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1. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa; Teräväinen, Joni (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract A conjecture of Erdős states that, for any large primeq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be written as $p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with $p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any $\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least $\left(\frac{2}{3}-\epsilon \right)\phi \left(q\right)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes $\left(\mathrm{mod}q\right)${(\operatorname{mod}q)}can be represented as a product $p_1{p}_2${p_{1}p_{2}}of two primes $p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of ${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character $\psi \left(\mathrm{mod}q\right)${\psi~{}(\operatorname{mod}\,q)}such that $\psi \left(p\right)=-1${\psi(p)=-1}for very many primes $p\le q${p\leq q}. 

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2. Supersingular elliptic curves over ℤ _𝑝 -extensions

   https://doi.org/10.1515/crelle-2023-0029  

   Çiperiani, Mirela (June 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $\mathrm{E}/\mathbb{Q}$\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for $\mathrm{E}$\mathrm{E}.Consider a quadratic extension $L/{\mathbb{Q}}_p$L/\mathbb{Q}_{p}and the corresponding anticyclotomic ${\mathbb{Z}}_p$\mathbb{Z}_{p}-extension $L_{\mathrm{\infty }}/L$L_{\infty}/L.We analyze the structure of the points $\mathrm{E}\left(L_{\mathrm{\infty }}\right)$\mathrm{E}(L_{\infty})and describe two global implications of our results. 

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3. Diffeomorphism groups of prime 3-manifolds

   https://doi.org/10.1515/crelle-2023-0069  

   Bamler, Richard H; Kleiner, Bruce (October 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract LetXbe acompact orientable non-Haken 3-manifold modeled on the Thurston geometry $\mathrm{Nil}${\operatorname{Nil}}. We show that the diffeomorphism group $\mathrm{Diff}\left(X\right)${\operatorname{Diff}(X)}deformation retracts to the isometry group $\mathrm{Isom}\left(X\right)${\operatorname{Isom}(X)}. Combining this with earlier work by many authors, this completes the determination the homotopy type of $\mathrm{Diff}\left(X\right)${\operatorname{Diff}(X)}for any compact, orientable, prime 3-manifoldX. 

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4. Negative moments of the Riemann zeta-function

   https://doi.org/10.1515/crelle-2023-0091  

   Bui, Hung M; Florea, Alexandra (January 2024, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Assuming the Riemann Hypothesis, we study negative moments of the Riemann zeta-function and obtain asymptotic formulas in certain ranges of the shift in $\zeta \left(s\right)${\zeta(s)}. For example, integrating ${\left|\zeta \left(\frac{1}{2}+\alpha +it\right)\right|}^{-2k}${|\zeta(\frac{1}{2}+\alpha+it)|^{-2k}}with respect totfromTto $2T${2T}, we obtain an asymptotic formula when the shift α is roughly bigger than $\frac{1}{\log T}${\frac{1}{\log T}}and $k<\frac{1}{2}${k<\frac{1}{2}}. We also obtain non-trivial upper bounds for much smaller shifts, as long as $\log \frac{1}{\alpha }\ll \log \log T${\log\frac{1}{\alpha}\ll\log\log T}. This provides partial progress towards a conjecture of Gonek on negative moments of the Riemann zeta-function, and settles the conjecture in certain ranges. As an application, we also obtain an upper bound for the average of the generalized Möbius function. 

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5. On Waring’s problem for larger powers

   https://doi.org/10.1515/crelle-2023-0072  

   Brüdern, Jörg; Wooley, Trevor D. (November 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let $G\left(k\right)${G(k)}denote the least numbershaving the property that everysufficiently large natural number is the sum of at mostspositive integralk-th powers.Then for all $k\in \mathbb{N}${k\in\mathbb{N}}, one has $G\left(k\right)⩽\lceil k\left(\log k+4.20032\right)\rceil .$G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when $k⩾14${k\geqslant 14}. 

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