We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We show that no $2$-superirreducible polynomials exist in $\mathbb F[t]$ when $p=2$ and that no such polynomials of odd degree exist when $p$ is odd. We address the remaining case in which $p$ is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree $d$. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

We prove distributional results for mixed character sums \begin{equation*} \sum_{n\le x }χ(n)e(nθ), \end{equation*} for fixed $θ\in [0,1]$ and random character $χ\pmod q$, as well as for a fixed character $χ$ and randomly sampled $θ\in [0,1].$ We present various applications of our results. For example, we construct Littlewood polynomials with large Mahler measure, thus establishing a new record in the Mahler problem (1963). We also show that $L_{2k}$ norms of well-known Turyn polynomials are asymptotically minimized at the shift $α=1/4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems.

In this note we prove (under mild hypotheses) that f is a nontrivial character of 𝔽_p if and only if the Fourier transform of f has magnitude 1 somewhere in 𝔽_p^×. This implies a converse to a theorem of Gauss on the magnitude of the Gauss sum, in addition to other consequences. The common theme in all our results is that extremal behavior on the Fourier side imposes multiplicative structure on the physical side.

Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in \mathbb{Z}[t]$ and any $\unicode[STIX]{x1D700}>0$, there are infinitely many $n\in \mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^{\unicode[STIX]{x1D700}}$.

We formulate a precise conjecture that, if true, extends the converse theorem of Hecke without requiring hypotheses on twists by Dirichlet characters or an Euler product. The main idea is to linearize the Euler product, replacing it by twists by Ramanujan sums. We provide evidence for the conjecture, including proofs of some special cases and under various additional hypotheses.

We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating quadratic exponential sums. In addition, we use a new simple multi-evaluation method to compute the zeta function in a very small range at little more than the cost of evaluation at a single point.

It is well-known that cancellation in short character sums (e.g. Burgess’ estimates) yields bounds on the least quadratic nonresidue. Scant progress has been made on short character sums since Burgess’ work, so it is desirable to find another approach to nonresidues. In this note we formulate a new line of attack on the least nonresidue via long character sums, an active area of research. Among other results, we demonstrate that improving the constant in the Pólya–Vinogradov inequality would lead to significant progress on nonresidues. Moreover, conditionally on a conjecture on long character sums, we show that the least nonresidue for any odd primitive character (mod k) is bounded by \((\log k)^{1.4}\).

We describe a method for bounding the rank of an elliptic curve under the assumptions of the Birch and Swinnerton-Dyer conjecture and the generalized Riemann hypothesis. As an example, we compute, under these conjectures, exact upper bounds for curves which are known to have rank at least as large as 20, 21, 22, 23, and 24. For the known curve of rank at least 28, we get a bound of 30.

We obtain explicit bounds on the moments of character sums, refining estimates of Montgomery and Vaughan. As an application we obtain results on the distribution of the maximal magnitude of character sums normalized by the square root of the modulus, finding almost double exponential decay in the tail of this distribution.

We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities.

We give a complete classification of a certain family of step functions related to the Nyman–Beurling approach to the Riemann hypothesis and previously studied by Vasyunin. Equivalently, we completely describe when certain sequences of ratios of factorial products are always integral. Essentially, once certain observations have been made, this comes down to an application of Beukers and Heckman's classification of the monodromy of the hypergeometric function nFn−1. We also note applications to the classification of cyclic quotient singularities.

We study certain step functions whose nonnegativity is related to the integrality of sequences of ratios of factorial products. In particular, we obtain a lower bound for the mean square of such step functions which allows us to give a restriction on when such a factorial ratio sequence can be integral. Additionally, we note that this work has applications to the classification of cyclic quotient singularities.