Jack Buttcane

Research

My current research is on the use of techniques from classical analytic number theory in the study of higher-rank automorphic forms. I hope to apply these methods to classical equidistribution problems and problems related to polynomials and primes. My interests and methods are in analytic number theory, harmonic analysis on symmetric spaces, complex analysis, the spectral theory of automorphic forms, trace formulas, exponential sums, and L-functions.

Research statement.

NSF Grant DMS-1601919

Publications

• A translation of Hadamard’s Sur la distribution des zéros de la function ζ(s) et ses conséquences arithmétiques.
• Bessel functions on GL(n), II, on the arXiv.
• Bessel functions on GL(n), I, in the Journal of Functional Analysis (arXiv).
• On sums of hyper-Kloosterman sums, in Forum Mathematicum (arXiv).
• The arithmetic Kuznetsov formula on GL(3), II: The general case, in Algebra & Number Theory (arXiv).
• Subconvexity for L-functions of non-spherical cusp forms on GL(3), with V. Blomer, in Acta Arithmetica (arXiv).
• The arithmetic Kuznetsov formula on GL(3), I: The Whittaker case, in Revista Matemática Iberoamericana (arXiv).
• Kuznetsov, Petersson and Weyl on GL(3), II: The generalized principal series forms, in the Mathematische Annalen (arXiv).
• Kuznetsov, Petersson and Weyl on GL(3), I: The principal series forms, in the American Journal of Mathematics (arXiv).
• Weights, raising and lowering operators, and K-types for automorphic forms on SL(3,ℝ), (book chapter) with S. D. Miller (arXiv).
• Higher weight on GL(3), II: The cusp forms, in Algebra & Number Theory (arXiv).
• Plancherel distribution of Satake parameters of Maass cusp forms on GL₃, with F. Zhou, in International Mathematics Research Notices (arXiv).
• On the fourth moment of Hecke Maass forms and the Random Wave Conjecture, with R. Khan, in Compositio Mathematica (arXiv).
• Higher weight on GL(3), I: The Eisenstein series, in Forum Mathematicum (arXiv).
• Global decomposition of GL(3) Kloosterman sums and the spectral large sieve, with V. Blomer, in Crelle’s Journal (arXiv).
• On the subconvexity problem for L-functions on GL(3), with V. Blomer, in the Annales scientifiques de l’École normale supérieure (arXiv).
• The spectral Kuznetsov formula on SL(3), in the Transactions of the American Mathematical Society (arXiv).
• A mean value of a triple product of L-functions, with R. Khan, in Mathematische Zeitschrift (arXiv).
• Applications of the Kuznetsov formula on GL(3), II: the level aspect, with V. Blomer and P. Maga, in the Mathematische Annalen (arXiv).
• L⁴ norms of Hecke newforms of large level, with R. Khan, in the Mathematische Annalen (arXiv).
• A Sato-Tate law for GL(3), with V. Blomer and N. Raulf, in Commentarii Mathematici Helvetici.
• On Sums of SL(3,ℤ) Kloosterman Sums, in the Ramanujan Journal (arXiv).
• UCLA Doctoral Thesis — On Sums of SL(3,ℤ) Kloosterman Sums.
• A note on the Waring-Goldbach problem, in the Journal of Number Theory.