On the linear complexity of binary threshold sequences derived from Fermat quotients

We determine the linear complexity of a family of p ²-periodic binary threshold sequences derived from Fermat quotients modulo an odd prime p, where p satisfies ${2^{p-1} \not\equiv 1 ({\rm mod}\, {p^2})}$ . The linear complexity equals p ² ? p or p ² ? 1, depending whether ${p \equiv 1}$ or 3 (mod 4). Our research extends the results from previous work on the linear complexity of the corresponding binary threshold sequences when 2 is a primitive root modulo p ². Moreover, we present a partial result on their linear complexities for primes p with ${2^{p-1} \equiv 1 ({\rm mod} \,{p^2})}$ . However such so called Wieferich primes are very rare.     

On the least primitive root in number fields

Let K be an algebraic number field and $ \mathfrak{O} $ _K its ring of integers. For any prime ideal $ \mathfrak{p} $ , the group $ (\mathfrak{O}_K /\mathfrak{p})* $ of the reduced residue classes of integers is cyclic. We call any element of a generator of the group $ (\mathfrak{O}_K /\mathfrak{p})* $ a primitive root modulo $ \mathfrak{p} $ . Stimulated both by Shoup’s bound for the rational improvement and Wang and Bauer’s generalization of the conditional result of Wang Yuan in 1959, we give in this paper a new bound for the least primitive root modulo a prime ideal $ \mathfrak{p} $ under the Grand Riemann Hypothesis for algebraic number field. Our results can be viewed as either the improvement of the result of Wang and Bauer or the generalization of the result of Shoup.     

Distribution of the coefficients of modular forms and the partition function

Let ? be an odd prime and j, s be positive integers. We study the distribution of the coefficients of integer and half-integral weight modular forms modulo an odd positive integer M. As an application, we investigate the distribution of the ordinary partition function p(n) modulo ? ^j and prove that for each integer 1?≤ r?<?? ^j , $$\sharp\{1\le n\le X\ |\ p(n)\equiv r\pmod{\ell^j} \}\gg_{s,r,\ell^j} \frac{\sqrt X}{\log X}(\log\log X)^s.$$      

Mass formula and structure of self-dual codes over {{\bf Z}_{2^s}}

We look at the structure of and give a mass formula for self-dual codes over the ring ${{\bf Z}_{2^s}}$ of integers modulo 2 ^s . Together with earlier work on the case of odd primes, this completes the mass formula for self-dual codes for ${{\bf Z}_{p^s}}$ , for all prime numbers p and positive integers s.     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

Hilbert genus fields of biquadratic fields

The Hilbert genus field of the real biquadratic field K=Q(δ~(1/2),d~(1/2)is described by Yue(2010)and Bae and Yue(2011)explicitly in the case=2 or p with p=1 mod 4 a prime and d a squarefree positive integer.In this article,we describe explicitly the Hilbert genus field of the imaginary biquadratic field K=Q(δ~(1/2),d~(1/2)),whereδ=-1,-2 or-p with p=3 mod 4 a prime and d any squarefree integer.This completes the explicit construction of the Hilbert genus field of any biquadratic field which contains an imaginary quadratic subfield of odd class number.     

Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

In two previous papers,the first named author jointly with Florian Luca and Henryk Iwaniec,have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers.In this note,we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values,proving the density modulo 1 of the sequencem[∑((m2+1))(m2+1)(m≤n)]n.This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.     

The polynomial multidimensional Szemerédi Theorem along shifted primes

If $\vec q_1 ,...,\vec q_m $ : ? → ? ^? are polynomials with zero constant terms and E ? ? ^? has positive upper Banach density, then we show that the set E ∩ (E ? $\vec q_1 $ (p ? 1)) ∩ … ∩ (E ? $\vec q_m $ (p ? 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.     

Fourier coefficients of GL(N) automorphic forms in arithmetic progressions

We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all \({N \geq 2}\) , satisfy a central limit theorem in a suitable range, generalizing the case N = 2 treated by Fouvry et al. (Commentarii Math Helvetici, 2014). Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.     

A note on a kind of character sums over the short interval

Let p be a prime, χ denote the Dirichlet character modulo p and $L(p)=\{a\in {\mathbb{Z}}^{+}|(a,p)=1,a\bar{a}\equiv 1(\bmod\ p),|a-\bar{a}|\le H\}$ . We study the distribution of elements in the set L(p) in character over the short interval. In this paper, we use the analytic method and show the distribution property of $$ \sum\limits_{n\le N\atop n\in L(p)}\chi(n), $$ and give a non-trivial estimate.     

The Frobenius Complex

Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers ${\mathbb Z}$ , that is, for a sub-semigroup ?? of the non-negative integers ( ${\mathbb N}$ , +), we define the order by n ??_?? m if ${{m-n \in \Lambda}}$ . When ?? is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when ?? is generated by the integers {a, a?+?d, a?+?2d, . . . , a?+?(a?1)d}, the order complex is homotopy equivalent to a wedge of spheres.     

Pairwise Balanced Designs with Prescribed Minimum Dimension

The dimension of a linear space is the maximum positive integer d such that any d of its points generate a proper subspace. For a set K of integers at least two, recall that a pairwise balanced design $\operatorname{PBD}(v,K)$ is a linear space on v points whose lines (or blocks) have sizes belonging to K. We show that, for any prescribed set of sizes K and lower bound d on the dimension, there exists a $\operatorname{PBD}(v,K)$ of dimension at least d for all sufficiently large and numerically admissible v.     

The Gauss–Wilson theorem for quarter-intervals

We define a Gauss factorial N _n ! to be the product of all positive integers up to N that are relatively prime to n. It is the purpose of this paper to study the multiplicative orders of the Gauss factorials $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!$ for odd positive integers n. The case where n has exactly one prime factor of the form p≡1(mod4) is of particular interest, as will be explained in the introduction. A fundamental role is played by p with the property that the order of  $\frac{p-1}{4}!$ modulo p is a power of 2; because of their connection to two different results of Gauss we call them Gauss primes. Our main result is a complete characterization in terms of Gauss primes of those n of the above form that satisfy $\left\lfloor\frac{n-1}{4}\right\rfloor_{n}!\equiv 1\pmod{n}$ . We also report on computations that were required in the process.     

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Given ${\Gamma \subset \mathbb{Q}^*}$ a multiplicative subgroup and ${m \in \mathbb{N}^+}$ , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which ind _p Γ = m, where ind _p Γ = (p ? 1)/|Γ _p | and Γ _p is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi–Pappalardi, Lenstra, Moree, Murata, Wagstaff, and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.     

Iterates of increasing sequences of positive integers

It is proved that for fixed integers ${K \ge 2, D \ge 2, {\rm if} (D - 1) \mid K}$ , then there exists a unique increasing sequence (a(n)) _{n ≥ K} of positive integers such that $$\underset{K {\rm times}}{\underbrace{a ( a ( \dots a(a}}(n)) \dots)) = Dn$$ otherwise, there are uncountably many increasing sequences of positive integers (a(n)) satisfying this iterated functional equation. This generalizes recent results of Propp and Allouche–Rampersad–Shallit.     

On the concentration of points of polynomial maps and applications

For a polynomial ${f\in{\mathbb {F}}_p[X]}$ , we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve ${f(x)\equiv y\, ({\rm mod}\, p)}$ , where ${f\in{\mathbb {F}}_p[X]}$ is a polynomial of degree d?≥ 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.     

Oscillation criteria for a class of nonlinear fourth order neutral differential equations

In this paper, sufficient conditions are obtained for oscillation of a class of nonlinear fourth order mixed neutral differential equations of the form (E) $$\left( {\frac{1} {{a\left( t \right)}}\left( {\left( {y\left( t \right) + p\left( t \right)y\left( {t - \tau } \right)} \right)^{\prime \prime } } \right)^\alpha } \right)^{\prime \prime } = q\left( t \right)f\left( {y\left( {t - \sigma _1 } \right)} \right) + r\left( t \right)g\left( {y\left( {t + \sigma _2 } \right)} \right)$$ under the assumption $$\int\limits_0^\infty {\left( {a\left( t \right)} \right)^{\tfrac{1} {\alpha }} dt} = \infty .$$ where α is a ratio of odd positive integers. (E) is studied for various ranges of p(t).     

Scattering of wave maps from {{\mathbb {R}^{2+1}}} to general targets

We show that smooth, radially symmetric wave maps U from ${\mathbb {R}^{2+1}}$ to a compact target manifold (N, where ? _r U and ? _t U have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for ${(\lambda^{\prime} \in (0,1),}$ energy does not concentrate in the set $$K_{\frac{5}{8}T,\frac{7}{8}T}^{\lambda^{\prime}} = \{(x,t) \in \mathbb{R}^{2+1} \vert \quad|x| \leq \lambda^{\prime} t, t \in [(5/8)T,(7/8)T] \}.$$