Using work of the first author [S. Bettin, High moments of the Estermann function. Algebra Number Theory 47(3) (2018), 659–684], we prove a strong version of the Manin–Peyre conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^{2}\times \mathbb{P}^{2}$ with bihomogeneous coordinates $[x_{1}:x_{2}:x_{3}],[y_{1}:y_{2},y_{3}]$ and in $\mathbb{P}^{1}\times \mathbb{P}^{1}\times \mathbb{P}^{1}$ with multihomogeneous coordinates $[x_{1}:y_{1}],[x_{2}:y_{2}],[x_{3}:y_{3}]$ defined by the same equation $x_{1}y_{2}y_{3}+x_{2}y_{1}y_{3}+x_{3}y_{1}y_{2}=0$. We thus improve on recent work of Blomer et al [The Manin–Peyre conjecture for a certain biprojective cubic threefold. Math. Ann. 370 (2018), 491–553] and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type $\mathbf{A}_{1}$ and three lines (the other existing proof relying on harmonic analysis by Chambert-Loir and Tschinkel [On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math. 148 (2002), 421–452]). Together with Blomer et al [On a certain senary cubic form. Proc. Lond. Math. Soc. 108 (2014), 911–964] or with work of the second author [K. Destagnol, La conjecture de Manin pour une famille de variétés en dimension supérieure. Math. Proc. Cambridge Philos. Soc. 166(3) (2019), 433–486], this settles the study of the Manin–Peyre conjectures for this equation.

We prove that, for any finite set $A\subset \mathbb{Q}$ with $|AA|\leqslant K|A|$ and any positive integer $k$, the $k$-fold product set of the shift $A+1$ satisfies the bound

$$\begin{eqnarray}|\{(a_{1}+1)(a_{2}+1)\cdots (a_{k}+1):a_{i}\in A\}|\geqslant \frac{|A|^{k}}{(8k^{4})^{kK}}.\end{eqnarray}$$

$K$

$c\log |A|$

$c=c(k)$

$\unicode[STIX]{x1D6EC}$

We study the number of non-trivial simple zeros of the Dedekind zeta-function of a quadratic number field in the rectangle $\{\unicode[STIX]{x1D70E}+\text{i}t:0<\unicode[STIX]{x1D70E}<1,0<t<T\}$. We prove that such a number exceeds $T^{6/7-\unicode[STIX]{x1D700}}$ if $T$ is sufficiently large. This improves upon the classical lower bound $T^{6/11}$ established by Conrey et al [Simple zeros of the zeta function of a quadratic number field. I. Invent. Math.86 (1986), 563–576].

We answer a challenge posed in Booker [$L$-functions as distributions. Math. Ann. 363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.

In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura. The result of the integral weight case is a product of an $L$-function and Riemann zeta functions, while the half-integral weight case is an infinite summation over negative fundamental discriminants with certain infinite products. To calculate an explicit formula for such Rankin–Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

We consider sequences of the form $(a_{n}\unicode[STIX]{x1D6FC})_{n}$ mod 1, where $\unicode[STIX]{x1D6FC}\in [0,1]$ and where $(a_{n})_{n}$ is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all $\unicode[STIX]{x1D6FC}$ in the sense of Lebesgue measure, we say that $(a_{n})_{n}$ has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of $(a_{n})_{n}$. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine’s criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence $(a_{n})_{n}$ having large additive energy which, however, maintains the metric pair correlation property.

In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than $1/2$. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate which was greater than $0.499$. This remained essentially the state of the art until a 2015 breakthrough by Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy rate. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain’s extractors which shows that it in fact extracts from sources with min-entropy rate near $\frac{21}{44}=0.477\ldots .$ Moreover, we construct a variant of this extractor which we show extracts from sources with min-entropy rate near $4/9=0.444\ldots .$ While this min-entropy rate is inferior to Chattopadhyay and Zuckerman’s construction, our extractors have the advantage of exponential small error, which is important in some applications. The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev’s point–plane incidence estimate, which in turn relies on Kollár’s generalization of the Guth–Katz incidence theorem.

A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$. Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

Following Wigert, various authors, including Ramanujan, Gronwall, Erdős, Ivić, Schwarz, Wirsing and Shiu, determined the maximal order of several multiplicative functions, generalizing Wigert’s result

$$\begin{eqnarray}\max _{n\leqslant x}\log d(n)=\frac{\log x}{\log \log x}(\log 2+o(1)).\end{eqnarray}$$

$\log f(f(n))$

$f$

$f$

$r_{2}$

$$\begin{eqnarray}\max _{n\leqslant x}\log r_{2}(r_{2}(n))=\frac{\sqrt{\log x}}{\log \log x}(c/\sqrt{2}+o(1))\end{eqnarray}$$

$c>0$

We say that a finite metric space $X$ can be embedded almost isometrically into a class of metric spaces $C$ if for every $\unicode[STIX]{x1D716}>0$ there exists an embedding of $X$ into one of the elements of $C$ with the bi-Lipschitz distortion less than $1+\unicode[STIX]{x1D716}$. We show that almost isometric embeddability conditions are equal for the following classes of spaces.

1. (a) Quotients of Euclidean spaces by isometric actions of finite groups.

2. (b) $L_{2}$-Wasserstein spaces over Euclidean spaces.

3. (c) Compact flat manifolds.

4. (d) Compact flat orbifolds.

5. (e) Quotients of connected compact bi-invariant Lie groups by isometric actions of compact Lie groups. (This one is the most surprising.)

We call spaces which satisfy these conditions finite flat spaces. Since Markov-type constants depend only on finite subsets, we can conclude that connected compact bi-invariant Lie groups and their quotients have Markov type 2 with constant 1.

In this paper, we improve the moment estimates for the gaps between numbers that can be represented as a sum of two squares of integers. We consider a certain sum of Bessel functions and prove the upper bound for its mean value. This bound provides estimates for the $\unicode[STIX]{x1D6FE}$th moments of gaps for all $\unicode[STIX]{x1D6FE}\leqslant 2$.

We prove that for every sufficiently large integer $n$, the polynomial $1+x+x^{2}/11+x^{3}/111+\cdots +x^{n}/111\ldots 1$ is irreducible over the rationals, where the coefficient of $x^{k}$ for $1\leqslant k\leqslant n$ is the reciprocal of the decimal number consisting of $k$ digits which are each $1$. Similar results following from the same techniques are discussed.

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

We study Piatetski-Shapiro sequences $(\lfloor n^{c}\rfloor )_{n}$ modulo $m$, for non-integer $c>1$ and positive $m$, and we are particularly interested in subword occurrences in those sequences. We prove that each block $\in \{0,1\}^{k}$ of length $k<c+1$ occurs as a subword with the frequency $2^{-k}$, while there are always blocks that do not occur. In particular, those sequences are not normal. For $1<c<2$, we estimate the number of subwords from above and below, yielding the fact that our sequences are deterministic and not morphic. Finally, using the Daboussi–Kátai criterion, we prove that the sequence $\lfloor n^{c}\rfloor$ modulo $m$ is asymptotically orthogonal to multiplicative functions bounded by 1 and with mean value 0.

We show that if a permutation $\unicode[STIX]{x1D70B}$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ of the interval $[1,\unicode[STIX]{x1D70B}]$ is zero. As a consequence, we prove that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^{2}\geqslant 0.3995$. This is the first result determining the value of $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]$ for an asymptotically positive proportion of permutations $\unicode[STIX]{x1D70B}$. We further establish other general conditions on a permutation $\unicode[STIX]{x1D70B}$ that ensure $\unicode[STIX]{x1D707}[1,\unicode[STIX]{x1D70B}]=0$, including the occurrence in $\unicode[STIX]{x1D70B}$ of any interval of the form $\unicode[STIX]{x1D6FC}\oplus 1\oplus \unicode[STIX]{x1D6FD}$.