For each integer ℓ ≥ 1, we prove an unconditional upper bound on the size of the ℓ-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of Q of degree d, for any fixed d ∈ {2, 3, 4, 5} (with the additional restriction in the case d = 4 that the field be non-D 4 ). For sufficiently large ℓ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-D 4 ) and quintic fields with chosen splitting types at a finite set of primes.for every ε > 0. (Throughout, ε > 0 is allowed to be arbitrarily small (possibly taking a different value in different occurrences), and A ≪ B indicates that |A| ≤ cB for an implied constant c, which we allow in any instance to depend on ℓ, d, ε.)It is conjectured thatfor every ε > 0, but improving on the trivial bound (1.1) has proved difficult. (Impetus for this conjecture may be found in Duke [Duk98], Zhang [Zha05, page 10], and Brumer and Silverman [BS96, "Question CL(ℓ, d)"].) For K quadratic, Gauss's genus theory [Gau01] implies (1.2) in the case ℓ = 2. Recently, [BST + 17] obtained nontrivial upper bounds for 2-torsion in fields of degree d for all d ≥ 3, proving |Cl K [2]| ≪ D 0.2784...+ε K for d = 3, 4 and |Cl K [2]| ≪ D 1/2−1/2d+ε K for d ≥ 5. For ℓ = 3, after initial incremental improvement in [HV06], [Pie05], [Pie06] over the trivial bound (1.1) for quadratic fields, Ellenberg and Venkatesh proved [EV07, Prop. 3.4, Cor. 3.7] that (1.3) |Cl K [3]| ≪ D 1/3+ε K 2010 Mathematics Subject Classification. 11R29, 11N36 11R45 .

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

We prove a new effective Chebotarev density theorem for Galois extensions L/Q that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of L); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of L, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal L-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal L-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of L-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for -torsion in class groups, for all integers ≥ 1, applicable to infinite families of fields of arbitrarily large degree.

Abstract. In this paper we prove a discrete version of Tanaka's theorem [19] for the Hardy-Littlewood maximal operator in dimension n = 1, both in the non-centered and centered cases. For the non-centered maximal operator M we prove that, given a function f : Z → R of bounded variation,where Var(f ) represents the total variation of f . For the centered maximal operator M we prove that, given a function f : Z → R such that f ∈ ℓ 1 (Z),This provides a positive solution to a question of Haj lasz and Onninen [6] in the discrete one-dimensional case.