D. A. Goldston, Ade Irma Suriajaya
We show that the error term in the asymptotic formula for the Ces{à}ro mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures cannot be too small and oscillates.
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A Cesàro average for an additive problem with an arbitrary number of prime powers and squares
该研究针对整数表示为素数幂与平方数之和的加性问题，拓展改进了带Cesàro权加权平均的现有结果，覆盖所有已研究情形并借助所选技术得到最优结论。
We continue our recent work on additive problems with prime summands: we already studied the \emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Cesàro weighted partial \emph{explicit} formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results.
Let $Λ\left(n\right)$ be the von Mangoldt function and $r_{SP}\left(n\right)=\sum_{m_{1}+m_{2}^{2}+m_{3}^{2}=n}Λ\left(m_{1}\right)Λ\left(m_{2}\right)Λ\left(m_{3}\right)$ be the counting function for the numbers that can be written as sum of a prime and two squares of primes. Let $N$ a sufficiently large integer, let $k>3/2$ and let $M_{i}\left(N,k\right),\, i=1,\dots,4$ suitable parameters depending on $Γ(s)$. We prove that $$\sum_{n\leq N}r_{SP}\left(n\right)\frac{\left(N-n\right)^{k}}{Γ\left(k+1\right)}=M_{1}\left(N,k\right)+M_{2}\left(N,k\right)+M_{3}\left(N,k\right)+M_{4}\left(N,k\right)+O…
In this paper we extend and improve our results on weighted averages for the number of representations of an integer as a sum of two powers of primes. Let $1\le \ell_1 \le \ell_2$ be two integers, $Λ$ be the von Mangoldt function and % \(r_{\ell_1,\ell_2}(n) = \sum_{m_1^{\ell_1} + m_2^{\ell_2}= n} Λ(m_1) Λ(m_2) \) % be the weighted counting function for the number of representation of an integer as a sum of two prime powers. Let $N \geq 2$ be an integer. We prove that the Cesàro average of weight $k > 1$ of $r_{\ell_1,\ell_2}$ over the interval $[1, N]$ has a development as a sum of terms depe…
We prove that a suitable explicit formula for the Cesaro-averaged number of representations of an integer as a sum of two primes holds in short intervals.
We obtain an asymptotic formula for a weighted sum of the square of the tail of the singular series for the Goldbach and prime-pair problems.
Orbit decomposition statistics for discrete dynamical systems: the Cesàro mean and a large deviation principle
本文通过Cesàro均值和动力系统zeta函数，证明了当不动点计数满足一定条件时，轨道分解中素轨道数目的计数满足大偏差原理，并推广到强可加函数。
We determine the conditions under which singular values of multiple $η$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index $2^{k' - 1}$ when $k' \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on $X_0^+ (…
Automorphic products that are singular modulo primes
本文利用O(n, 2)上的Rankin--Cohen括号，证明了反射Borcherds乘积的傅里叶系数通常满足模某些素数的同余关系。
Gross and Smith have put forward generalizations of Hardy - Littlewood twin prime conjectures for algebraic number fields. We estimate the behavior of sums of a singular series that arises in these conjectures, up to lower order terms. More exactly, we find asymptotic formulas for smoothed sums of the singular series minus one. Based upon Gross and Smith's conjectures, we use our result to suggest that for large enough 'short intervals' in an algebraic number field K, the variance of counts of prime elements in a random short interval deviates from a Cramer model prediction by a universal fact…
In the present work a new simple proof of the theorem of Gallagher about the average of the singular series in the Hardy-Littlewood prime k-tuple conjecture is proved (in an even stronger form) which is uniform with respect to k (if the length of the interval $H$ is sufficiently large as a function of $k$). This result of Gallagher played a key role in our original joint work with D. A. Goldston and C. Y. Yildirim, where we showed the existence of infinitely many small gaps between consecutive primes. In the present work some weaker variants of Gallagher`s result are also proved (in an even ea…
Sums of singular series with large sets and the tail of the distribution of primes
本文研究k相对于h较大时区间内集合的奇异级数常数平均值，将其应用于素数分布尾部，在Hardy--Littlewood猜想下得出含至少k个素数的对数长度区间数量的上界估计。
We prove an explicit formula, analogous to the classical explicit formula for $ψ(x)$, for the Cesàro-Riesz mean of any order $k>0$ of the number of representations of $n$ as a sum of two primes. Our approach is based on a double Mellin transform and the analytic continuation of certain functions arising therein.
Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove that there are infinitely many solutions to $\mathbf{F}(\mathbf{x})=\mathbf{0}$ in prime coordinates if (i) $\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution over the $p$-adic units $\U_p$ for all prime numbers $p$, and (ii) $\mathbf{F}(\mathbf{x})=\mathbf{0}$ has a non-singular solution in the open cube $(0,1)^n$.
On the saturation number for singular cubic surfaces
本文利用万有挠子、圆法和素数线性方程的结果，研究了奇异三次曲面上坐标具有较少素因子的有理点分布问题。
We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the $k$-tuple conjecture and more general problems of polynomial representation of primes. We show that the singular series for the $k$-tuple conjecture have a limiting distribution when taken over $k$-tuples with (distinct) entries of growing size, and observe that its moments have a curious symmetry property. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in…
Let $Λ\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}Λ\left(m_{1}\right)Λ\left(m_{2}\right), \] \[ r_{PT}\left(N,h\right)=\sum_{n=0}^{N}Λ\left(n\right)Λ\left(n+h\right),\,h\in\mathbb{N} \] be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_{G}\left(n\right)$ and $r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Beta function $B_{z}\left(a,b\…
We address a question posed by Ono, prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results coincides with a recent result of Edixhoven, and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.
Let $n$ be an integer such that $n = 5$ or $n \geq 7$. In this article, we introduce a recipe for a certain infinite family of non-singular plane curves of degree $n$ which violate the local-global principle. Moreover, each family contains infinitely many members which are not geometrically isomorphic to each other. Our construction is based on two arithmetic objects; that is, prime numbers of the form $X^{3}+NY^{3}$ due to Heath-Brown and Moroz and the Fermat type equation of the form $x^{3}+Ny^{3} = Lz^{n}$, where $N$ and $L$ are suitably chosen integers. In this sense, our construction is a…
We give a simple method for estimating the average of singular series' associated with prime k-tuples.
We prove a refinement of the results of Gross and Zagier on prime factorizations of singular moduli.
The Multidimensional Bateman Horn Conjecture: Effective Estimates for the Tail of the Singular Series and an Algorithm for Computation with Guaranteed Accuracy
本文基于概率启发式和多元巴特曼-霍恩猜想，研究多项式取素数值的数量渐进行为，通过代数几何和解析数论方法，获得奇异乘积尾部的估计，并对对角多项式改进了衰减指数和常数。
We prove, in a unified way, $r$-variational estimates, $r>2$, on $\ell^{s}(\mathbb{Z})$ spaces, $s \in (1, \infty)$, for averages and truncated singular integrals along the set of prime numbers.
We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod $\frak{p}$ singular modular forms and discuss some relations between their weights and the corresponding prime $p$. We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finall…
Let $p$ be an irregular prime. Let $K=\Q(ζ)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classic…
We use the circle method to evaluate the behavior of limit-periodic functions on primes. For those limit-periodic functions that satisfy a kind of Barban-Davenport-Halberstam condition and whose singular series converge fast enough, we can evaluate their average value on primes. As an application, this result is used to show how tuples of different k-free numbers behave when linear shifts are applied.
The Euler product formula relates Dirichlet $L(s,χ)$ functions to an infinite product over primes, and is known to be valid for $\Re (s) >1$, where it converges absolutely. We provide arguments that the formula is actually valid for $\Re (s) > 1/2$ in a specific sense. Namely, the logarithm of the Euler product, although formally divergent, is meaningful because it is Cesàro summable, and its Cesàro average converges to $\log L (s,χ)$. Our argument relies on the prime number theorem, an Abel transform, and a central limit theorem for the Random Walk of the Primes, the series $\sum_{n=1}^N \cos…
We calculate the triple correlations for the truncated divisor sum $λ_{R}(n)$. The $λ_{R}(n)$'s behave over certain averages just as the prime counting von Mangoldt function $Λ(n)$ does or is conjectured to do. We also calculate the mixed (with a factor of $Λ(n)$) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation $Λ_{R}(n)$. However, when $λ_{R}(n)$ is used the error in the singular series app…
The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacob…
Let p be an odd prime. Let F_p^* be the no-null part of the finite field of p elements. Let K=\Q(zeta) be a p-cyclotomic field and O_K be its ring of integers. Let pi be the prime ideal of K lying over p. Let sigma : zeta --> zeta^v be the Q-isomorphism of K for a primitive root v mod p. The subgroup of exponent p of the class group of K can be seen as a direct sum oplus_{i=1}^r Gamma_i of groups of order p where the group Gamma_i is annihilated by a polynomial sigma-mu_i with mu_i \in F_p^*. Let Gamma be one of the Gamma_i. From Kummer, there exist prime non-principal prime ideals Q of inerti…
Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 in Z[G], where v^n is understood (mod p). We apply Stickelberger relation to odd prime numbers q different of p and to some singular integers A of K_p connected with the p-class group C_p of K_p and prove the pi-adic congruences: 1) pi^{2p-1} | A^{P(σ)} if q = 1 (mod p), 2) pi^{2p-1} || A^{P(σ)} if q = 1 (mod p) and p^{…
Exponential sums over singular binary quartic forms and applications
本文研究奇异二元四次型上的指数和，给出其有限域傅里叶变换的显式公式，并利用初等代数几何与经典不变量理论证明点计数函数的突出出现，进而证明判别式无平方因子且至多含四个素因子的椭圆曲线存在大量2-Selmer元素。
On the series expansion of the prime zeta function about $s=1$ and its coefficients
本文推导了素数ζ函数在s=1处的级数展开，并证明了其系数的通用公式，类似于黎曼ζ函数的Stieltjes系数。结果可视为对梅特恩斯定理的推广，并通过数值验证计算了高精度公式。
Répartition conjointe de trois nombres premiers et applications
本文研究三个素数的联合分布，证明相关奇异级数的渐近结果，并在哈代-利特尔伍德素数三元组猜想的定量假设下推导渐近公式，改进并补全已有结果，进而给出短区间素数分布的猜想性应用。
We show that the Riesz mean of the singular series in the Goldbach and the Hardy-Littlewood prime-pair conjectures has an asymptotic formula with an error term that can be expressed as an explicit formula that depends on the zeros of the Riemann zeta-function. Unconditionally this error term can be shown to oscillate, while conditionally it can be shown to oscillate between sharp bounds.
One of the most significant discrete invariants of a quadratic form $ϕ$ over a field $k$ is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behaviour of $ϕ$ under scalar extension to arbitrary overfields of $k$. A similarly important, but more accessible variant of this notion is that of the Knebusch splitting pattern of $ϕ$, which captures the isotropy behaviour of $ϕ$ as one passes over a certain prescribed tower of $k$-overfields. In this paper, we determine all possible values of this latter invariant in the case where $ϕ$ is totally singul…
For each prime $p\equiv 1\pmod{4}$ consider the Legendre character $χ=(\frac{\cdot}{p})$. Let $p_\pm(n)$ be the number of partitions of $n$ into parts $λ>0$ such that $χ(λ)=\pm 1$. Petersson proved a beautiful limit formula for the ratio of $p_+(n)$ to $p_-(n)$ as $n\to\infty$ expressed in terms of important invariants of the real quadratic field $\mathbb{Q}(\sqrt{p})$. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz-Cesàro theorem. In this paper we suggest an approach to address Grosswald's conjecture. We discuss a monot…
Sums of the singular series constants that appear in the Hardy--Littlewood $k$-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo $r$ or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo $r$ of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats.
Recently Yu. Bilu, P. Habegger and L. Kühne proved that no singular modulus can be a unit in the ring of algebraic integers. In this paper we study for which sets S of prime numbers there is no singular modulus that is an S-units. Here we prove that when the set S contains only primes congruent to 1 modulo 3 then no singular modulus can be an S-unit. We then give some remarks on the general case and we study the norm factorizations of a special family of singular moduli.
There are at most finitely many singular moduli that are S-units
本文证明对任意有限素数集S，至多有限多个奇异模数是S-单位，并推广至相关模函数，且特化出无穷多个代数单位的模函数必为弱模单位。
Given a singular modulus $j_0$ and a set of rational primes $S$, we study the problem of effectively determining the set of singular moduli $j$ such that $j-j_0$ is an $S$-unit. For every $j_0 \neq 0$, we provide an effective way of finding this set for infinitely many choices of $S$. The same is true if $j_0=0$ and we assume the Generalized Riemann Hypothesis. Certain numerical experiments will also lead to the formulation of a "uniformity conjecture" for singular $S$-units.
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