Abstract: We present in this article a new method for dealing with automatic sequences. This method allows us to prove a Mobius randomness principle for automatic sequences from which we deduce the Sarnak conjecture for this class of sequences. Furthermore, we can show a prime number theorem for automatic sequences that are generated by strongly connected automata where the initial state is fixed by the transition corresponding to 0.

Abstract: In this paper we first establish new explicit estimates for Chebyshev's $\vartheta$-function. Applying these new estimates, we derive new upper and lower bounds for some functions defined over the prime numbers, for instance the prime counting function $\pi(x)$, which improve the currently best ones. Furthermore, we use the obtained estimates for the prime counting function to give two new results concerning the existence of prime numbers in short intervals.

Abstract: We use a variant of Vinogradov’s method to show that the density of the set of prime numbers p ≡ −1 mod 4 for which the class group of the imaginary quadratic number field Q( √−8p) has an element of order 16 is equal to 1/16, as predicted by the Cohen–Lenstra heuristics.

Abstract: We use pseudodeformation theory to study Mazur's Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for $\Gamma_0(N)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel's result on this rank and of Mazur's results on the structure of the Hecke algebra.

Abstract: Let p be a prime number. We study the slopes of \(U_p\)-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation happens over a definite quaternion algebra by Jacquet–Langlands correspondence; it generalizes a prior work of Jacobs (Slopes of compact hecke operators. Thesis, University of London, Imperial College, 2004) who treated the case of \(p=3\) with a particular level. In case when the modular forms have a finite character of conductor highly divisible by p, we improve the lower bound to show that the slopes of \(U_p\)-eigenvalues grow roughly like arithmetic progressions as the weight k increases. This is the first very positive evidence for Buzzard–Kilford’s conjecture on the behavior of the eigencurve near the boundary of the weight space, that is proved for arbitrary p and general level. We give the exact formula of a fraction of the slope sequence.

Abstract: Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-representations attached to Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties $\mathrm{mod}$ $p^m$, extending a previous result of M. Emerton and the authors which required $p$ to be unramified in $F$.

Abstract: We study pseudodeterministic constructions, i.e., randomized algorithms which output the same solution on most computation paths. We establish unconditionally that there is an infinite sequence {pn} of primes and a randomized algorithm A running in expected sub-exponential time such that for each n, on input 1|pn|, A outputs pn with probability 1. In other words, our result provides a pseudodeterministic construction of primes in sub-exponential time which works infinitely often. This result follows from a more general theorem about pseudodeterministic constructions. A property Q ⊆ {0,1}* is ϒ-dense if for large enough n, |Q ∩ {0,1}n| ≥ ϒ2n. We show that for each c > 0 at least one of the following holds: (1) There is a pseudodeterministic polynomial time construction of a family {Hn} of sets, Hn ⊆ {0,1}n, such that for each (1/nc)-dense property Q E DTIME(nc) and every large enough n, Hn ∩ Q ≠ ∅ or (2) There is a deterministic sub-exponential time construction of a family {H′n} of sets, H′n ∩ {0,1}n, such that for each (1/nc)-dense property Q E DTIME(nc) and for infinitely many values of n, H′n ∩ Q ≠ ∅. We provide further algorithmic applications that might be of independent interest. Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.

Abstract: In classical prime number theory several asymptotic relations are considered to be “equivalent” to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.

Abstract: Let p be a prime number and F a totally real number eld. For each prime p of F above p we construct a Hecke operator Tp acting on (modp m ) Katz Hilbert modular classes which agrees with the classical Hecke operator at p for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of F. Calegari and D. Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight 1 are unramied at p when (F : Q) = 2. Some partial and some conjectural results are obtained when (F : Q) > 2.

Abstract: We prove that for $n \gt 2$ there exists a quandle of cyclic type of size $n$ if and only if $n$ is a power of a prime number. This establishes a conjecture of S. Kamada, H. Tamaru and K. Wada. As a corollary, every finite quandle of cyclic type is an Alexander quandle. We also prove that finite doubly transitive quandles are of cyclic type. This establishes a conjecture of H. Tamaru.

Abstract: For a hyperbolic rational map f of degree at least two on the Riemann sphere, we obtain estimates for the number of primitive periodic orbits of f ordered by their multiplier, and establish equidistribution of the associated holonomies, both with power saving error terms.

Abstract: We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension pnm, where p is a prime number, m is a square-free natural number and gcd(p,m) = 1. We prove that, if n ≤ 5 or m is prime with m < p, then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than 1125 is group-theoretical.

Abstract: We recast Euclid's proof of the infinitude of prime numbers as a Euclidean criterion for a domain to have infinitely many atoms. We make connections with Furstenberg's “topological“ proof of the infinitude of prime numbers and show that our criterion applies even in certain domains in which not all nonzero nonunits factor into products of irreducibles.

Abstract: In this paper we discuss the generalizations of the concept of Chebyshev's bias from two perspectives. First we give a general framework for the study of prime number races and Chebyshev's bias attached to general $L$-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet $L$-functions and Hasse--Weil $L$-functions of elliptic curves over $\mathbf{Q}$. This also apply to new Chebyshev's bias phenomena that were beyond the reach of the previously known cases. In addition we weaken the required hypotheses such as GRH or linear independence properties of zeros of $L$-functions. In particular we establish the existence of the logarithmic density of the set $\lbrace x\geq 2 : \sum_{p\leq x} \lambda_{f}(p) \geq 0 \rbrace$ for coefficients $(\lambda_{f}(p))$ of general $L$-functions conditionally on a much weaker hypothesis than was previously known.

Abstract: Recently, it was shown that the dithered relative prime interleavers and quadratic permutation polynomial (QPP) interleavers can be expressed in terms of almost regular permutation (ARP) interleavers. In this paper, the conditions for a QPP interleaver to be equivalent to an ARP interleaver are extended for cubic permutation polynomial (CPP) interleavers. It is shown that the CPP interleavers are always equivalent to an ARP interleaver with disorder degree greater than one and smaller than the interleaver length, when the prime factorization of the interleaver length contains at least one prime number to a power higher than one and it fulfills the conditions for which there are true CPPs for the considered length. When the prime factorization of the interleaver length contains only prime numbers to the power of one, with at least two prime numbers $p_{i}$ , fulfilling the conditions $p_{i}>3$ and $3 mid (p_{i}-1)$ , values of disorder degree smaller than the interleaver length are possible under some conditions on the coefficients of the second and third degree terms of the CPP.

Abstract: This chapter discusses the basis for the ring of polynomials integer-valued on prime numbers. It also describes the search for a canonical basis and presents a proposition to check whether a polynomial which takes integral values on prime numbers necessarily takes integral values at other integers.

Abstract: — Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group Γ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting σ be a generator of Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic action of σ on Γ, focusing on the critical role played by the fixed points of this action, and their relation to the ramification in L/K. The method of Boston works only when there are no non-trivial fixed points for this action. We show that even in the presence of arbitrarily many fixed points, the action of σ places severe arithmetic conditions on the existence of finitely and tamely ramified uniform p-adic analytic extensions over K, which in some instances leads us to be able to deduce the non-existence of such extensions over K from their non-existence over k.

Abstract: ‎‎In this paper‎, ‎we prove that Ree group ${}^2G_2(q)$‎, ‎where $qpmsqrt{3q}+1$ is a prime number can be uniquely determined by the order of group and the number of elements with the same order‎.

Abstract: We consider the Diophantine inequality \[ \left| p_1^{c} + p_2^{c} + p_3^c- N \right| 0$ is an arbitrarily large constant. We prove that the above inequality has a solution in primes $p_1$, $p_2$, $p_3$ such that each of the numbers $p_1 + 2, p_2 + 2, p_3 + 2$ has at most $\left[ \frac{369}{180 - 168 c} \right]$ prime factors, counted with the multiplicity.

Abstract: Let p be a prime number. We study the dimensions of Brauer constructions of Young and Young permutation modules with respect to p-subgroups of the symmetric groups. They depend only on partitions labelling the modules and the orbits of the action of the p-subgroups, and are related to their generic Jordan types. We obtain some reductive formulae and, in the case of two-part partitions, make some explicit calculation.

Abstract: In this paper, we generalize Mauduit and Rivat's theorem on the Rudin-Shapiro sequence. Weakening the hypothesis needed in their theorem, we prove a prime number theorem for a large class of functions defined on the digits. Our result covers the case of generalized Rudin-Shapiro sequences as well as bloc-additive sequences on finite and infinite expansions. We also give a partial answer to a question posed by Kalai.