Journal of Number Theory最新文献

For a prime p larger than 7, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight $k\ge 4$ for the full modular group as the modular forms for which the first $\dim \left(M_k\right)$ Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.

For $j=1,2$, let $f_j\left(z\right)={\sum }_{n=1}^{\infty }{a}_j\left(n\right)e^{2\pi inz}$ be a holomorphic, non-CM cuspidal newform of even weight $k_j\ge 2$ with trivial nebentypus. For each prime p, let ${\theta }_j\left(p\right)\in [0,\pi ]$ be the angle such that $a_j\left(p\right)=2p^{(k-1)/2}\cos {\theta }_j\left(p\right)$. The now-proven Sato–Tate conjecture states that the angles $\left({\theta }_j\right(p\left)\right)$ equidistribute with respect to the measure $d{\mu }_{ST}=\frac{2}{\pi }{\sin }^2\theta d\theta$. We show that, if $f_1$ is not a character twist of $f_2$, then for subintervals $I_1,I_2\subseteq [0,\pi ]$, there exist infinitely many bounded gaps between the primes p such that ${\theta }_1\left(p\right)\in I_1$ and ${\theta }_2\left(p\right)\in I_2$. We also prove a common generalization of the bounded gaps with the Green–Tao theorem.

The primary purpose of this paper is to generalize the classical Riemann zeta function to the setting of Krull monoids with torsion class groups. We provide a first study of the same generalization by extending Euler's classical product formula to the more general scenario of Krull monoids with torsion class groups. In doing so, the Decay Theorem is fundamental as it allows us to use strong atoms instead of primes to obtain a weaker version of the Fundamental Theorem of Arithmetic in the more general setting of Krull monoids with torsion class groups. Several related examples are exhibited throughout the paper, in particular, algebraic number fields for which the generalized Riemann zeta function specializes to the Dedekind zeta function.

Let $N\ge 3$ and $r\ge 1$ be integers and $p\ge 2$ be a prime such that $p\nmid N$. One can consider two different integral structures on the space of modular forms over $Q$, one coming from arithmetic via q-expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level $\Gamma \left(N{p}^r\right)$ over $Q_p\left({\zeta }_{N{p}^r}\right)$ to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level $p^r$ whenever $p^r>3$, allowing us to compute a lower bound which agrees with the upper bound. Hence we compute the exponent precisely.

Let $Q\left(x\right)$ be a positive definite integral quadratic form with the determinant D being squarefree, and $r(n,Q)$ denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function $d\left(n\right)$ and Fourier coefficients $r(n,Q)$. With more efforts, our method should have a number of applications for other multiplicative functions.

For a function $f:N\to N$, let$N_f^{+}\left(x\right)=\{n⩽x:n=k+f(k\left)\text{ for some }k\right\}.$ Let $\tau \left(n\right)={\sum }_{d|n}1$ be the divisor function, $\omega \left(n\right)={\sum }_{p|n}1$ be the prime divisor function, and $\phi \left(n\right)=\#\{1⩽k⩽n:\gcd (k,n)=1\}$ be Euler's totient function. We show that$\left(1\right)x\ll N_{\omega }^{+}\left(x\right),\left(2\right)x\ll N_{\tau }^{+}\left(x\right)⩽0.94x,\left(3\right)x\ll N_{\phi }^{+}\left(x\right)⩽0.93x.$

Recently Bilu, Marques and Togbé [4] gave a general effective finiteness result on the equation$F_n^{\left(k\right)}=C_m,$ where $F_n^{\left(k\right)}$ denotes the k-generalized Fibonacci-sequence and $C_m$ the sequence of Cullen numbers, by giving explicit absolute bounds for $n,k,m$. However, the authors in [4] explained that their bounds were too large to use Dujella-Pethő reduction to completely solve the equation in question. In the present paper, using the bounds established by Bilu, Marques and Togbé in [4] and a different approach based on 2-adic analysis, we completely solve this equation. Further, using the same approach we also solve the corresponding equation for Woodall numbers.

The $(k,i)$-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to 6k. In this paper we consider the parity of the number ${\bar{C}}_{k,i}\left(n\right)$ of $(k,i)$-singular overpartitions of n. In particular, we give a sufficient condition on even values of k so that the values of ${\bar{C}}_{4k,k}\left(n\right)$ are almost always even. Furthermore, we show that for odd values of $k\le 23$, $k\ne 19$, certain subsequences of ${\bar{C}}_{4k,k}\left(n\right)$ are almost always even.

In this article, we mainly obtain a Schmidt's Subspace Theorem for arbitrary families of moving hypersurface targets in algebraic varieties, which is an extension of the Schmidt's Subspace Theorem due to Son-Tan-Thin [17] and Quang [10], etc.

In this article we prove an explicit sub-Weyl bound for the Riemann zeta function $\zeta \left(s\right)$ on the critical line $s=1/2+it$. In particular, we show that $\left|\zeta \right(1/2+it\left)\right|\le 66.7t^{27/164}$ for $t\ge 3$. Combined, our results form the sharpest known bounds on $\zeta (1/2+it)$ for $t\ge \exp \left(61\right)$.