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Автор: Grafiati

Опубліковано: 18 травня 2024

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1

Dr. Rituraj Trivedi. "A. K. Ramanujan: A Leading Indo-Anglican Poet." Creative Launcher 7, no. 2 (April 30, 2022): 115–21. http://dx.doi.org/10.53032/tcl.2022.7.2.15.

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Ramanujan is one of the prominent Indo-Anglican Poets. Some critics consider him to be one of the three great Indo-Anglican poets, the other two being Nissim Ezekiel and Kamala Das. Ramanujan’s poetry is largely autobiographical and thought-provoking. The themes Ramanujan considers in his poetry are limited in scope, but some other passages of his poetry largely compensate for that inadequacy. Inversely important as a theme in Ramanujan’s poetry is his Hindu heritage. Ramanujan has shown a sharp and intense textual sensitivity in his poetry. Ramanujan is one of the most competent and professed craftsmen in Indo-Anglican poetry. Among the silent features of Ramanujan’s poetry is its cerebral literalism. His poetry abounds in boons of world and expression. Ramanujan generally writes in free verse without the importance of punctuation, but he does relatively frequently introduce rhyme and assonance into his poems. Another striking point of Ramanujan’s poetry is the ascendance in it of irony. Irony too is a device that is employed by nearly every Indo-Anglican poet, but Ramanujan makes use of this device in nearly every poem. Ramanujan’s poetry contains distinctive and distinguishable imagery from the imagery of other Indo-Anglican Poets.

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2

Bernd, Bruce C., and Heng Huat Chan. "Some Values for the Rogers-Ramanujan Continued Fraction." Canadian Journal of Mathematics 47, no. 5 (October 1, 1995): 897–914. http://dx.doi.org/10.4153/cjm-1995-046-5.

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AbstractIn his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujan's beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions.

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3

Anamika Kumari. "The Poetry of A. K. Ramanujan: In Search of Self." Creative Launcher 4, no. 6 (February 29, 2020): 71–76. http://dx.doi.org/10.53032/tcl.2020.4.6.12.

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Self at the centre of all kinds of search manifests itself in Ramanujan’s art from the very beginning of his creative life and the artist has all through assumed an elusive character till his vision clears; well, but his vision is gained through experience. His vision of the self permeates most of his elusive poems, the poems which have so far been faulted on one count or another. First, perhaps is “The Stridess” which is not by chance, the first poem of Ramanujan’s first volume of poems, and this volume The Striders is also entitled after this poem. Ramanujan concern with the self and hence his idea of the individuality of beings is very much there but misted with an uncanny subject like waterbug and mare, gone veiled under an objectivist style of the moderns.

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4

Anamika Kumari. "Diasporic Concerns in A. K. Ramanujan’s Writings." Creative Launcher 4, no. 2 (June 30, 2019): 39–43. http://dx.doi.org/10.53032/tcl.2019.4.2.06.

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Ramanujan appears to be poised and perched between two worlds. The world in which he is born and the other which he has acquired. It then becomes very obvious that the perception of Ramanujan “is not just that of Hindu or merely an Indian in the sense that he sees only those. His perceptive eye roves wider and the limit of his perception is encompassing wider area.” His perception is pluralistic absorbing other culture. This does not Point towards assimilation or integration of the others into the Indian or the Indian into the global. Ramanujan used to describe his position as “being the hyphen in Indian-American Identifying with E. M Forester’s great urge to “connect” Ramanujan also makes his greatest work out of disconnections. His life's mission seems to be “to keep the dialogues and corals alive and to make something of them.” His aim is to achieve a synthesis between warring cultural coordinates, “It looks as if I live between things all the time two (or more) languages, two countries, and two disciplines. In all his writing translations, critical essays or poetic compositions, there is an invisible thread which lends homogeneity to his writings. In his encounter with different cultures, Ramanujan feels “himself translated a little in each encounter” and learns “a good deal about myself and about Indian arts”.

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5

Chen, Kwang-Wu. "Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion." Mathematics 10, no. 12 (June 12, 2022): 2033. http://dx.doi.org/10.3390/math10122033.

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Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.

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6

Dharwadker, Vinay. "A. K. Ramanujan: Author, Translator, Scholar." World Literature Today 68, no. 2 (1994): 279. http://dx.doi.org/10.2307/40150143.

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7

McIntosh, Richard J. "The H and K Family of Mock Theta Functions." Canadian Journal of Mathematics 64, no. 4 (August 1, 2012): 935–60. http://dx.doi.org/10.4153/cjm-2011-066-0.

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AbstractIn his last letter to Hardy, Ramanujan defined 17 functionsF(q), |q| < 1, which he calledmockθ-functions. He observed that asqradially approaches any root of unity ζ at whichF(q) has an exponential singularity, there is aθ-functionTζ(q) withF(q) −Tζ(q) =O(1). Since then, other functions have been found that possess this property. These functions are related to a functionH(x,q), wherexis usuallyqrore2πirfor some rational numberr. For this reason we refer toHas a “universal” mockθ-function. Modular transformations ofHgive rise to the functionsK,K1,K2. The functionsKandK1appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mockθ-functions of even order andHare listed.

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8

Chidambaraswamy, J., and P. V. Krishnaiah. "An identity for a class of arithmetical functions of two variables." International Journal of Mathematics and Mathematical Sciences 11, no. 2 (1988): 351–54. http://dx.doi.org/10.1155/s0161171288000419.

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For a positive integerr, letr∗denote the quotient ofrby its largest squarefree divisor(1∗=1). Recently, K. R. Johnson proved that(∗)∑d|n|C(d,r)|=r∗∏pa‖nr∗p+r(a+1)∏pa‖nr∗p|r(a(p−1)+1) or 0according asr∗|nor not whereC(n,r)is the well known Ramanujan's sum. In this paper, using a different method, we generalize(∗)to a wide class of arithmetical functions of2variables and deduce as special cases(∗)and similar formulae for several generalizations of Ramanujan''s sum.

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9

G., E., Vinay Dharwadker, and A. K. Ramanujan. "The Collected Essays of A. K. Ramanujan." Journal of the American Oriental Society 121, no. 3 (July 2001): 537. http://dx.doi.org/10.2307/606720.

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10

YAMADA, TOMOHIRO. "A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION." Glasgow Mathematical Journal 61, no. 03 (August 22, 2018): 535–44. http://dx.doi.org/10.1017/s0017089518000344.

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AbstractWe shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}.

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11

Zhang, Zhongfeng, and Alain Togbé. "On the Ramanujan-Nagell type Diophantine equation \(Dx^2+k^n=B\)." Glasnik Matematicki 56, no. 2 (December 23, 2021): 263–70. http://dx.doi.org/10.3336/gm.56.2.04.

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12

Christian Axler and Thomas Leßmann. "On the First k-Ramanujan Prime." American Mathematical Monthly 124, no. 7 (2017): 642. http://dx.doi.org/10.4169/amer.math.monthly.124.7.642.

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13

Lee, Yoonjin, та Yoon Kyung Park. "Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity". Open Mathematics 18, № 1 (1 січня 2020): 1727–41. http://dx.doi.org/10.1515/math-2020-0105.

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Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

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14

Ramazani, Jahan, and A. K. Ramanujan. "Metaphor and Postcoloniality: The Poetry of A. K. Ramanujan." Contemporary Literature 39, no. 1 (1998): 27. http://dx.doi.org/10.2307/1208920.

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15

Ray, Chiranjit, and Rupam Barman. "Infinite families of congruences for k-regular overpartitions." International Journal of Number Theory 14, no. 01 (November 21, 2017): 19–29. http://dx.doi.org/10.1142/s1793042118500021.

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Let [Formula: see text] be the number of overpartitions of [Formula: see text] into parts not divisible by [Formula: see text]. In this paper, we find infinite families of congruences modulo 4, 8 and 16 for [Formula: see text] and [Formula: see text] for any [Formula: see text]. Along the way, we obtain several Ramanujan type congruences for [Formula: see text] and [Formula: see text]. We also find infinite families of congruences modulo [Formula: see text] for [Formula: see text].

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16

Candelpergher, B., H. Gopalkrishna Gadiyar, and R. Padma. "Ramanujan summation and the exponential generating function $\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\zeta^{\prime}(-k)$." Ramanujan Journal 21, no. 1 (December 2, 2009): 99–122. http://dx.doi.org/10.1007/s11139-009-9166-0.

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17

WANG, LIUQUAN. "ARITHMETIC PROPERTIES OF -REGULAR BIPARTITIONS." Bulletin of the Australian Mathematical Society 95, no. 3 (December 1, 2016): 353–64. http://dx.doi.org/10.1017/s0004972716000964.

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Let$B_{k,\ell }(n)$denote the number of$(k,\ell )$-regular bipartitions of$n$. Employing both the theory of modular forms and some elementary methods, we systematically study the arithmetic properties of$B_{3,\ell }(n)$and$B_{5,\ell }(n)$. In particular, we confirm all the conjectures proposed by Dou [‘Congruences for (3,11)-regular bipartitions modulo 11’,Ramanujan J.40, 535–540].

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18

Dharwadker, Vinay, and Bruce King. "Three Indian Poets: Nissim Ezekiel, A. K. Ramanujan, Dom Moraes." World Literature Today 66, no. 4 (1992): 781. http://dx.doi.org/10.2307/40148812.

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19

Hughes, Kim. "Ramanujan Congruences For p-k(n) Modulo Powers Of 17." Canadian Journal of Mathematics 43, no. 3 (June 1, 1991): 506–25. http://dx.doi.org/10.4153/cjm-1991-031-0.

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For each integer r we define the sequence pr(n) by We note that p-1(n) = p(n), the ordinary partition function. On account of this some authors set r = — k to make positive values of k correspond to positive powers of the generating function for p(n): We follow this convention here. In [3], Atkin proved the following theorem.

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20

Takeda, Wataru. "Existence of the solutions to the Brocard–Ramanujan problem for norm forms." Proceedings of the American Mathematical Society, Series B 10, no. 35 (November 16, 2023): 413–21. http://dx.doi.org/10.1090/bproc/181.

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The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions ( x , ℓ ) (x,\ell ) of x 2 − 1 = ℓ ! x^2-1=\ell ! . Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions ( x , ℓ ) (x, \ell ) , up to a unit factor, to the equations N K ( x ) = ℓ ! N_K(x) = \ell ! , where N K N_K are the norms of number fields K / Q K/\mathbf Q . In this paper, we construct infinitely many number fields K K such that N K ( x ) = ℓ ! N_K(x) = \ell ! has at least 22 22 solutions for positive integers ℓ \ell .

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21

Vipin Kumar and Dr. Vivek Kumar Dwivedi. "A.K. Ramanujan: A Poet of Different Cultures and Languages." Creative Launcher 8, no. 2 (April 30, 2023): 51–58. http://dx.doi.org/10.53032/tcl.2023.8.2.07.

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The paper explores the impact of different cultures and languages in the poetic writings of A. K. Ramanujan. He has a full command over Indian culture, scriptures and rituals. Tamil, Kannad, Sanskrit and English languages are well known to him. Language is a very important tool in the formulation of a culture and its aesthetics, as it is a medium of expression. Without language no human culture can be imagined. Culture is a manifestation of the ideas, customs, and social behavior of a particular group of human society. It is a code of conduct which guides and control a certain human society. Ramanujan was deeply rooted in Indian culture and tradition, which is evident in his work. However, his exposure to Western education, particularly his studies in the United States, also influenced his literary style and themes. As a result, Ramanujan's work reflects a unique blend of different cultures and languages, and he is known for bridging the gap between Indian and Western literary traditions. Oxford Advanced Learner Dictionary defines culture as “the customs and beliefs; ways of life and social organization of a particular country or group” (373). The cultural and linguistic influences are evident in the literature of any nation, therefore, it always becomes a perfect source of information. Literature of any nation keeps the record of its history, geography, culture and tradition. For instance, we have to study Leo Tolstoy to know the history and geography of nineteenth century Russian literature; similarly, if we want to know something about the English culture, we have to study English literature as literature is a part of culture. In the same way, there are several languages and cultures that are observed in India and each of them are closely connected with the theme of Indianness and this is how it paves the way of unity in diversity.

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22

Ismail, Mourad E. H., Xin Li, and M. Rahman. "Landau constants and their q-analogues." Analysis and Applications 13, no. 02 (March 2015): 217–31. http://dx.doi.org/10.1142/s0219530514500201.

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We derive inequalities and a complete asymptotic expansion for the Landau constants Gn, as n → ∞ using the asymptotic sequence n!/(n + k)!. We also introduce a q-analogue of the Landau constants and calculate their large degree asymptotics. In the process, we also establish q-analogues of identities due to Ramanujan and Bailey.

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23

Craveiro, Irene Magalhães, Elen Viviani Pereira Spreafico, and Mustapha Rachidi. "New approaches of (q,k)-Fibonacci–Pell sequences via linear difference equations. Applications." Notes on Number Theory and Discrete Mathematics 29, no. 4 (October 11, 2023): 647–69. http://dx.doi.org/10.7546/nntdm.2023.29.4.647-669.

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In this paper we establish some explicit formulas of (q,k)-Fibonacci–Pell sequences via linear difference equations of order 2 with variable coefficients, and explore some of their new properties. More precisely, our results are based on two approaches, namely, the determinantal and the nested sums approaches, and their closed relations. As applications, we investigate the q-analogue Cassini identities and examine a pair of Rogers–Ramanujan type identities.

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24

Kurodi, Shashikant, and Deepthi R. "Challenges of Translation with a Special Reference to A. K. Ramanujan." POETCRIT 34, no. 1 (January 18, 2021): 86–95. http://dx.doi.org/10.32381/poet.2020.34.01.12.

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25

Guo, Victor J. W., and Su-Dan Wang. "Some congruences involving fourth powers of central q-binomial coefficients." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 3 (January 30, 2019): 1127–38. http://dx.doi.org/10.1017/prm.2018.96.

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AbstractWe prove some congruences on sums involving fourth powers of central q-binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]: $$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {\bmod p^{r + 3}} \right),$$where p⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.

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26

Bilu, Yuri F., Jean-Marc Deshouillers, Sanoli Gun, and Florian Luca. "Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms." Compositio Mathematica 154, no. 11 (October 18, 2018): 2441–61. http://dx.doi.org/10.1112/s0010437x18007455.

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Let $\unicode[STIX]{x1D70F}(\cdot )$ be the classical Ramanujan $\unicode[STIX]{x1D70F}$-function and let $k$ be a positive integer such that $\unicode[STIX]{x1D70F}(n)\neq 0$ for $1\leqslant n\leqslant k/2$. (This is known to be true for $k<10^{23}$, and, conjecturally, for all $k$.) Further, let $\unicode[STIX]{x1D70E}$ be a permutation of the set $\{1,\ldots ,k\}$. We show that there exist infinitely many positive integers $m$ such that $|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(1))|<|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(2))|<\cdots <|\unicode[STIX]{x1D70F}(m+\unicode[STIX]{x1D70E}(k))|$. We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

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27

CHAN, HENG HUAT, LIUQUAN WANG, and YIFAN YANG. "CONGRUENCES MODULO 5 AND 7 FOR 4-COLORED GENERALIZED FROBENIUS PARTITIONS." Journal of the Australian Mathematical Society 103, no. 2 (December 21, 2016): 157–76. http://dx.doi.org/10.1017/s1446788716000616.

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Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.

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28

Altuğ, S. Ali. "BEYOND ENDOSCOPY VIA THE TRACE FORMULA – III THE STANDARD REPRESENTATION." Journal of the Institute of Mathematics of Jussieu 19, no. 4 (November 12, 2018): 1349–87. http://dx.doi.org/10.1017/s1474748018000427.

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Анотація:

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at: arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

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29

Venkateswaran, Pramila, and Ralph Nazareth. "Rescuing Literary Aesthetics from Ideology: A. K. Ramanujan, Chitra Divakaruni, and Ginu Kamani." South Asian Review 23, no. 2 (December 2002): 44. http://dx.doi.org/10.1080/02759527.2002.11932291.

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30

Merca, Mircea. "On a nonlinear relation for computing the overpartition function." Mathematica Slovaca 71, no. 3 (June 1, 2021): 535–42. http://dx.doi.org/10.1515/ms-2021-0002.

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Abstract In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p (n). Computing p (n) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p (n) that requires only the values of p (k) with k ≤ n/2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p (n) to obtain a simple and fast computation of the value of p (n). This new method uses only (large) integer arithmetic and it is simpler to program.

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31

Yu, Yahui, and Jiayuan Hu. "On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)." AIMS Mathematics 6, no. 10 (2021): 10596–601. http://dx.doi.org/10.3934/math.2021615.

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<abstract><p>Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

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32

Balakrishnan, Jennifer S., Sorina Ionica, Kristin Lauter, and Christelle Vincent. "Constructing genus-3 hyperelliptic Jacobians with CM." LMS Journal of Computation and Mathematics 19, A (2016): 283–300. http://dx.doi.org/10.1112/s1461157016000322.

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Given a sextic CM field $K$, we give an explicit method for finding all genus-$3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field $K$, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field $\mathbb{F}_{p}$ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo $p$.

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33

Lemke Oliver, Robert J., and Jesse Thorner. "Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture." International Mathematics Research Notices 2019, no. 22 (February 2, 2017): 6988–7036. http://dx.doi.org/10.1093/imrn/rnx309.

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Abstract Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.

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34

Chen, William Y. C., Qing-Hu Hou, and Yan-Ping Mu. "Non-Terminating Basic Hypergeometric Series and the q-Zeilberger Algorithm." Proceedings of the Edinburgh Mathematical Society 51, no. 3 (October 2008): 609–33. http://dx.doi.org/10.1017/s0013091506001313.

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AbstractWe present a systematic method for proving non-terminating basic hypergeometric identities. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non-terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formulae, including the Sears–Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers–Fine identity and the limiting case of Watson's formula that implies the Rogers–Ramanujan identities.

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35

Hiltebeitel, Alf. "Folklore and HinduismAnother Harmony: Essays on the Folklore of India. Stuart H. Blackburn , A. K. Ramanujan." History of Religions 27, no. 2 (November 1987): 216–18. http://dx.doi.org/10.1086/463113.

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36

Chen, Xigeng, and Maohua Le. "On the number of solutions of the generalized Ramanujan--Nagell equation $x^2-D=k^n$." Publicationes Mathematicae Debrecen 49, no. 1-2 (July 1, 1996): 85–92. http://dx.doi.org/10.5486/pmd.1996.1688.

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37

DEKA BARUAH, NAYANDEEP, and KALLOL NATH. "INFINITE FAMILIES OF ARITHMETIC IDENTITIES FOR 4-CORES." Bulletin of the Australian Mathematical Society 87, no. 2 (June 7, 2012): 304–15. http://dx.doi.org/10.1017/s0004972712000378.

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AbstractLetu(n) andv(n) be the number of representations of a nonnegative integernin the formsx2+4y2+4z2andx2+2y2+2z2, respectively, withx,y,z∈ℤ, and leta4(n) andr3(n) be the number of 4-cores ofnand the number of representations ofnas a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we prove that$u(8n+5)=8a_4(n)=v(8n+5)=\frac {1}{3}r_3(8n+5)$. With the help of this and a classical result of Gauss, we find a simple proof of a result ona4(n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’,Acta Arith.80(1997), 249–272]. We also find some new infinite families of arithmetic relations involvinga4(n) .

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38

Le, Maohua. "A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$." Acta Arithmetica 78, no. 1 (1996): 11–18. http://dx.doi.org/10.4064/aa-78-1-11-18.

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39

HIRSCHHORN, MICHAEL D., and JAMES A. SELLERS. "ELEMENTARY PROOFS OF PARITY RESULTS FOR 5-REGULAR PARTITIONS." Bulletin of the Australian Mathematical Society 81, no. 1 (July 2, 2009): 58–63. http://dx.doi.org/10.1017/s0004972709000525.

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AbstractIn a recent paper, Calkin et al. [N. Calkin, N. Drake, K. James, S. Law, P. Lee, D. Penniston and J. Radder, ‘Divisibility properties of the 5-regular and 13-regular partition functions’, Integers8 (2008), #A60] used the theory of modular forms to examine 5-regular partitions modulo 2 and 13-regular partitions modulo 2 and 3; they obtained and conjectured various results. In this note, we use nothing more than Jacobi’s triple product identity to obtain results for 5-regular partitions that are stronger than those obtained by Calkin and his collaborators. We find infinitely many Ramanujan-type congruences for b5(n), and we prove the striking result that the number of 5-regular partitions of the number n is even for at least 75% of the positive integers n.

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40

Kwak, Jin, and Alexander Mednykh. "Enumeration of branched coverings of closed orientable surfaces whose branch orders coincide with multiplicity." Studia Scientiarum Mathematicarum Hungarica 44, no. 2 (April 1, 2007): 215–23. http://dx.doi.org/10.1556/sscmath.2007.1014.

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The number Nn , g , r of nonisomorphic n -fold branched coverings of a given closed orientable surface S of genus g with r ≧ 1 branch points of order n is determined. The result is given in terms of the Euler characteristic of the surface S with r points removed and the von Sterneck-Ramanujan function ϕ( k,n ) = Σ ( d , n )=1 exp (\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{{2\pi ikd}}{n}$$ \end{document}). More precisely, if v = 2 g − 2 + r then \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$N_{n,g,r} = \sum\limits_{\ell |n,\ell m = n} {(m!\ell ^m )^\nu } \sum\limits_{k = 1} {\left( {\frac{{\Phi (k,\ell )}}{n}} \right)^r } \sum\limits_{s = 0}^{m - 1} {( - 1)^{sr} \left( {\begin{array}{*{20}c} {m - 1} \\ s \\ \end{array} } \right)^{ - \nu } } .$$ \end{document}.

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41

S, Hema, Hitha Haridas, and Nimisha P. "Rivers in Deluge: Degradation and Ecological Restoration in the Poems of A. K. Ramanujan and Keki N. Daruwalla." International Journal of English Literature and Social Sciences 6, no. 3 (2021): 284–87. http://dx.doi.org/10.22161/ijels.63.38.

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42

Ostrowski, Roman, Jan Kusiński, Krzysztof Czyż, Antoni Rycyk, Antoni Sarzyński, Wojciech Skrzeczanowski, Marek Strzelec, and Olaf Czyż. "The influence of periodical, interference laser micromachining on nano-crystallization of amorphous magnetic ribbons." Photonics Letters of Poland 9, no. 3 (September 30, 2017): 91. http://dx.doi.org/10.4302/plp.v9i3.762.

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The main problem connected with crystallization of amorphous magnetic materials from the FeSiB(X) group is the increase of their fragility with the increasing content of crystalline phase. To overcome this defect the possibilities offered by the interference, pulse laser heating of FeSiB ribbons were utilized. A simple, interference prismatic system at the output of Nd:YAG laser (8-10 ns, 2 J, 1064 nm) was used. The investigations included the influence of laser radiation energy density, number of laser pulses and their periodical, spatial surface arrangement on the crystallization process. Full Text: PDF ReferencesY.R. Zhang and R.V. Ramanujan, "A study of the crystallization behavior of an amorphous Fe77.5Si13.5B9 alloy", Mater. Sci. Eng. A 416, 161 (2006). CrossRef T. Alam, T. Borkar, S. Joshi, S. Katakamb, X. Chenc and N.B. Dahotre, "Influence of niobium on laser de-vitrification of Fe-Si-B based amorphous magnetic alloys", J. Non-Cryst. Solids 428, 75 (2015). CrossRef K.G. Pradeep, G. Herzer, P. Choi and D. Raabe, "Atom probe tomography study of ultrahigh nanocrystallization rates in FeSiNbBCu soft magnetic amorphous alloys on rapid annealing", Acta Mater. 68, 295 (2014). CrossRef S. Katakam, J.Y. Hwang, H. Vora, S.P. Harimkar, R. Banerjee and N.B. Dahotre, "Laser-induced thermal and spatial nanocrystallization of amorphous Fe-Si-B alloy", Scr. Mater. 66, 538 (2012). CrossRef J. Kusiński, A. Sypień, G. Kusiński and C. Nilson, "Microstructure and nanochemistry of Ca-doped cobalt oxide single crystals", Mat. Chem. Phys. 81, 390 (2003). CrossRef C. Smith, S. Katakam, S. Nag, X. Chen, R.V. Ramanujan, N.B. Dahotre and R. Banerjee, "Improved soft magnetic properties by laser de-vitrification of Fe-Si-B amorphous magnetic alloys", Mater. Lett. 122, 155 (2014). CrossRef Ch. Lu and R. H. Lipson, "Interference lithography: a powerful tool for fabricating periodic structures", Laser Photonics Rev. 4, 568 (2010). CrossRef K. Czyż, J. Marczak, R. Major, A. Mzyk, A. Rycyk, A. Sarzyński and M. Strzelec, "Selected laser methods for surface structuring of biocompatible diamond-like carbon layers", Diam. Relat. Mater. 67, 26 (2016). CrossRef A. Sarzyński, J. Marczak, M. Strzelec, A. Rycyk, K. Czyż and D. Chmielewska, "Laser micro-structuring of surfaces for applications in materials and biomedical science", Proc. SPIE, 10159, 101590A (2016). CrossRef

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43

M, Kavitha. "A Study of Selected Stories of Ki. Ra." International Research Journal of Tamil 4, S-4 (July 7, 2022): 29–36. http://dx.doi.org/10.34256/irjt22s45.

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Srikrishnaraja Narayana Perumal Ramanujan alias K. Rajanarayanan is the name of a man who has a deep concern and love for the community. He is a writer who has beautifully recorded the Karisal forest, the oak trees, the cotton forest, the people familiar to everyone, the realities that stand out in reality and are sewn into the chest. Writer Boomani Ki. Ra. explains how this thing, which went unnoticed by many legends in Tamil literature, was only caught by that nonchan like a chicken trapped in a cage. Wonders about. Ezra refers to Ki. Ra as the Bischoff of Tamil literature. Writers Ekalaivan Model. Ki. Ra's claim that they did not learn their profession by keeping any instrument, nor did they want to teach anyone by instrument informs us of many thoughts about the creativity of the Creator. Based on that statement, the purpose of this article is to express Ki. Ra's creativity, especially through KR's selected stories.

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44

Faried, Nashat, and Mona Fathey. "S-nuclearity and n-diameters of infinite Cartesian products of bounded subsets in Banach spaces." Acta et Commentationes Universitatis Tartuensis de Mathematica 7 (December 31, 2003): 45–55. http://dx.doi.org/10.12697/acutm.2003.07.05.

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In this paper we classify compact subsets of a normed space according to the rate of convergence to zero of its sequence {δn(B)} of Kolmogorov diameters. We introduce σ-compact sets to satisfy that {δn(B)}∈σ where σ is an ideal of convergent to zero sequences. Examples of sequence ideals are the ideals of rapidly decreasing sequences {λn} satisfying limn→∞λnnα=0 or radically decreasing sequences satisfying limn→∞(|λn|)1/n=0. In the case that σ is the ideal of rapidly decreasing sequences, this notion is identical to the S-nuclearity introduced by K. Astala and M. S. Ramanujan in 1987. We show that the infinite Cartesian product ∏i=1∞Bi of compact sets Bi is ℓp-compact in ℓp(Xi), for all p>0, if (δ0(Bi))∈S. In this case, we give upper estimates for the n-th diameters of ∏i=1∞Bi in ℓp(Xi) for any p>0.

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45

MC LAUGHLIN, JAMES. "FURTHER RESULTS ON VANISHING COEFFICIENTS IN INFINITE PRODUCT EXPANSIONS." Journal of the Australian Mathematical Society 98, no. 1 (November 11, 2014): 69–77. http://dx.doi.org/10.1017/s1446788714000536.

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AbstractWe extend results of Andrews and Bressoud [‘Vanishing coefficients in infinite product expansions’, J. Aust. Math. Soc. Ser. A27(2) (1979), 199–202] on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if $$\begin{eqnarray}\frac{(q^{r-tk},q^{mk-(r-tk)};q^{mk})_{\infty }}{(\pm q^{r},\pm q^{mk-r};q^{mk})_{\infty }}=:\mathop{\sum }_{n=0}^{\infty }c_{n}q^{n}\end{eqnarray}$$ for certain integers $k$, $m$, $s$ and $t$, where $r=sm+t$, then $c_{kn-rs}$ is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon [‘Vanishing coefficients in the expansion of products of Rogers–Ramanujan type’, in: The Rademacher Legacy to Mathematics (University Park, PA, 1992), Contemporary Mathematics, 166 (American Mathematical Society, Providence, RI, 1994), 129–139], but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms of integer partitions.

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46

Patel, Geeta. "Three Indian Poets: Nissim Ezekiel, A. K. Ramanujan, Dom Moraes. By Bruce King. Madras: Oxford University Press, 1991. 147 pp. $7.95." Journal of Asian Studies 51, no. 4 (November 1992): 960–61. http://dx.doi.org/10.2307/2059108.

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47

Richman, Paula. "When God Is a Customer: Telugu Courtesan Songs by Kṣetrayya and Others. A. K. Ramanujan , Velcheru Narayana Rao , David Shulman , Kṣetrayya". History of Religions 36, № 1 (серпень 1996): 58–60. http://dx.doi.org/10.1086/463445.

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48

BRINGMANN, KATHRIN, KARL MAHLBURG, and ROBERT C. RHOADES. "Taylor coefficients of mock-Jacobi forms and moments of partition statistics." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 2 (July 9, 2014): 231–51. http://dx.doi.org/10.1017/s0305004114000292.

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AbstractWe develop a new technique for deriving asymptotic series expansions for moments of combinatorial generating functions that uses the transformation theory of Jacobi forms and “mock” Jacobi forms, as well as the Hardy-Ramanujan Circle Method. The approach builds on a suggestion of Zagier, who observed that the moments of a combinatorial statistic can be simultaneously encoded as the Taylor coefficients of a function that transforms as a Jacobi form. Our use of Jacobi transformations is a novel development in the subject, as previous results on the asymptotic behavior of the Taylor coefficients of Jacobi forms have involved the study of each such coefficient individually using the theory of quasimodular forms and quasimock modular forms.As an application, we find asymptotic series for the moments of the partition rank and crank statistics. Although the coefficients are exponentially large, the error in the series expansions is polynomial, and have the same order as the coefficients of the residual Eisenstein series that are undetectable by the Circle Method. We also prove asymptotic series expansions for the symmetrized rank and crank moments introduced by Andrews and Garvan, respectively. Equivalently, the former gives asymptotic series for the enumeration of Andrews k-marked Durfee symbols.

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49

Sengupta, Sagaree. "The Oxford Anthology of Modern Indian Poetry. Edited by Vinay Dharwadker and A. K. Ramanujan. Delhi: Oxford University Press, 1995. xx, 265 pp. $18.95 (cloth)." Journal of Asian Studies 55, no. 2 (May 1996): 496–98. http://dx.doi.org/10.2307/2943416.

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50

Nimeshika Venkatesan. "Sita’s Story: Intertextuality and Folkloric Allusions in the Creation of a Desi Feminist Discourse in Nandini Sahu’s Sita." Creative Saplings 1, no. 01 (October 22, 2023): 12–25. http://dx.doi.org/10.56062/gtrs.2023.1.01.410.

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The story of Sita in the Indian epic Ramayana has over time been subject to several transformations, reinterpretations, and recontextualization. Sita the fiercely dedicated wife of Lord Rama has evolved to become a woman protagonist and, in some cases, even a feminist idol. Although there have been several modern interpretations of the Ramayana, Nandini Sahu’s Sita composed as a poetic memoir running into 25 cantos, in its form, content, and context is situated in a liminal space between the real world and the mythical world. This liminality is otherwise referred to as the “permeable membrane” in the words of A. K Ramanujan provides scope for many voices to emerge; from orality, from marga and desi mediums all of which oscillate between the temporal zones of the past, present, and future, constantly engaging with one another. Furthermore, the figure of Sita and her narratives extend to what Sahu refers to as the “Sitaness” in every woman whose agency has been snatched. For instance, she yokes together women protagonists from the Literary domain such as Desdemona, mythical namely, Trijada, and historical such as Meerabai, Mother Teresa, Kalpna Chawla, and even Nirbhaya respectively to create a uniquely Indian feminist discourse highlighting various instances of injustice meted towards women. Moreover, the poem is crafted using self-reflexive storytelling inspired by oral tradition and folklore. It also implements multilayered intertextual allusions to reimagine Sita as a woman protagonist transcending time and space. Therefore, this paper will investigate the intertextual and folkloric allusions in Nandini Sahu’s Sita consequently exploring the relevance of this composition as a contribution towards the creation of a desi-feminist discourse.

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