zeros must be distinct. Thus, Corollary VI.2.3 provides an improvement in this case as well and appears to be the first result to show that the proportion of distinct zeros is positive for all Dedekind zeta functions.
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