A remark on the Lang-Trotter and Artin conjectures

Murty, M.; Vatwani, Akshaa

doi:10.1090/proc/13900

A remark on the Lang-Trotter and Artin conjectures
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Proc. Amer. Math. Soc. 146 (2018), 3191-3202 Request permission

Abstract:

We use recent advances in sieve theory to show that conditional upon the generalized Elliott-Halberstam conjecture, at least one of the following is true:

Artin’s primitive root conjecture holds for all $a$ not equal to $\pm 1$ or a perfect square
The Lang-Trotter conjecture holds for all CM elliptic curves $E/\mathbb {Q}$ with rank $E(\mathbb {Q}) \geq 1$ and CM field $k \neq \mathbb {Q}(\omega ) , \mathbb {Q}(i)$, where $\omega$ is a primitive cube root of unity.

References

Emil Artin, Collected papers, Springer-Verlag, New York-Berlin, 1982. Edited by Serge Lang and John T. Tate; Reprint of the 1965 original. MR 671416
E. Bombieri, J. B. Friedlander, and H. Iwaniec, Primes in arithmetic progressions to large moduli, Acta Math. 156 (1986), no. 3-4, 203–251. MR 834613, DOI 10.1007/BF02399204
Rajiv Gupta and M. Ram Murty, A remark on Artin’s conjecture, Invent. Math. 78 (1984), no. 1, 127–130. MR 762358, DOI 10.1007/BF01388719
Rajiv Gupta and M. Ram Murty, Primitive points on elliptic curves, Compositio Math. 58 (1986), no. 1, 13–44. MR 834046
D. R. Heath-Brown, Almost-prime $k$-tuples, Mathematika 44 (1997), no. 2, 245–266. MR 1600529, DOI 10.1112/S0025579300012584
D. R. Heath-Brown, Artin’s conjecture for primitive roots, Quart. J. Math. Oxford Ser. (2) 37 (1986), no. 145, 27–38. MR 830627, DOI 10.1093/qmath/37.1.27
Christopher Hooley, On Artin’s conjecture, J. Reine Angew. Math. 225 (1967), 209–220. MR 207630, DOI 10.1515/crll.1967.225.209
S. Lang and H. Trotter, Primitive points on elliptic curves, Bull. Amer. Math. Soc. 83 (1977), no. 2, 289–292. MR 427273, DOI 10.1090/S0002-9904-1977-14310-3
Serge Lang and Hale Trotter, Frobenius distributions in $\textrm {GL}_{2}$-extensions, Lecture Notes in Mathematics, Vol. 504, Springer-Verlag, Berlin-New York, 1976. Distribution of Frobenius automorphisms in $\textrm {GL}_{2}$-extensions of the rational numbers. MR 0568299
James Maynard, Small gaps between primes, Ann. of Math. (2) 181 (2015), no. 1, 383–413. MR 3272929, DOI 10.4007/annals.2015.181.1.7
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
L. J. Mordell, On the rational solutions of indeterminate equations of the third and fourth degree, Proc. Cambridge Philos. Soc, 21 (1922), 179-192.
Yoichi Motohashi, An induction principle for the generalization of Bombieri’s prime number theorem, Proc. Japan Acad. 52 (1976), no. 6, 273–275. MR 422179
M. Ram Murty, Artin’s conjecture for primitive roots, Math. Intelligencer 10 (1988), no. 4, 59–67. MR 966133, DOI 10.1007/BF03023749
M. Ram Murty and V. Kumar Murty, A variant of the Bombieri-Vinogradov theorem, Number theory (Montreal, Que., 1985) CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 243–272. MR 894326
M. Ram Murty and A. Vatwani, A higher rank Selberg sieve with an additive twist and applications, Funct. Approx. Comment. Math., 2017, to appear.
M. Ram Murty and Akshaa Vatwani, Twin primes and the parity problem, J. Number Theory 180 (2017), 643–659. MR 3679820, DOI 10.1016/j.jnt.2017.05.011
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), Art. 12, 83. MR 3373710, DOI 10.1186/s40687-014-0012-7
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
A. Vatwani, A higher rank Selberg sieve and applications, Czechoslovak Mathematical Journal, 2017, to appear.
Akshaa Vatwani, Bounded gaps between Gaussian primes, J. Number Theory 171 (2017), 449–473. MR 3556693, DOI 10.1016/j.jnt.2016.07.008
Hartmut Siebert and Dieter Wolke, Über einige Analoga zum Bombierischen Primzahlsatz, Math. Z. 122 (1971), no. 4, 327–341. MR 409391, DOI 10.1007/BF01110168
Yitang Zhang, Bounded gaps between primes, Ann. of Math. (2) 179 (2014), no. 3, 1121–1174. MR 3171761, DOI 10.4007/annals.2014.179.3.7