Christian Axler
In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $π(x)$ holds for every $x \geq N_{\mathcal{R}}$.
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In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $π(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.
A highly strong upper estimate in the modified asymptotic formula for sums of the primes' reciprocals is proved to be necessary (as well as sufficient) in order the Ramanujan inequality holds true. Some other criteria in similar terms are also obtained.
Ramanujan proved that the inequality $π(x)^2 < \frac{e x}{\log x} π\Big(\frac{x}{e}\Big)$ holds for all sufficiently large values of $x$. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that it holds if $x \geq \exp(9658)$. Furthermore, we solve the inequality completely on the Riemann Hypothesis, and show that $x=38, 358, 837, 682$ is the largest integer counterexample.
We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a new sequence of prime numbers, which we call derived Ramanujan primes. For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then the number of Ramanujan primes in the interval from x to x/2 is greater than or equal to…
A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n<=x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant gamma=0.57721566.... This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (an…
On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$
本文利用 Chebyshev Theta 函数 $\vartheta(x)$ 的估计导出素数计数函数 $π(x)$ 的更紧界，从而无条件证明了 Ramanujan 不等式对所有 $x\geq \exp(43.5102147)$ 成立。
We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.
A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.
The Hardy--Ramanujan inequality for sifted sets and its applications
本文建立了适用于筛集的Hardy-Ramanujan不等式加权版本，并将其应用于多个数论问题，包括大偏差、乘法表问题和移位素数因数等。
The aim of our present work here is to present few results in the theory of Mellin transforms using the method that S. Ramanujan used in proving his Master Theorem. Further applications of our results for some number-theoretic functions such as the prime counting function and the zeta function are established.
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(π(R_n))_{n \in \mathbb{N}}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $π(R_n)$ conjectured by Yang and Togbé. In the second part of this paper, we use these explicit estimates to derive a r…
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.
Let $P$ be a polynomial with integer coefficients and degree at least two. We prove an upper bound on the number of integer solutions $n\leq N$ to $n! = P(x)$ which yields a power saving over the trivial bound. In particular, this applies to a century-old problem of Brocard and Ramanujan. The previous best result was that the number of solutions is $o(N)$. The proof uses techniques of Diophantine and Padé approximation.
We note that the inequalities $0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $π(x)$ came up recently in papers on algebraic number theory.
We prove a bound on the number of primes with a given splitting behaviour in a given field extension. This bound generalises the Brun-Titchmarsh bound on the number of primes in an arithmetic progression. The proof is set up as an application of Selberg's Sieve in number fields. The main new ingredient is an explicit counting result estimating the number of integral elements with certain properties up to multiplication by units. As a consequence of this result, we deduce an explicit estimate for the number of ideals of norm up to $x$.
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $σ(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a calculus to associate a generating function with each of these divisor sums. This yields analogues of Ramanujan's recurrence relation for several partition-theoretic functions as well as $r_k(n)$ and $t_k(n)$, functions counting the number of ways of writing a number as a sum of squares (respectively, triangular) numbers. As by-products of this association, we ob…
We use the celebrated circle method of Hardy and Ramanujan to develop convergent formulae for counting a restricted class of partitions that arise from the Göllnitz--Gordon identities.
For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin's InEquality $X(n):=\frac{σ(n)}{n\ln\ln n}<e^γ$ (RIE). The largest known violation of RIE is $n_8=5040$. In this paper Robin's multipliers are split into logarithmic terms $\mathcal{L}$ and relative divisor sums $\mathcal{G}$. A violation of RIE above $n_{8}$ is proposed to imply oscillations that cause $\mathcal{G}$ to exceed $\mathcal{L}$. To this aim Alaoglu and Erdős's conjecture for the CA numbers algorithm is used and the paper's key points are in section 4.2
We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.
In this paper we use a theorem first proved by S.W.Golomb and a famous inequality by J.B. Rosser and L.Schoenfeld in order to prove that there exists an exact formula for $π(n)$ which holds infinitely often.
Let $p\geq 5$ be a prime and $\ell\neq p$ be a prime not dividing the tame level of a $p$-ordinary Hida family. We prove that the actions of the Frobenius element at $\ell$ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar. If $k_f$ denotes the weight of a cusp form $f(z)= \sum_{n\geq 1} a_\ell(f) e^{2πi n z}$, then the inequality $$|a_\ell(f) | \leq 2 \ell^{(k_f-1)/2},$$ predicted by the Ramanujan conjecture, is a strict inequality for almost all members $f$ of the family.
Counting solutions to the quadratic determinant equation
本文研究了二次行列式方程x₁x₂ - x₃x₄ = h在整数解中的渐进行为，通过组合与分析方法结合 Kloosterman 和界，证明了当h = N² + O(N)时的平方根消去误差项，验证了Dhanda-Haynes-Prasala的猜想。
We prove a Diophantine approximation inequality for rational points in varieties of any dimension, in the direction of Vojta's conjecture with truncated counting functions. Our results also provide a bound towards the $abc$ conjecture which in several cases is subexponential. The main theorem gives a lower bound for the truncated counting function relative to a divisor with sufficiently many components, in terms of the proximity to an algebraic point. Furthermore, we show that the Lang-Waldschmidt conjecture implies a special case of Vojta's conjecture with truncation in arbitrary dimension. O…
In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log n}\right\rfloor\leqπ\big((n+1)^2\big)-π(n^2). $$
Let $Ω(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $ω(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $ω(N)$ and $Ω(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $Ω\geq \frac{8}{3}ω(N)-\frac{7}{3}$ and if $3 |N$ then $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$
The sequence $3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\dots$ consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős--Ford--Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy--Ramanujan inequality.
Alder-type partition inequality at the general level
本文证明了在仅有有限个例外的情况下，任意水平 $a$ 的Alder型分拆不等式成立，且无需排除部分 $d+3-a$，推广了已知的不等式结论。
Counting Polynomial-type Exceptional Units on Algebraic Varieties over Number Fields
本文研究了代数数域上代数簇上的多项式型例外单位的计数问题，通过中国剩余定理和Hensel提升技术，推导出在良好降解条件下精确计数公式，并利用Lang-Weil不等式建立渐进行为估计，特别是对二次度数以下的簇改进了误差项。
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…
Let $Γ$ be a cocompact discrete subgroup of $\mathrm{PSL}_{2}(\mathbb{C})$ and denote by $\mathcal{H}$ the three dimensional upper half-space. For a $p\in\mathcal{H}$, we count the number of points in the orbit $Γp$, according to their distance, $\operatorname{arccosh} X$, from a totally geodesic hyperplane. The main term in $n$ dimensions was obtained by Herrmann for any subset of a totally geodesic submanifold. We prove a pointwise error term of $O(X^{3/2})$ by extending the method of Huber and Chatzakos-Petridis to three dimensions. By applying Chamizo's large sieve inequalities we obtain t…
Graphs embedded into surfaces have many important applications, in particular, in combinatorics, geometry, and physics. For example, ribbon graphs and their counting is of great interest in string theory and quantum field theory (QFT). Recently, Koch, Ramgoolam, and Wen [Nuclear Phys.\,B {\bf 870} (2013), 530--581] gave a refined formula for counting ribbon graphs and discussed its applications to several physics problems. An important factor in this formula is the number of surface-kernel epimorphisms from a co-compact Fuchsian group to a cyclic group. The aim of this paper is to give an expl…
A relative trace formula and counting geodesic arcs in the hyperbolic plane
研究双曲圆问题变体，考虑两个双曲子群的双陪集及闭测地线间公共垂线计数，证明显式相对迹公式，利用大筛法不等式得到均方误差阶 $O(X^{1/2}\log X)$ 的新界，并给出点态误差界 $O(X^{2/3})$ 的新证明。
The infinitude is established of the set ${\bf U_1}$ of positive integers $N>5$ such that $G(N)\le \min(G(N/q), G(Np))$ where $q, p$ are primes, $q\ | N$ and $G(N):=\frac{σ(N)}{N\log \log N}$ stands for Gronwall number, $σ(N)$ being the sum of all divisors of $N$. The constructive algorithm is proposed which successively calculates the elements of ${\bf U_1}$, the least of them $N_1^*=2^5\cdot 3^3 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 =160\ 626\ 866\ 400, \ G(N_1^*)=1.7374\dots$ Some interesting properties of these numbers are studied which may occur useful for the pro…
d-Fold Partition Diamonds
本文定义$d$重分拆菱形及Schmidt型变体，利用分拆分析求得生成函数并联系Eulerian多项式，从而证明无穷多个Ramanujan型同余式。
Let r_j(π,s) denote the number of cells, colored j, in the s-residue diagram of partition π. The GBG-rank of πmod s is defined as r_0+r_1*w_s+r_2*w_s^2+...+r_(s-1)*w_s^(s-1), where w_s=exp(2*Π*I/s). We will prove that for (s,t)=1, v(s,t) <= binomial(s+t,s)/(s+t), where v(s,t) denotes a number of distinct values that GBG-rank mod s of t-core may assume. The above inequality becomes an equality when s is prime or when s is composite and t<=2p_s, where p_s is a smallest prime divisor of s. We will show that the generating functions for 4-cores with the prescribed values of GBG-rank mod 3 are all …
Non-relativistic field theories with anisotropic scale invariance in (1+1)-d are typically characterized by a dispersion relation $E\sim k^{z}$ and dynamical exponent $z>1$. The asymptotic growth of the number of states of these theories can be described by an extension of Cardy formula that depends on $z$. We show that this result can be recovered by counting the partitions of an integer into $z$-th powers, as proposed by Hardy and Ramanujan a century ago. This gives a novel relationship between the characteristic energy of the dispersion relation with the cylinder radius and the ground state…
Introducing and Applying S.C.E Model Under Dusart's Inequality to Prove Goldbach's Strong Conjecture for 74 Typical Structures out of All 75 Structural Types of Even Number
本文通过引入S.C.E模型并结合Dusart不等式，证明了74种典型结构满足Goldbach强猜想，并提出了三个未证明的不等式以进一步验证该猜想。
Counting rational points on elliptic and hyperelliptic curves over function fields
结合2-下降、Riemann-Roch和Bézout定理，给出函数域上椭圆与超椭圆曲线有界高度有理点及S-整点数的上界，并应用于小有限域上雅可比簇挠部分，改进了已知上界。
In this paper we give effective estimates for some classical arithmetic functions defined over prime numbers. First we find the smallest real number $x_0$ so that some inequality involving Chebyshev's $\vartheta$-function holds for every $x \geq x_0$. Then we give some new results concerning the existence of prime numbers in short intervals. Also we derive new upper and lower bounds for some functions defined over prime numbers, for instance the prime counting function $π(x)$, which improve current best estimates of similar shape.
We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski densit…
It is the purpose of this thesis to enunciate and prove a collection of explicit results in the theory of prime numbers. First, the problem of primes in short intervals is considered. We prove that there is a prime between consecutive cubes $n^3$ and $(n+1)^3$ for all $n \geq \exp(\exp(33.3))$. To prove this, we first derive an explicit version of the Riemann--von Mangoldt explicit formula. We then assume the Riemann hypothesis and show that there will be a prime in the interval $(x-4/ π\sqrt{x} \log x, x]$ for all $x > 2$. Moreover, we show that the constant $4/π$ can be reduced to $(1+ε)$ fo…
GCD and ABC inequalities for abelian varieties over function fields
该研究证明了特征0函数域上阿贝尔簇有理点的GCD不等式与几何abc猜想的特殊情形，采用三步证明模式，借助Xie-Yuan高度等工具解决了Buium的相关猜想。
More than 83.69% of the zeros of the Riemann zeta function are distinct
本文通过引入带自由裁剪参数的矩阵不等式，将Riemann zeta函数互异非平凡零点的下渐近比例从0.83625提升至0.83699，并在Lean 4中形式化证明，展示了AI辅助数学研究的潜力。
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