Strong complete minors in digraphs

Kostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_{\!\!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_{\!\!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $\overrightarrow{K}_{\!\!r}$ minor, and any tournament with minimum out-degree $\Omega(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_{\!\!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f\;:\;\mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $\overrightarrow{K}_{\!\!r}$ minor, but such a function exists when considering dichromatic number.

The explicit Sato–Tate conjecture for primes in arithmetic progressions

Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.

Nonvanishing for cubic L-functions

We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

A note on the Brown–Erdős–Sós conjecture in groups

We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some k, or the entire multiplication table of a certain large abelian group, as a subgrid. As a consequence, we show that triples systems coming from a finite group contain configurations with t triples spanning $ O(\sqrt t )$ vertices, which is the best possible up to the implied constant. We confirm that for all t we can find a collection of t triples spanning at most t + 3 vertices, resolving the Brown–Erdős–Sós conjecture in this context. The proof applies well-known arithmetic results including the multidimensional versions of Szemerédi’s theorem and the density Hales–Jewett theorem.This result was discovered simultaneously and independently by Nenadov, Sudakov and Tyomkyn [5], and a weaker result avoiding the arithmetic machinery was obtained independently by Wong [11].

Bounds for GL3L-functions in depth aspect

Let f be a Hecke–Maass cusp form for {\mathrm{SL}_{3}(\mathbb{Z})} and χ a primitive Dirichlet character of prime power conductor {\mathfrak{q}=p^{\kappa}}, with p prime. We prove the subconvexity boundL\Big{(}\frac{1}{2},\pi\otimes\chi\Big{)}\ll_{p,\pi,\varepsilon}\mathfrak{q}^{% 3/4-3/40+\varepsilon}for any {\varepsilon>0}, where the dependence of the implied constant on p is explicit and polynomial.

Mean value theorem connected with Fourier coefficients of Hecke-Maass forms for SL(m, ℤ)

Let F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF(n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish $$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$ where Q = xϑ−ϵ with some ϑ > 0, the implied constant depends on F, A, ϵ.

GENERALIZING THE TITCHMARSH DIVISOR PROBLEM

Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem [Formula: see text] where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: [Formula: see text] where Ck is a constant dependent on k and a, and the implied constant is dependent on k. We also apply it a question related to Artin's conjecture for primitive roots.

Two Linnik-type problems for automorphic L-functions

Let m ≥ 2 be an integer, and π an irreducible unitary cuspidal representation for GLm(), whose attached automorphic L-function is denoted by L(s, π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s, π) in the half-plane ℜs > 1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then the first sign change in the sequence {λπ(n)}n=1∞ occurs at some n ≪ Qπ1 + ϵ, where Qπ is the conductor of π, and the implied constant depends only on m and ϵ. This improves the previous bound with the above exponent 1 + ϵ replaced by m/2 + ϵ. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −(L′/L)(s, π) in the half-plane ℜs > 1.

REPRESENTATIONS OF INTEGERS BY TERNARY QUADRATIC FORMS

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space [Formula: see text], where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.

THE DENSITY OF RATIONAL POINTS ON NON-SINGULAR HYPERSURFACES, II

For any integers $d,n \geq 2$, let $X \subset \mathbb{P}^{n}$ be a non-singular hypersurface of degree $d$ that is defined over the rational numbers. The main result in this paper is a proof that the number of rational points on $X$ which have height at most $B$ is $O(B^{n - 1 + \varepsilon})$, for any $\varepsilon > 0$. The implied constant in this estimate depends at most upon $d$, $\varepsilon$ and $n$.