We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Advertisement

Patterns of primes in the Sato–Tate conjecture

  • 13 Accesses

Abstract

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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Notes

  1. 1.

    This notation is introduced in [18].

References

  1. 1.

    Baker, R.C., Zhao, L.: Gaps between primes in Beatty sequences. Acta Arith. 172(3), 207–242 (2016)

  2. 2.

    Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi-Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011)

  3. 3.

    Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over \(Q\): wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001)

  4. 4.

    Clozel, L., Thorne, J.A.: Level raising and symmetric power functoriality, II. Ann. Math. (2) 181(1), 303–359 (2015)

  5. 5.

    Clozel, L., Thorne, J.A.: Level-raising and symmetric power functoriality, III. Duke Math. J. 166(2), 325–402 (2017)

  6. 6.

    Conrad, B., Diamond, F., Taylor, R.: Modularity of certain potentially Barsotti-Tate Galois representations. J. Am. Math. Soc. 12(2), 521–567 (1999)

  7. 7.

    Davenport, H.: Multiplicative Number Theory. Graduate Texts in Mathematics, vol. 74, 2nd edn. Springer, Berlin (1982)

  8. 8.

    David, C., Gafni, A., Malik, A., Prabhu, N., Turnage-Butterbaugh, C.: Extremal primes of elliptic curves without complex multiplication. arXiv:1807.05255, to appear in Proc. Am. Math. Soc.

  9. 9.

    Diamond, F.: On deformation rings and Hecke rings. Ann. Math. (2) 144(1), 137–166 (1996)

  10. 10.

    Dusart, P.: The \(k\)th prime is greater than \(k(\ln k+\ln \ln k-1)\) for \(k\ge 2\). Math. Comput. 68(225), 411–415 (1999)

  11. 11.

    Gallagher, P.X.: A large sieve density estimate near \(\sigma =1\). Invent. Math. 11, 329–339 (1970)

  12. 12.

    Gelbart, S., Jacquet, H.: A relation between automorphic representations of \({\rm GL}(2)\) and \({\rm GL}(3)\). Ann. Sci. École Norm. Sup. (4) 11(4), 471–542 (1978)

  13. 13.

    Goldston, D.A., Pintz, J., Yıldırım, C.Y.: Primes in tuples. I. Ann. Math. (2) 170(2), 819–862 (2009)

  14. 14.

    Humphries, P.: Standard zero-free regions for Rankin-Selberg L-functions via sieve theory. Math. Z. 292(3–4), 1105–1122 (2019). With an appendix by Farrell Brumley

  15. 15.

    Iwaniec, H., Kowalski, E.: Analytic Number Theory. American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence, RI (2004)

  16. 16.

    Kim, H.H.: Functoriality for the exterior square of \({\rm GL}_4\) and the symmetric fourth of \({\rm GL}_2\). J. Am. Math. Soc. 16(1), 139–183 (2003). With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak

  17. 17.

    Kim, H.H., Shahidi, F.: Functorial products for \({\rm GL}_2\times {\rm GL}_3\) and the symmetric cube for \({\rm GL}_2\). Ann. Math. (2) 155(3), 837–893 (2002). With an appendix by Colin J. Bushnell and Guy Henniart

  18. 18.

    Robert, J., Oliver, L., Thorner, J.: Effective log-free zero density estimates for automorphic l-functions and the Sato-Tate conjecture. Int. Math. Res. Notices 02, 55–88 (2018)

  19. 19.

    Maynard, J.: Small gaps between primes. Ann. Math. (2) 181(1), 383–413 (2015)

  20. 20.

    Molteni, Giuseppe: L-functions: Siegel-type theorems and structure theorems. PhD thesis, University of Milan, Milan, (1999)

  21. 21.

    Murty, M.R., Murty, V.K: A variant of the Bombieri-Vinogradov theorem. In: Number Theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, pp. 243–272. American Mathematical. Society, Providence, RI (1987)

  22. 22.

    Norton, K.K.: Upper bounds for sums of powers of divisor functions. J. Number Theory 40(1), 60–85 (1992)

  23. 23.

    Pintz, J.: Are there arbitrarily long arithmetic progressions in the sequence of twin primes? In: Barany, I. (ed.) An Irregular Mind, Janos Bolyai Mathematical Society, vol. 21, pp. 525–559. János Bolyai Mathematical Society, Budapest (2010)

  24. 24.

    Pintz, J.: Polignac numbers, conjectures of Erdõs on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture. In: Sander, J. (ed.) From Arithmetic to Zeta-Functions, pp. 367–384. Springer, Cham (2016)

  25. 25.

    Pintz, J.: Patterns of primes in arithmetic progressions. In: Elsholtz, C. (ed.) Number Theory-Diophantine Problems, Uniform Distribution and Applications, pp. 369–379. Springer, Cham (2017)

  26. 26.

    Polymath, D.H.J.: Variants of the Selberg sieve, and bounded intervals containing many primes. Res. Math. Sci. 1, 83 (2014)

  27. 27.

    Rademacher, H.: On the Phragmén-Lindelöf theorem and some applications. Math. Z., 72:192–204 (1959/1960)

  28. 28.

    Ramachandra, K.: A simple proof of the mean fourth power estimate for \(\zeta (1/2+it)\) and \(L(1/2+it,\, X)\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1(81–97), 1974 (1975)

  29. 29.

    Rouse, J.: Atkin-Serre type conjectures for automorphic representations on \({\rm GL}(2)\). Math. Res. Lett. 14(2), 189–204 (2007)

  30. 30.

    Rouse, J., Thorner, J.: The explicit Sato-Tate conjecture and densities pertaining to Lehmer-type questions. Trans. Am. Math. Soc. 369(5), 3575–3604 (2017)

  31. 31.

    Rudnick, Z., Sarnak, P.: Zeros of principal \(L\)-functions and random matrix theory. Duke Math. J. 81(2), 269–322 (1996). A celebration of John F. Nash, Jr

  32. 32.

    Soundararajan, K., Thorner, J.: Weak subconvexity without a Ramanujan hypothesis. Duke Math. J. 168(7), 1231–1268 (2019). With an appendix by Farrell Brumley

  33. 33.

    Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. Math. (2) 141(3), 553–572 (1995)

  34. 34.

    Thorner, J.: Bounded gaps between primes in Chebotarev sets. Res. Math. Sci. 1, 16 (2014)

  35. 35.

    Vatwani, A., Wong, P.-J.: Patterns of primes in Chebotarev sets. Int. J. Number Theory 13(7), 1651–1677 (2017)

  36. 36.

    Wiles, A.: Modular elliptic curves and Fermat’s last theorem. Ann. Math. (2) 141(3), 443–551 (1995)

  37. 37.

    Zhang, Y.: Bounded gaps between primes. Ann. Math. (2) 179(3), 1121–1174 (2014)

Download references

Author information

Correspondence to Nate Gillman.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Proof

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.

Proof

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.

Proof

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. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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