Andrés Chirre, Oswaldo Velásquez Castañón
In this paper, we review the study of the distribution of the zeros of certain approximations for the Ramanujan $Ξ-$function given by Haseo Ki, and we provide a new proof of his results. Our approach is motivated by the ideas of Velásquez in the study of the zeros of certain sums of entire functions with some condition of stability related to the Hermite-Biehler theorem.
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In this short note, we aim to discuss some summations due to Ramanujan, their generalizations and some allied series
We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd's, the Gamma function, the Bernoulli polynomials, and binomial coefficients.
In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in one of these manuscripts, namely a transformation formula involving the Gamma function and Riemann Zeta function. This formula elegantly yields a modular relation. Later we also give another proof of the first equality of this transformation formula along the lines suggested by Andrew P. Guinand.
We give a brief review of the work of Ramanujan on cranks that is found in the Lost Notebook. Recent work by Bruce Berndt and his coauthors have brought to light many interesting results of Ramanujan on cranks, which we highlight in this article.
Generalizations (in the spirit of Koshliakov) of some formulas from Ramanujan's Lost Notebook
本文在科什利亚科夫理论框架下推广了拉马努金遗失笔记中关于除数函数和戴德金η函数对数变换的若干恒等式。
In his lost notebook, Ramanujan listed 5 identities related to the false theta function $$f(q)=\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}.$$ A new combinatorial interpretation and proof of one of these identities is given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtained a combinatorial interpretation of another one of the identities.
In this note, we shall give a brief survey of the results that are found in Ramanujan's Lost Notebook related to cranks. Recent work by B. C. Berndt, H. H. Chan, S. H. Chan and W. -C. Liaw have shown conclusively that cranks was the last mathematical object that Ramanujan studied. We shall closely follow the work of Berndt, Chan, Chan and Liaw and give a brief description of their work.
The Ramanujan Machine project detects new expressions related to constants of interest, such as $ζ$ function values, $γ$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even and odd $ζ$ function values, logarithms etc. We show that many relations detected by the Ramanujan Machine Project stem from a specific algebraic observation and show how to generate infinitely many. This provides an automated proof and/or an explanation of many of the relations listed as conjectures by the project (although not all of them).
On Ramanujan's lost notebook and new tenth-order like identities for second-, sixth-, and eighth-order mock theta functions
该研究依托拉马努金遗失的笔记本，借助Hickerson和Mortenson的Appell函数性质，发现并证明了3个六阶仿theta函数的类十阶新恒等式，还补充了19个不同阶仿theta函数组合的类十阶恒等式。
In this short research note, we aim to establish an interesting extension of a summation due to Ramanujan.The result is derived with the help of an extension of Gauss's summation theorem available in the literature.
Ramanujan's approximation to the exponential function is reexamined with the help of Perron's saddle-point method. This allows for a wide generalization that includes the results of Buckholtz, and where all the asymptotic expansion coefficients may be given in closed form. Ramanujan's approximation to the exponential integral is treated similarly.
Absolute convergence of Ramanujan expansions admits coefficients' coexistence of Ramanujan expansions
本文证明每个算术函数$F$都存在无穷多个拉马努金系数$G$使其拉马努金展开绝对收敛，揭示了固定$F$时$G$的无穷多非唯一性即“系数共存”现象。
Ninth degree analogue of Ramanujan's septic theta function identity
本文提出了Ramanujan七次θ函数恒等式的九次类似，并应用其笔记簿中的三次方程条目、用θ函数解释三角恒等式，最后计算了五个φ(e^{-π√n})的例子。
In this short note, we show an analogue of one of Alladi's and Dawsey's formulas with respect to the Ramanujan sum $c_n(m)$ for $m\geq1$. Their formulas may be viewed as the case $m=1$ in our result.
We derive two new analogues of a transformation formula of Ramanujan involving the Gamma and Riemann zeta functions present in the Lost Notebook. Both involve infinite series consisting of Hurwitz zeta functions and yield modular relations. As a special case of the first formula, we obtain an identity involving polygamma functions given by Andrew P. Guinand and as a limiting case of the second formula, we derive the transformation formula of Ramanujan.
A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to Ramanujan's approximation coefficients. A more analytic second proof of the main result is also included in an appendix.
Period Function of Maass forms from Ramanujan's Lost Notebook
本文证明拉马努金失散笔记第220页的函数$\mathcal{F}_1(x)$满足Maass形式的周期关系，称为拉马努金周期函数，并发现其为Hecke本征函数，推广至参数$s$，连接更广泛的周期函数理论。
This is a review of the 5-volumes of Ramanujan's Notebooks, as worked over by Bruce C. Berndt over the last quarter of the XX-th Century. To illustrate how useful Ramanujan's insights could be for anyone who indulges in the wild pleasure of savoring Ramanujan's genius, the reviewer takes one formula of Ramanujan from the Notebooks, analyses it carefully, and arrives at the beginnings of what could be fairly called "Quantum Number Theory."
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the functions that appear in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in turn to find identities involving binary quadratic forms.
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.
In 1730 James Stirling, building on the work of Abraham de Moivre, published what is known as Stirling's approximation of $n!$. He gave a good formula which is asymptotic to $n!$. Since then hundreds of papers have given alternative proofs of his result and improved upon it, including notably by Burside, Gosper, and Mortici. However Srinivasa Ramanujan gave a remarkably better asymptotic formula. Hirschhorn and Villarino gave a nice proof of Ramanujan's result and an error estimate for the approximation. In recent years there have been several improvements of Stirling's formula including by Ne…
The hypergeometric formulae designed by Ramanujan more than a century ago for efficient approximation of $π$, Archimedes' constant, remain an attractive object of arithmetic study. In this note we discuss some $q$-analogues of Ramanujan-type evaluations and of related supercongruences.
In this Note we show that given any cusp form πon GL(3) over the rationals, there exist an infinite number of primes p which are Ramanujan for π, i.e., that the local components π_p are tempered for an infinite number of p. It turns out that the key structural device which is needed is the degree 8 adjoint L-function of π.
Cubic and quintic analogues of Ramanujan's septic theta function identity
受Ramanujan七次θ函数恒等式补全的启发，本文提出三次和五次类似，并结合Ramanujan类不变量，为其著名θ函数φ(q)提供了多种求值。
Bruinier and Ono recently developed the theory of generalized Borcherds products, which uses coefficients of certain Maass forms as exponents in infinite product expansions of meromorphic modular forms. Using this, one can use classical results on congruences of modular forms to obtain congruences for Maass forms. In this note we work out the example of Ramanujan's mock theta functions f and ω in detail.
We study two aspects of Hecke symmetry in this note: first, we conjecture a generalization of the Ramanujan identities to the case of automorphic forms of Hecke groups; second, we conjecture a generalization of an inversion formula from the theory of elliptic functions.
We prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in Ramanujan's notebooks. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant, gamma.
Rogers-Ramanujan continued fraction and approximations to $\mathbf{2π}$
本文利用 Rogers–Ramanujan 连分数与 Ramanujan、Rogers 的模方程构造 $2π$ 的高速递归逼近，分别实现正确位数五倍和十一倍增长。
On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable product of theta-functions $a_{m, n}$. Also he recorded eighteen explicit values depending on two parameters, namely, $m$, and $n$, where these are odd integers. All these values were established by Berndt et al. In this paper, we initiate to study explicit evaluations of $a_{m, n}$ for even values of $m$. We establish a new general theorems for the explicit evaluations of $a_{m, n}$ involving class invariant of Ramanujan. For this purpose, we establish several new P-Q mixed modular equations. Using these theorems, we…
A number field extension of Ramanujan's identity and its integral representation via the generalized Riemann $Ξ$-function
本文建立拉马努金恒等式的数域类似物，得到戴德金ζ函数高次幂变换公式的积分表示，重现Hardy的雅可比θ关系积分表示，还得到扩展拉马努金-科什利亚科夫公式的数域类似物。
A series of Ramanujan, two-term dilogarithm identities and some Lucas series
本文研究一类与Ramanujan遗失笔记中级数相关的初等级数，利用反双曲反正切、多对数及Clausen函数给出闭式，并证明新的双项二重对数恒等式，求出若干Lucas数级数的闭式。
The notion of cubic partitions is introduced by Hei-Chi Chan and named by Byungchan Kim in connection with Ramanujan's cubic continued fractions. Chan proved that cubic partition function has Ramanujan Type congruences modulo powers of $3$. In a recent paper, William Y.C. Chen and Bernard L.S. Lin studied the congruent property of the cubic partition function modulo $5$. In this note, we give Ramanujan type congruences for cubic partition function modulo powers of $5$.
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 $x \geq H_0$.
Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either inconvenient for most people (not majored in math) who do not want do write programs, or unsatisfying in accuracy. By bringing in two parameters in the Hardy-Ramanujan's Asymptotic formula and fitting the data of the two parameters by least square method, iteration method and some other special designed methods, several revised elementary estimation formulae with high accuracy for p(n) are obtained. With th…
This paper provides a survey of particular values of Ramanujan's theta function $φ(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-π\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.
In this paper, we introduce some explicit approximations for the summation $\sum_{k\leq n}Ω(k)$, where $Ω(k)$ is the total number of prime factors of $k$.
Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms
本文通过计算 Euler-Kronecker 常数得到除数和函数计数函数渐近展开的二阶项，并确定 Ramanujan 逼近优于 Landau 逼近的条件，进而研究六个尖形式 Fourier 系数的不可整除性。
We present q-series proofs of four identities involving sixth order mock theta functions from Ramanujan's lost notebook. We also show how Ramanujan's identities can be used to give a quick proof of four sixth order identities of Berndt and Chan.
On a function of Ramanujan twisted by a logarithm
本文利用拉马努金《遗失笔记》中的模关系和德宁格函数导数的近期结果，推导出涉及双伽马函数与对数因子的无穷级数函数方程，并研究了其中起关键作用的积分 $\mathscr{H}(x)$ 的多种表示。
Many of the fastest known algorithms to compute $π$ involve generalized hypergeometric series, such as the Ramanujan-Sato series. In this paper, we investigate the rates of convergence for several such series and we give asymptotic expansions for the error of finite approximation. For example, when using the first $n$ terms of the Chudnovskys' series, we obtain the finite approximation $π_n\approx π$. It is known that the truncation error satisfies $|π_n-π|\approx 53360^{-3n}.$ In this paper, we prove that the asymptotic expansion for the truncation error in the Chudnovskys' series is $$\left|…
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of Appell--Lerch sums and partial theta functions and on constructing bilateral $q$-series with mixed mock modular behaviour is well-suited for such radial limits. We present five more radial limit results which follow from mixed mock modular bilateral $q$-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function…
In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $φ(q)$ and $ψ(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evaluated.
Elliptic Functions
本研究围绕拉马努金著作中的椭圆函数展开探讨
Zeros of Ramanujan-type Polynomials
本文针对拉马努金型多项式$R_{2k+1,p}(z)$的非实零点分布猜想，给出了该多项式所有非实零点均位于圆$|z|=1/p$上的证明，推进了相关数论多项式零点性质的研究。
An extension of Ramanujan-Guinand identity for the Dedekind zeta function and a new formula for $ζ_\mathbb{F}(2)$ and $ζ_\mathbb{F}(2m+1)$
本文为戴德金ζ函数建立拉马努金-吉南恒等式的数域类似物，得到ζ_F(2)的新公式及ζ_F(2m+1)的恒等式，还获戴德金η函数对数的变换公式。
Andrews studied a function which appears in Ramanujan's identities. In Ramanujan's "Lost" Notebook, there are several formulas involving this function, but they are not as simple as the identities with other similar shape of functions. Nonetheless, Andrews found out that this function possesses combinatorial information, odd-even partition. In this paper, we provide the asymptotic formula for this combinatorial object. We also study its companion odd-even overpartitions.
Splitting Appell functions in terms of single quotients of theta functions
围绕拉马努金仿θ函数的恒等式研究，本文提出了若干族适用于其构建基块Appell函数的类十阶恒等式，拓展了仿θ函数恒等式的研究范畴。
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x^2+5y^2. Making use of Ramanujan's_1ψ_1 summation formula we establish a new Lambert series identity for \sum_{n,m=-\infty}^{\infty} q^{n^2+5m^2}. Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don't stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity we obtaina collection of new Lambert series for certain infinite products associated with quadratic forms such as x^2+6y^2, 2x^2+3y^2, x^2+15y^2, 3x^…
Recently, Amdeberhan and Merca proved some arithmetic properties of the crank parity function $C(n)$ defined as the difference between the number of partitions of $n$ with even cranks and those with odd cranks and the sequence $a(n)$ whose generating function is the reciprocal of that of $C(n)$. The function $C(n)$ was first studied by Choi, Kang, and Lovejoy. In this note, we give new elementary proofs of some of their main results and extend them. In particular, we establish Ramanujan-type congruences modulo $5$ and $25$ for certain finite sums involving $C(n)$ and $a(n)$. Our proofs employ …
A remark on modular equations involving Rogers-Ramanujan continued fraction via $5$-dissections
本文通过研究Ramanujan theta函数的5-剖分并建立与Rogers-Ramanujan连分数相关的恒等式，重新证明了Ramanujan遗失笔记本中的一个模方程，并进一步建立了涉及$R(q)$至$R(q^{16})$的新模方程。
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $φ(q)$ and $ψ(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.
The Rogers-Ramanujan dissection of a theta function
本文借助参数$s$推广了Ramanujan失落笔记簿中的恒等式，证明了theta函数可被分拆为广义Rogers-Ramanujan函数乘积的有限和，并通过渐近分析揭示了其超越模关系的特性。
Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for $ζ(2m+1)$. We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.
The three missing terms in Ramanujan's septic theta function identity
本文给出了 Ramanujan 遗失笔记簿中关于其 theta 函数 $φ(q)$ 的七次恒等式里缺失的三个具体项，并进一步计算了类不变量 $G_{343}$ 及 $φ(e^{-nπ})$ 在特定整数下的特殊值。
We focus on three pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong -- each is vitiated by divergent series. We concentrate on only one of the two incorrect "identities", which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of …
On the Hardy-Ramanujan Theorem
本文证明了Hardy-Ramanujan定理的有效版本，通过Poisson分布点态控制ω(n)的移位经验分布，并推导了平方自由数类似结果、Chernoff估计、高斯窗口估计、中偏差上界和一致矩估计。
In this article, we define functions analogous to Ramanujan's function $f(n)$ defined in his famous paper "Modular equations and approximations to $π$". We then use these new functions to study Ramanujan's series for $1/π$ associated with the classical, cubic and quartic bases.
ABC implies that Ramanujan's tau function misses almost all primes
在 ABC 猜想假设下，证明了 tau 函数的素数值密度为 O(X^{13/22})，表明 tau 函数错过大部分素数，同时提出 S(X) 可能无限且有预测阶数。
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