Our work owes its origin to a recent note of Ram Murty [‘Irrationality of zeros of the digamma function’, Number Theory in Memory of Eduard Wirsing (eds. H. Maier, R. Steuding and J. Steuding) (Springer, Cham, 2023), 237–243], in which he proves that all the zeros of the digamma function are irrational with at most one possible exception. We extend this investigation to higher-order polygamma functions.

In this work, we develop an integral representation for the partial L-function of a pair $\pi \times \tau $ of genuine irreducible cuspidal automorphic representations, $\pi $ of the m-fold covering of Matsumoto of the symplectic group $\operatorname {\mathrm {Sp}}_{2n}$ and $\tau $ of a certain covering group of $\operatorname {\mathrm {GL}}_k$, with arbitrary m, n and k. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Our global results are subject to certain conjectures, but when $k=1$ they are unconditional for all m. One possible future application of this work will be a Shimura-type lift of representations from covering groups to general linear groups. In a recent work, we used the present results in order to provide an analytic definition of local factors for representations of the m-fold covering of $\operatorname {\mathrm {Sp}}_{2n}$.

Let X be a smooth, projective and geometrically connected curve defined over a finite field ${\mathbb {F}}_q$ of characteristic p different from $2$ and $S\subseteq X$ a subset of closed points. Let $\overline {X}$ and $\overline {S}$ be their base changes to an algebraic closure of ${\mathbb {F}}_q$. We study the number of $\ell $-adic local systems $(\ell \neq p)$ in rank $2$ over $\overline {X}-\overline {S}$ with all possible prescribed tame local monodromies fixed by k-fold iterated action of Frobenius endomorphism for every $k\geq 1$. In all cases, we confirm conjectures of Deligne predicting that these numbers behave as if they were obtained from a Lefschetz fixed point formula. In fact, our counting results are expressed in terms of the numbers of some Higgs bundles.

In this article, we extend, with a great deal of generality, many results regarding the Hausdorff dimension of certain dynamical Diophantine coverings and shrinking target sets associated with a conformal iterated function system (IFS) previously established under the so-called open set condition. The novelty of the result we present is that it holds regardless of any separation assumption on the underlying IFS and thus extends to a large class of IFSs the previous results obtained by Beresnevitch and Velani [A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. of Math. (2) 164(3) (2006), 971–992] and by Barral and Seuret [The multifractal nature of heterogeneous sums of Dirac masses. Math. Proc. Cambridge Philos. Soc. 144(3) (2008), 707–727]. Moreover, it will be established that if S is conformal and satisfies mild separation assumptions (which are, for instance, satisfied for any self-similar IFS on $\mathbb {R}$ with algebraic parameters, no exact overlaps and similarity dimension smaller than $1$), then the classical result of Hill–Velani regarding the shrinking target problem associated with a conformal IFS satisfying the open set condition (and for which the Hausdorff measure was later computed by Allen and Barany [On the Hausdorff measure of shrinking target sets on self-conformal sets. Mathematika 67 (2021), 807–839]) can be extended.

Let $E/\mathbb {Q}$ be an elliptic curve and let p be a prime of good supersingular reduction. Attached to E are pairs of Iwasawa invariants $\mu _p^\pm $ and $\lambda _p^\pm $ which encode arithmetic properties of E along the cyclotomic $\mathbb {Z}_p$-extension of $\mathbb {Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $\mu _p^\pm =0$. We provide support for this conjecture by proving that for any $\ell \geq 0$, we have $\mu _p^\pm \leq 1$ for all but finitely many primes p with $\lambda _p^\pm =\ell $. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $\mu _p^\pm \leq 1$ holds on a density 1 set of good supersingular primes for E.

H. H. Chan, K. S. Chua and P. Solé [‘Quadratic iterations to $\pi $ associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc. 355(4) (2002), 1505–1520] found that, for each positive integer d, there are theta series $A_d, B_d$ and $C_d$ of weight one that satisfy the Pythagoras-like relationship $A_d^2=B_d^2+C_d^2$. In this article, we show that there are two collections of theta series $A_{b,d}, B_{b,d}$ and $C_{b,d}$ of weight one that satisfy $A_{b,d}^2=B_{b,d}^2+C_{b,d}^2,$ where b and d are certain integers.

We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.

We show that for any set D of at least two digits in a given base b, almost all even integers taking digits only in D when written in base b satisfy the Goldbach conjecture. More formally, if $\mathcal {A}$ is the set of numbers whose digits base b are exclusively from D, almost all elements of $\mathcal {A}$ satisfy the Goldbach conjecture. Moreover, the number of even integers in $\mathcal {A}$ which are less than X and not representable as the sum of two primes is less than $|\mathcal {A}\cap \{1,\ldots ,X\}|^{1-\delta }$.

Let E be an elliptic curve defined over $\mathbb {Q}$ with good ordinary reduction at a prime $p\geq 5$ and let F be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell–Weil group of E over the $\mathbb {Z}_{p}^2$-extension of F is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb {Z}_{p}^2$.

Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$. Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.

We compute primes $p \equiv 5 \bmod 8$ up to $10^{11}$ for which the Pellian equation $x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes $p\leqslant x$ is well approximated by

$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$

where $\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.

We prove structural results for measure-preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions, these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function, all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwiłł, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that a trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro averages.

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integer and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on the central characters. To do this, we use the local Gan–Gross–Prasad conjecture, the local Rankin–Selberg integrals and the local theta correspondence.

We develop the theory and algorithms necessary to be able to verify the strong Birch–Swinnerton-Dyer Conjecture for absolutely simple modular abelian varieties over ${\mathbf {Q}}$. We apply our methods to all 28 Atkin–Lehner quotients of $X_0(N)$ of genus $2$, all 97 genus $2$ curves from the LMFDB whose Jacobian is of this type and six further curves originally found by Wang. We are able to verify the strong BSD Conjecture unconditionally and exactly in all these cases; this is the first time that strong BSD has been confirmed for absolutely simple abelian varieties of dimension at least $2$. We also give an example where we verify that the order of the Tate–Shafarevich group is $7^2$ and agrees with the order predicted by the BSD Conjecture.

In [2], Pillay introduced definable Galois cohomology, a model-theoretic generalization of Galois cohomology. Let M be an atomic and strongly $\omega $-homogeneous structure over a set of parameters A. Let B be a normal extension of A in M. We show that a short exact sequence of automorphism groups $1 \to \operatorname {\mathrm {Aut}}(M/B) \to \operatorname {\mathrm {Aut}}(M/A) \to \operatorname {\mathrm {Aut}}(B/A) \to 1$ induces a short exact sequence in definable Galois cohomology. We also discuss compatibilities with [3]. Our result complements the long exact sequence in definable Galois cohomology developed in [4].

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$. Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G, corresponding to a nef and big $\mathbf {R}$-divisor, satisfying the Northcott property. By characterizing the zero locus of $\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.

Let (K, v) be a valued field and $\phi\in K[x]$ be any key polynomial for a residue-transcendental extension w of v to K(x). In this article, using the ϕ-Newton polygon of a polynomial $f\in K[x]$ (with respect to w), we give a lower bound for the degree of an irreducible factor of f. This generalizes the result given in Jakhar and Srinivas (On the irreducible factors of a polynomial II, J. Algebra 556 (2020), 649–655).

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

Let p be a fixed prime number, and let F be a global function field with characteristic not equal to p. In this article, we shall study the variation properties of the Sylow p-subgroups of the even K-groups in a p-adic Lie extension of F. When the p-adic Lie extension is assumed to contain the cyclotomic $\mathbb {Z}_p$-extension of F, we obtain growth estimate of these groups. We also establish a duality between the direct limit and inverse limit of the even K-groups.