Using a recent verification of the Riemann hypothesis up to height $3\cdot 10^{12}$, we provide strong estimates on $π(x)$ and other prime counting functions for finite ranges of $x$. In particular, we get that $|π(x)-\text{li}(x)|<\sqrt{x}\log x/8π$ for $2657\leq x\leq 1.101\cdot 10^{26}$. We also provide weaker bounds that hold for a wider range of $x$, and an application to an inequality of Ramanujan.
In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vallée Poussin's form: $$ π(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) $$ Instead of performing asymptotic expansion on Chebyshev functions as in conventional analytic methods, this new approach uses contour-integration method to analyze Riemann's prime counting function $J(x)$, which only differs from $π(x)$ by $\mathcal O(\sqrt x/\log x)$.
In this article, we show that $$ \left| N (T) - \frac{T}{ 2 π} \log \left( \frac{T}{2πe}\right) \right| \le 0.1038 \log T + 0.2573 \log\log T + 9.3675 $$ where $N(T)$ denotes the number of non-trivial zeros $ρ$, with $0<\Im(ρ) \le T$, of the Riemann zeta function. This improves the previous result of Trudgian for sufficiently large $T$. The improvement comes from the use of various subconvexity bounds and ideas from the work of Bennett $et$ $al.$ on counting zeros of Dirichlet $L$-functions.
Prime number theorem asserts that (at large $x$) the prime counting function $π(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{Li}(x^{1/n})$ deviates from $π(x)$ by the asymptotically vanishing sum $\sum_ρ\mbox{Ri}(x^ρ)$ depending on the critical zeros $ρ$ of the Riemann zeta function $ζ(s)$. We find a fit $π(x)\approx \mbox{Ri}^{(3)}[ψ(x)]$ [with three to four new exact digits compared to $\mbox{li}(x)$] by making use of the Von Mangoldt explicit formula for the Cheby…
Let $X(D,1) =Γ(D,1) \backslash \mathbb{H}$ denote the Shimura curve of level $N=1$ arising from an indefinite quaternion algebra of fixed discriminant $D$. We study the discrete average of the error term in the hyperbolic circle problem over Heegner points of discriminant $d <0$ on $X(D,1)$ as $d \to -\infty$. We prove that if $|d|$ is sufficiently large compared to the radius $r \approx \log X$ of the circle, we can improve on the classical $O(X^{2/3})$-bound of Selberg. Our result extends the result of Petridis and Risager for the modular surface to arithmetic compact Riemann surfaces.
We present many novel results in number theory, including a double series formula for the natural logarithm and a proof concerning the Hölder mean based on the functional equation for the Riemann zeta function. We find a harmonic mean analogue of Chebyshev's inequality for the prime counting function involving the Euler-Mascheroni constant. Furthermore, we define a function taking the Hölder mean of all positive integers up to a given number and investigate its asymptotic behavior, finding two different patterns which are separated by the harmonic mean. Additionally, we discuss the behavior of…
We provide approximations to the prime counting function by various discretized versions of the logarithmic integral function, expressed solely in terms of the harmonic numbers. We demonstrate with explicit error bounds that these approximations are at least as good as the logarithmic integral approximation. As a corollary, we provide some reformulations of the Riemann hypothesis in terms of the prime counting function and the harmonic numbers.
It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.
Under the Riemann Hypothesis, we improve the error term in the asymptotic formula related to the counting lattice problem studied in a first part of this work. The improvement comes from the use of Weyl's bound for exponential sums of polynomials and a device due to Popov allowing us to get an improved main term in the sums of certain fractional parts of polynomials.
We prove several asymptotic continued fraction expansions of $π(x)$, $Π(x)$, $\operatorname{li}(x)$, $\operatorname{Ri}(x)$, and related functions, where $π(x)$ is the prime counting function, $Π(x) = \sum_{k = 1}^\infty \frac{1}{k}π(\sqrt[k]{x})$ is the Riemann prime counting function, and $\operatorname{Ri}(x) = \sum_{k=1}^\infty \frac{ μ(k)}{k} \operatorname{li}(\sqrt[k]{x})$ is Riemann's approximation to the prime counting function. We also determine asymptotic continued fraction expansions of the function $\sum_{p \leq x} p^s$ for all $s \in \mathbb{C}$ with $\operatorname{Re}(s) > -1$, a…
If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion $(4\times10^{18})$.
We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.
On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces
W. Luo研究了与紧致双曲黎曼曲面相关的Selberg zeta函数导数的零点分布。实质上，罗文的主要结果包括以下三点：临界线左边半平面中零点个数的渐近性;测量零点垂直分布的计数函数的渐近展开;测量零点到临界线水平距离的计数函数的渐近展开。在本文中，我们研究了更复杂的设置的Selberg zeta函数的导数的零点分布的非紧，有限体积双曲黎曼曲面。有许多困难存在于非紧的情况下，不存在于紧凑的设置，开始的事实，即在非紧的情况下，塞尔伯格zeta函数不满足类似的黎曼假设。更具体地说，我们实际上研究了ZH的导数的零点，其中Z是Selberg zeta函数，H是散射矩阵的狄利克雷级数分量，两者都与任意有限体积双曲黎曼曲面相关联。我们的主要结果地址有限的零在半平面的左边的临界线，零的垂直分布的渐近计数，和零的水平距离的渐近计数。
The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$ , respectively, $\mbox{li}(x)$ is the logarithmic integral , $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$ , $ε_θ$ and $ε_ψ$ may potentially occur only at $x+…
In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $π(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $ζ(s)$.
The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $π(x)$. This is done by proving the well-known explicit Schoenfeld bound on the RH to hold as long as $4.92 \sqrt{x/\log(x)} \leq T$. Similar statements are proven for the Riemann prime-counting function and the Chebyshov functions $ψ(x)$ and $\vartheta(x)$. Apart from that, we also improve some of the existing bounds of Chebyshov type for the function $ψ(x)$.
We prove several results regarding the distribution of numbers that are the product of a prime and a $k$-th power. First, we prove an asymptotic formula for the counting function of such numbers; this generalises a result of E. Cohen. We then show that the error term in this formula can be sharpened on the assumption of the Riemann hypothesis. Finally, we prove an asymptotic formula for these counting functions in short intervals.
We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies $π(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = ρx + Ω_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ for some $c>0$, where $ρ>0$ is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and co…
The harmonic sawtooth map w(x) of the unit interval onto itself is defined where it is shown that its fixed points are enumerated by generating functions involving the golden ratio in their parameters. The appropriately scaled Mellin transform of w(x) is an analytic continuation of the Riemann zeta function ζ(s) valid for all -Re(s) not an integer. The series expansion of the inverse scaling function which makes the Mellin transform of w(x) equal to the zeta function has coefficients enumerating the Large Schroder Numbers S_n, the number of perfect matchings in a triangular grid of n squares. …
Let $p\ge5$ be a prime and $T$ a Kodaira type of the special fiber of an elliptic curve. We estimate the number of elliptic curves over $\mathbb Q$ up to height $X$ with Kodaira type $T$ at $p$. This enables us find the proportion of elliptic curves over $\mathbb Q$, when ordered by height, with Kodaira type $T$ at a prime $p\ge5$ inside the set of all elliptic curves. This proportion is a rational function in $p$. For instance, we show that $\displaystyle\frac{p^8(p-1)}{p^9-1}$ of all elliptic curves with bad reduction at $p$ are of multiplicative reduction. Furthermore, we prove that the pri…
An equivalent, but variant form of the Riemann functional equation is explored, and several discoveries are made. Properties of the Riemann zeta function $ζ(s)$ from which a necessary and sufficient condition for the existence of zeros in the critical strip are deduced. This in turn, by an indirect route, eventually produces a simple, solvable, differential equation for $arg(ζ(s))$ on the critical line $s=1/2+iρ$, the consequences of which are explored, and the "LogZeta" function is introduced. A singular linear transform between the real and imaginary components of $ζ$ and $ζ^\prime$ on the c…
Let $m,n\in \Bbb{N}$. We represent the additive subgroups of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$, which are also (unital) subrings, and deduce explicit formulas for $N^{(s)}(m,n)$ and $N^{(us)}(m,n)$, denoting the number of subrings of the ring $\Bbb{Z}_m \times \Bbb{Z}_n$ and its unital subrings, respectively. We show that the functions $(m,n)\mapsto N^{(s)}(m,n)$ and $(m,n)\mapsto N^{(us)}(m,n)$ are multiplicative, viewed as functions of two variables, and their Dirichlet series can be expressed in terms of the Riemann zeta function. We also establish an asymptotic formula for the sum $\su…
We discuss new relations among string theory, four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) and the Riemann hypothesis. It is known that the Riemann hypothesis is equivalent to an inequality for the sum of divisors function $σ(n)$. Based on previous results in literature, we focus on the fact that $σ(n)$ appears in a problem of counting supersymmetric states in the $\mathcal{N}=4$ SYM with $SU(3)$ gauge group: the Schur limit of the superconformal index plays a role of a generating function of $σ(n)$. Then assuming the Riemann hypothesis gives bounds on information on…
We consider the non-trivial zeros of the Riemann $ζ$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critic…
