For a fixed non-integral α>1, let PS(α)={⌊nα⌋:n=1,2,…}. We show that x+y=z has only finitely many solutions (x,y,z)∈PS(α)3 for almost every α>3. Furthermore, we show that PS(α) contains only finitely many arithmetic progressions of length 3 for almost every α>10. In addition, we give upper bounds for the Hausdorff dimension of the set of α∈[s,t] such that y=a1x1+⋯+anxn has infinitely many solutions in PS(α).

Let f be a normalized newform of weight 2 on Γ0(N) whose coefficients lie in Q and let χM be a primitive quadratic Dirichlet character with conductor M. Under mild assumptions on M, we give a sharp lower bound for the 2-adic valuation of the algebraic part of the L-value L(f,χM,1) and evaluate the 2-adic valuation for infinitely many M.

Let D>3, D≡3(mod4) be a prime, and let C be an ideal class in the field Q(−D−−−√). We give a new proof that p(D,C), the smallest norm of a split prime p∈C, satisfies p(D,C)≪DL for some absolute constant L. Our proof is sieve-theoretic. In particular, this allows us to avoid the use of log-free zero-density estimates (for class group L-functions) and the repulsion properties of exceptional zeros, two crucial inputs to previous proofs of this result.

Let f∈L1(N), where N=L∞(Gm)⊗ˉM, Gm is a bounded Vilenkin group and M is a semifinite von Neumann algebra. We prove the noncommutative weak type maximal inequality ∥(Dn(f))n≥1∥Λ1,∞(N,ℓ∞)≤C∥f∥L1(N), where Dn(f) represents the Vilenkin derivative of the integral function If. The main strategy in the proof is to exploit the recent advances on the noncommutative Calderón–Zygmund decomposition established by Cadilhac, Conde-Alonso and Parcet.

We prove lower bounds of the form ≫N/(logN)3/2 for the number of primes up to N primitively represented by a shifted positive definite integral binary quadratic form, and under the additional condition that primes are from an arithmetic progression. This extends the sieve methods of Iwaniec, who showed such lower bounds without the primitivity and congruence conditions. Imposing primitivity adds some subtle\-ties to the local criteria for representation of a shifted prime: for example, some shifted quadratic forms of discriminant 5(mod8) do not primitively represent infinitely many primes. We also provide a careful list of the local conditions under which a genus of an integral binary quadratic form represents an integer, verified by computer, and correcting some minor errors in previous statements. The motivation for this work is as a tool for the study of prime components in Apollonian circle packings.

Let K/Q be a Galois extension of number fields. We study the ideal classes of primes p of K of residue degree bigger than 1 in the class group of K. In particular, we explore those extensions K/Q for which there exists an integer f>1 such that the ideal classes of primes p of K of residue degree f generate the full class group of K. We show that there are many such fields. Then we use this approach to obtain information on the class group of K, like the rank of the ℓ-torsion subgroup of the class group, factors of the class number, fields with class group of certain exponents, and even structure of the class group in some cases. Moreover, such f can be used to construct annihilators of the class groups. In fact, for any extension K/F (even non-abelian), if the class group of K is generated by primes of relative degree f for the extension K/F and the class group of F is trivial, this method can be used to construct ‘relative’ annihilators.

Let p>3 be a prime, a1,a2,a3∈Z, and let Np(x3+a1x2+a2x+a3) denote the number of solutions to the congruence x3+a1x2+a2x+a3≡0(modp). We give an explicit criterion for Np(x3+a1x2+a2x+a3)=3 via binary quadratic forms.

We give an upper bound for the distribution of base-a pseudoprimes that is uniform in the base and does not require coprimality to the base. In addition we show that there are infinitely many “near Carmichael numbers” meaning that they are pseudoprimes for a positive proportion of bases, but not all bases.

A proper Euler’s magic matrix is an integer n×n matrix M∈Zn×n such that M⋅Mt=γ⋅I for some nonzero constant γ, the sum of the squares of the entries along each of the two main diagonals equals γ, and the squares of all entries in M are pairwise distinct. Euler constructed such matrices for n=4. In this work, we use multiplication matrices of the octonions to construct examples for n=8, and prove that no such matrix exists for n=3.

We realize infinitely many covering groups 2.An (where An is the alternating group) as the Galois groups of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works investigating special cases or proving conditional results in this direction, these are the first unramified realizations of infinitely many of these groups.