Lahoucine Elaissaoui
We give here a new proof of a Tauberian Theorem of complex Laplace transform using the Theory of measure and theory of function with bounded variations. However we deduce the simple proof of Prime Number Theorem.
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The prime number theorem through one-sided Tauberian theorems
本文提供了单边Ingham-Karamata Tauberian定理的优雅证明，并以此为基础简短推导出素数定理。
This is an exposition, in 12 pages including all prerequisites and a generalization, of Karamata's little known elementary proof of the Landau-Ingham Tauberian theorem, a result in real analysis from which the Prime Number Theorem follows in a few lines. We believe this to be the most transparent of the elementary proofs of the PNT.
We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberi…
It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)=\int_{0}^{\infty}ρ(x)e^{-s x}\:\mathrm{d}x$ of a bounded function $ρ$ has analytic continuation through every point of the segment $(-iλ,iλ)$ of the imaginary axis, then $$ \limsup_{x\to\infty} \left|\int_{0}^{x}ρ(u)\:\mathrm{d}u - G(0)\right|\leq \frac{ \mathfrak{C}}λ \: \limsup_{x\to\infty} |ρ(x)|. $$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=π/2$ and that this value is best po…
A general quantified Ingham-Karamata Tauberian theorem
本文提出一个一般化的定量 Ingham-Karamata Tauber 定理，采用灵活单边条件，改进 Stahn 定理并证明速率在几乎所有情形下最优。
This is a new proof of the tauberian theorem of complex Laplace transformation for getting a simple proof of the prime numbers theorem $π(x) \sim \frac{x}{\log(x)}$ for a largest real $x$.
An elementary Tauberian proof of the Prime Number Theorem
本文用初等实分析给出素数定理的 Tauber 证明，避免 zeta 函数解析延拓，通过推广 Wiener-Ikehara 定理至 Laplace 变换实现。
A Tauberian characterization of the Riemann hypothesis through the floor function
本文通过正则算术函数建立新的陶伯型框架，将取整函数与黎曼zeta函数零点分布关联，给出黎曼假设的陶伯型等价刻画，并拓展至组合数论。
Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory
本文改进了新曼泰贝定理，并应用其研究抽象解析半群，证明了素数定理及梅特恩函数的估计。
An Explicit Tauberian Theorem taking Averaged Inputs with an Application to Counting Abelian Number Fields
本文提出仅依赖 L-函数平均大小的显式陶伯定理，并将其应用于证明若干循环扩张数域计数问题中的平方根节省误差界。
Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the conclusion in a direct and elegant manner". Hardy's proof is also quite adaptable, and it is readily combined with well-known results from prime number theory. We demonstrate this by proving a version of the theorem for primes in arithmetic progressions with uniformity in the modulus, as well as a non-abelian analogue of this.
In this short note, we establish an operator theoretic version of the Wiener-Ikehara tauberian theorem, and point out how this leads to a new proof of the Prime number theorem that should be accessible to anyone with a basic knowledge of operator theory.
On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem
本文研究一类 Laplace 变换可整延拓的连续可微函数所构成的 Fréchet 空间的逼近性质，并利用其与开映射定理证明 Ingham–Karamata Tauberian 定理的最优性。
Associated with a given suitable function, or a measure, on $\mathbb{R}$, we introduce a correlation function, so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on $\mathbb{R}$ are positive definite functions and the Wronskians of the Laplace transform of a nonnegative function on $\mathbb{R}_+$ are completely monotone functions. Then we establish necessary…
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…
Erdős-Kac theorems for discriminants of number fields
本文证明了当 G 为阿贝尔群时，随机 G-扩张中分歧素数个数的中心极限定理，推广了 Lemke Oliver 和 Thorne 在 G=S_d (2≤d≤5) 情形的工作，并首次给出不同素数处分歧事件不独立的例子。
How often is $x\mapsto x^3$ one-to-one in $\mathbb{Z}/n\mathbb{Z}$?
该研究刻画了使映射$x\to x^3$在$\mathbb{Z}/n\mathbb{Z}$中为双射的整数$n$的特征，确定其频率并给出计数常数$C$的显式值，证明基于Delange的Tauberian定理。
We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple l…
A Subconvex Metaplectic Prime Geodesic Theorem and the Shimura Correspondence
该论文证明了素数测地线定理的次凸误差界，并首次为最优指数1+ε提供理论证据，通过利用主项与最大剩余拉普拉斯特征值的关系。
A Wiener-Ikehara type theorem and its application to Chebyshev bounds for Beurling primes
本文提出了一种改进的Wiener-Ikehara定理，用于在更弱条件下验证Beurling素数系统的Chebyshev界。
In this paper, we discuss P(n), the number of ways in which a given integer n may be written as a sum of primes. In particular, an asymptotic form P_as(n) valid for n towards infinity is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This directly gives the asymptotic number of …
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