Abstract We study lower bounds for the Riemann zeta function ζ ⁢ ( s ) {\zeta(s)} along vertical arithmetic progressions in the right-half of the critical strip. We show that the lower bounds obtained in the discrete case coincide, up to the constants in the exponential, with the ones known for the continuous case, that is when the imaginary part of s ranges on a given interval. Our methods are based on a discretization of the resonance method for estimating extremal values of ζ ⁢ ( s ) {\zeta(s)} .

Abstract We show that, for any given q ≥ 0 {q\geq 0} , any Sasakian structure on a closed manifold M is approximated in the C q {C^{{q}}} -norm by structures induced by CR embeddings into weighted Sasakian spheres. In order to obtain this result, we also strengthen the approximation of an orbifold Kähler form by projectively induced ones given in [J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differential Geom. 88 2011, 1, 109–159] in the C 0 {C^{0}} -norm to a C q {C^{{q}}} -approximation.

Abstract In this paper, the weighted estimates for multilinear pseudo-differential operators were systematically studied in rearrangement invariant Banach and quasi-Banach spaces. These spaces contain the Lebesgue space, the classical Lorentz space and Marcinkiewicz space as typical examples. More precisely, the weighted boundedness and weighted modular estimates, including the weak endpoint case, were established for multilinear pseudo-differential operators and their commutators. As applications, we show that the above results also hold for the multilinear Fourier multipliers, multilinear square functions, and a class of multilinear Calderón–Zygmund operators.

Abstract Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

Abstract Relative Rota–Baxter groups are generalizations of Rota–Baxter groups and have been introduced recently in the context of Lie groups. In this paper, we explore connections of relative Rota–Baxter groups with skew left braces, which are well known to give bijective non-degenerate set-theoretical solutions of the Yang–Baxter equation. We prove that every relative Rota–Baxter group gives rise to a skew left brace, and conversely, every skew left brace arises from a relative Rota–Baxter group. It turns out that there is an isomorphism between the two categories under some mild restrictions. We propose an efficient GAP algorithm, which would enable the computation of relative Rota–Baxter operators on finite groups. In the end, we introduce the notion of isoclinism of relative Rota–Baxter groups and prove that an isoclinism of these objects induces an isoclinism of corresponding skew left braces.

Abstract Recently, Guo and Tang independently established some q-supercongruences from Rahman’s quadratic transformation. In this paper, by applying the method of creative microscoping devised by Guo and Zudilin together with Rahman’s quadratic transformation again, we provide proofs for eight conjectures on q-supercongruences proposed by Guo and by Tang.

Abstract We study the Moore–Penrose inverse of perturbations by a proper symmetrically-normed ideal of a closed range operator on a Hilbert space. We show that the notion of essential codimension of projections gives a characterization of subsets of such perturbations in which the Moore–Penrose inverse is continuous with respect to the metric induced by the operator ideal. These subsets are maximal satisfying the continuity property, and they carry the structure of real analytic Banach manifolds, which are acted upon transitively by the Banach–Lie group consisting of invertible operators associated with the ideal. This geometric construction allows us to prove that the Moore–Penrose inverse is indeed a real bianalytic map between infinite-dimensional manifolds. We use these results to study the polar decomposition of closed range operators from a similar geometric perspective. At this point we prove that operator monotone functions are real analytic in the norm of any proper symmetrically-normed ideal. Finally, we show that the maps defined by the operator modulus and the polar factor in the polar decomposition of closed range operators are real analytic fiber bundles.

Abstract This paper is devoted to studying the pointwise convergence problem and nonlinear smoothing of the generalized Zakharov–Kuznetsov equation. Firstly, we present an alternative proof of Theorem 1.5 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936] and Theorem 1.8 of Linares and Ramos [The Cauchy problem for the L 2 L^{2} -critical generalized Zakharov–Kuznetsov equation in dimension 3, Comm. Partial Differential Equations 46 (2021), 9, 1601–1627]. Secondly, we give an alternative proof of Theorem 1.1 of Ribaud and Vento [A note on the Cauchy problem for the 2D generalized Zakharov–Kuznetsov equations, C. R. Math. Acad. Sci. Paris 350 (2012), 9–10, 499–503] and present the nonlinear smoothing and uniform convergence of two-dimensional generalized Zakharov–Kuznetsov equation. Thirdly, we give an alternative proof of Theorem 1.4 of Linares and Ramos [Maximal function estimates and local well-posedness for the generalized Zakharov–Kuznetsov equation, SIAM J. Math. Anal. 53 (2021), 1, 914–936]. Finally, we study the nonlinear smoothing and uniform convergence of 𝑛-dimensional generalized Zakharov–Kuznetsov equation with n ≥ 3 n\geq 3 .

Abstract LetG⁢(g;x):=∑n≤xg⁢(n){G(g;x):=\sum_{n\leq x}g(n)}be the summatory function of an arithmetical functiong⁢(n){g(n)}. In this paper, we prove that we can write weighted averages of an arbitrary fixed numberNof arithmetical functionsgj⁢(n),j∈{1,…,N}{g_{j}(n),\,j\in\{1,\dots,N\}}as an integral involving the convolution (in the sense of Laplace) ofGj⁢(x){G_{j}(x)},j∈{1,…,N}{j\in\{1,\dots,N\}}. Furthermore, we prove an identity that allows us to obtain known results about averages of arithmetical functions in a very simple and natural way, and overcome some technical limitations for some well-known problems.

Abstract Define an infinite matrix D α = ( d n , v α ) \mathfrak{D}^{\alpha}=(d^{\alpha}_{n,v}) by d n , v α = { v α σ ( α ) ⁢ ( n ) , v ∣ n , 0 , v ∤ n , d^{\alpha}_{n,v}=\begin{cases}\dfrac{v^{\alpha}}{\sigma^{(\alpha)}(n)},&v\mid n,\\ 0,&v\nmid n,\end{cases} where σ ( α ) ⁢ ( n ) \sigma^{(\alpha)}(n) is defined to be the sum of the 𝛼-th power of the positive divisors of n ∈ N n\in\mathbb{N} , and construct the matrix domains ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) ( 0 < p < ∞ 0<p<\infty ), c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) defined by the matrix D α \mathfrak{D}^{\alpha} . We develop Schauder bases and determine 𝛼-, 𝛽- and 𝛾-duals of these new spaces. We characterize some matrix transformation from ℓ p ⁢ ( D α ) \ell_{p}(\mathfrak{D}^{\alpha}) , c 0 ⁢ ( D α ) c_{0}(\mathfrak{D}^{\alpha}) , c ⁢ ( D α ) c(\mathfrak{D}^{\alpha}) and ℓ ∞ ⁢ ( D α ) \ell_{\infty}(\mathfrak{D}^{\alpha}) to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} and ℓ 1 \ell_{1} . Furthermore, we determine some criteria for compactness of an operator (or matrix) from X ∈ { ℓ p ⁢ ( D α ) , c 0 ⁢ ( D α ) , c ⁢ ( D α ) , ℓ ∞ ⁢ ( D α ) } X\in\{\ell_{p}(\mathfrak{D}^{\alpha}),c_{0}(\mathfrak{D}^{\alpha}),c(\mathfrak{D}^{\alpha}),\ell_{\infty}(\mathfrak{D}^{\alpha})\} to ℓ ∞ \ell_{\infty} , 𝑐, c 0 c_{0} or ℓ 1 \ell_{1} .