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Patterns of primes in the Sato–Tate conjecture

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Fix a non-CM elliptic curve \(E/\mathbb {Q}\), and let \(a_E(p) = p + 1 - \#E(\mathbb {F}_p)\) denote the trace of Frobenius at p. The Sato–Tate conjecture gives the limiting distribution \(\mu _{ST}\) of \(a_E(p)/(2\sqrt{p})\) within \([-1, 1]\). We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval \(I\subseteq [-1, 1]\), let \(p_{I,n}\) denote the nth prime such that \(a_E(p)/(2\sqrt{p})\in I\). We show \(\liminf _{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty \) for all \(m\ge 1\) for “most” intervals, and in particular, for all I with \(\mu _{ST}(I)\ge 0.36\). Furthermore, we prove a common generalization of our bounded gap result with the Green–Tao theorem. To obtain these results, we demonstrate a Bombieri–Vinogradov type theorem for Sato–Tate primes.

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Appendix A. Minorization of indicator functions

Appendix A. Minorization of indicator functions

Lemma A.1

Let \([\alpha ,\beta ]\subseteq [-1,1]\). We consider the following polynomials:

  1. (1)

    \(f_1(x):=-(x-\alpha )(x-\beta )[(x-x_1)(x-x_2)(x-x_3)]^2\),

  2. (2)

    \(f_2(x):=-(x-\alpha )(x-\beta )(1-x^2)[(x-x_1)(x-x_2)]^2\).

If for some choice of \(x_1,x_2,x_3\in [-1,1]\) we have \(\int _{-1}^1f_i(x)\mu _{ST}(dx)>0\) for some \(i\in \{1,2\}\), then \([\alpha ,\beta ]\) is \({{\,\mathrm{Sym}\,}}^8\)-minorizable.


This is clear from the definition of \({{\,\mathrm{Sym}\,}}^8\)-minorization. \(\square \)

Example A.2

Let \(I = [-1,-5/6]\). Then \(\mu _{ST}(I) = 0.0398\) and I can be \({{\,\mathrm{Sym}\,}}^8\)-minorized by the polynomial \(f(x) = (x-1)(x+5/6)(x+0.4)^2(x-0.16)^2(x-0.68)^2\) with corresponding \(b_0 = 0.001017\). To achieve bounded gaps, by Proposition 3.2, we need \(b_0M_k\theta /2 > 1\), i.e., \(M_k>13766\). By the bound on \(M_k\) given in [34, Proposition 4.5], we want \(k\ge 213\) such that

$$\begin{aligned} \log k - 2\log \log k -2 > 13766, \end{aligned}$$

so it suffices to pick \(k =\lceil e^{13787.1}\rceil \) and take \(\mathcal {H}\) to be the first k prime numbers greater than k. By [10], for \(n\ge 6\), the nth prime number satisfies the bound

$$\begin{aligned} n(\log n + \log \log n-1)\le p_n\le n(\log n + \log \log n). \end{aligned}$$

In particular, this shows that the number of primes \(\le k\) is at most

$$\begin{aligned} \frac{k}{\log k-\log \log k-1} \le \frac{k}{13776}. \end{aligned}$$

Therefore, the largest number in \(\mathcal {H}\) is at most \(p_{k+k/13776} \le 10^{5991.81}\), hence we have

$$\begin{aligned} \liminf _{n\rightarrow \infty } (p_{I,n+1} - p_{I,n}) \le \sup _{x,y\in \mathcal {H}}|x-y| \le 10^{5992}. \end{aligned}$$

Lemma A.3

If we sample endpoints of an interval \(I\subseteq [-1, 1]\) according to the Sato–Tate measure, then with at least a \(50.74\%\) chance, I can be \({{\,\mathrm{Sym}\,}}^8\)-minorized.


We use a brute-force computer program that provides a lower bound on the proportion of intervals \([\alpha ,\beta ]\) which satisfy the hypothesis of Lemma A.1. The idea behind our program is to use the two forms of polynomials described in Lemma A.1 as candidate minorizations of I. Our implementation is as follows, with \(\eta _1 = 0.01\) and \(\eta _2 = 0.0025\). The notation \(var\leftarrow val\) means that the value val is assigned to the variable var.

  1. 1.

    Initialize \(S = 0\) and \(\alpha =\beta =x_1=x_2=x_3 = -1\).

  2. 2.

    Augment each parameter \(x_1,x_2\), and \(x_3\) by \(\eta _1\) in nested loops from \(-1\) to 1, until one of the following cases happens:

    1. a.

      All \(x_i\) reach 1. If \(\beta <1\), set \(\beta \leftarrow \beta + \eta _2\), reset \(x_i \leftarrow -1\) for all i, and repeat step 2. If \(\beta =1\), then set \(\alpha \leftarrow \alpha + \eta _2\) and reset \(x_i \leftarrow -1\) for all i, and then set \(\beta \leftarrow \alpha \). Then repeat step 2.

    2. b.

      The assumptions of Lemma A.1 are satisfied for the current values of \(\alpha ,\beta ,x_1,x_2\), and \(x_3\). Accordingly, increment S by the quantity \(\mu _{ST}([\alpha - \eta _2,\alpha ])\mu _{ST}([\beta ,1])\) if \(\alpha \ne -1\). Then increment \(\alpha \leftarrow \alpha + \eta _2\) and reset \(x_1 ,x_2,x_3 \leftarrow -1\) and \(\beta \leftarrow \alpha \). Then repeat step 2.

  3. 3.

    When \(\alpha \) reaches 1, return the final value of S.

Here, 2S is a lower bound for the proportion of all closed subintervals \(I\subseteq [-1,1]\) in Sato–Tate measure that are \({{\,\mathrm{Sym}\,}}^8\)-minorizable. (As we only consider the case when \(\alpha < \beta \); such a sample space has Sato–Tate measure 1 / 2.) \(\square \)

Lemma A.4

If \(I=[\alpha ,\beta ]\subseteq [-1,1]\) has Sato–Tate measure \(\mu _{ST}(I) \ge 0.36\), then I is \({{\,\mathrm{Sym}\,}}^8\)-minorizable.


Modify the algorithm in the proof of Lemma A.3 as follows:

  1. 1.

    Each time that the assumptions of Lemma A.1 are satisfied for the current values of \(\alpha ,\beta ,x_1,x_2\), and \(x_3\), we replace S by the quantity \(\max \{S, \mu _{ST}([\alpha -\eta _2,\beta ])\}\) if \(\alpha \ne -1\).

  2. 2.

    Return the final value of S.

Here, S is the measure of the smallest interval (in our search space) which is \({{\,\mathrm{Sym}\,}}^8\)-minorizable. \(\square \)

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Gillman, N., Kural, M., Pascadi, A. et al. Patterns of primes in the Sato–Tate conjecture. Res. number theory 6, 9 (2020). https://doi.org/10.1007/s40993-019-0184-8

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  • Prime gaps
  • Sato–Tate conjecture
  • Symmetric power L-functions
  • Green–Tao theorem