Abstract
Fermat showed that every prime p = 1 mod 4 is a sum of two squares: p = a2 + b2. To any of the 8 possible representations (a, b) we associate an angle whose tangent is the ratio b/a. In 1919 Hecke showed that these angles are uniformly distributed as p varies, and in the 1950’s Kubilius proved uniform distribution in somewhat short arcs. We study fine scale statistics of these angles, in particular the variance of the number of such angles in a short arc. We present a conjecture for this variance, motivated both by a random matrix model, and by a function field analogue of this problem, for which we prove an asymptotic form for the corresponding variance.
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References
L. Bary-Soroker, Y. Smilansky and A. Wolf, On the function field analogue of Landau’s theorem on sums of squares, Finite Fields and their Applications 39 (2016), 195–215.
H. M. Bui, J. P. Keating and D. J. Smith, On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class, Journal fo the London Mathematical Society 94 (2016), 161–185.
F. J. Dyson, Statistical theory of the energy levels of complex systems, I, II and III, Journal of Mathematical Physics 3 (1962), 140–175.
G. Harman and P. A. Lewis, Gaussian primes in narrow sectors, Mathematika 48 (2001), 119–135.
E. Hecke, Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen. I, Mathematische Zeitschrift 1 (1918), 357–376; II, Mathematische Zeitschrift 6 (1920), 11–51.
H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, Vol. 53, American Mathematical Society, Providence, RI, 2004.
N. M. Katz, Witt vectors and a question of Rudnick and Waxman, International Mathematics Research Notices 2017 (2017), 3377–3412.
J. P. Keating and B. E. Odgers, Symmetry transitions in random matrix theory & L-functions, Communications in Mathematical Physics 281 (2008), 499–528.
J. Keating and Z. Rudnick, The variance of the number of prime polynomials in short intervals and in residue classes, International Mathematics Research Notices 2014 (2014), 259–288.
F. B. Koval’chik, Density theorems for sectors and progressions, Lietuvos Matematikos Rinkinys 15 (1975), 133–151.
I. Kubilyus, The distribution of Gaussian primes in sectors and contours, Leningradskiĭ Gosudarstvennyĭ Ordena Lenina Universitet imeni A. A. Zhdanova. Uchenye Zapiski. Seriya Matematicheskikh Nauk 137 (1950), 40–52.
J. Kubilius, On a problem in the n-dimensional analytic theory of numbers, Vilniaus Valst. Univ. Mokslo Darbai. Mat. Fiz. Chem. Mokslu Ser. 4 (1955), 5–43.
P. Lévy, Sur la division d’un segment par des points choisis au hasard, Comptes rendus hebdomadaires des séances de l’Académie des Sciences 208 (1939), 147–149.
M. Maknys, Zeros of Hecke Z-functions and the distribution of primes of an imaginary quadratic field, Lithuanian Mathematical Journal 15 (1975), 140–149.
M. Maknys, Density theorems for Hecke Z functions and the distribution of primes of an imaginary quadratic field, Lithuanian Mathematical Journal 16 (1976), 105–110.
M. Maknys, Refinement of the remainder term in the law of the distribution of prime numbers of an imaginary quadratic field in sectors, Lithuanian Mathematical Journal 17 (1977), 90–93.
O. Parzanchevski and P. Sarnak, Super-golden-gates for PU(2), Advances in Mathematics 327 (2018), 869–901.
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Rudnick, Z., Waxman, E. Angles of Gaussian primes. Isr. J. Math. 232, 159–199 (2019). https://doi.org/10.1007/s11856-019-1867-5
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DOI: https://doi.org/10.1007/s11856-019-1867-5