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Journal articles on the topic 'Ramanujan's lost notebook'

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Published: 3 June 2025

Last updated: 31 July 2025

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1

BERNDT, BRUCE C., BYUNGCHAN KIM, and KENNETH S. WILLIAMS. "EULER PRODUCTS IN RAMANUJAN'S LOST NOTEBOOK." International Journal of Number Theory 09, no. 05 (2013): 1313–49. http://dx.doi.org/10.1142/s1793042113500292.

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In his famous paper, "On certain arithmetical functions", Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan's tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for the Euler products appearing in his lost notebook, we provide or sketch proofs using elementary methods, binary quadratic forms, and modular forms.

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2

Nil, Ratan Bhattacharjee, and Das Sabuj. "GENERALIZATIONS OF RAMANUJAN'S RANK FUNCTIONS COLLECTED FROM RAMANUJAN'S LOST NOTEBOOK." International Journal of Research – Granthaalayah 4, no. 3 (2017): 1–20. https://doi.org/10.5281/zenodo.846728.

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In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers. We prove the Mock Theta Conjectures with the help of Dyson’s rank and S. Ramanujan’s Mock Theta functions. These functions were quoted in Ramanujan’s lost notebook and unpublished papers. In1916, Ramanujan stated the theta series in x like A(x), B(x), C(x), D(x). We discuss the Ramanujan’s functions with the help of Dyson’s rank symbols. These functions are useful to prove the Mock Theta Conjectures. Now first Mock Theta Conjecture is “The number of partitions of 5n with rank congruent to

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3

BERNDT, BRUCE C., HENG HUAT CHAN, and YOSHIO TANIGAWA. "Two Dirichlet series evaluations found on page 196 of Ramanujan's Lost Notebook." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 2 (2012): 341–60. http://dx.doi.org/10.1017/s0305004112000151.

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AbstractOn page 196 in his lost notebook, S. Ramanujan offers evaluations of two particular Dirichlet series. In this paper, we establish Ramanujan's evaluations and more general results by various approaches. The different evaluations arising from different methods yield intriguing, unsuspecting identities.

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4

E. Andrews, George. "Ramanujan's “Lost” Notebook VIII: the entire Rogers–Ramanujan function." Advances in Mathematics 191, no. 2 (2005): 393–407. http://dx.doi.org/10.1016/j.aim.2004.03.012.

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5

BERNDT, BRUCE C., and PING XU. "An Integral Analogue of Theta Functions and Gauss Sums in Ramanujan's Lost Notebook." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 2 (2009): 257–65. http://dx.doi.org/10.1017/s0305004109002552.

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AbstractOne page in Ramanujan's lost notebook is devoted to claims about a certain integral with two parameters. One claim gives an inversion formula for the integral that is similar to the transformation formula for theta functions. Other claims are remindful of Gauss sums. In this paper we prove all the claims made by Ramanujan about this integral.

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6

CHAN, HENG HUAT, WEN-CHIN LIAW та VICTOR TAN. "RAMANUJAN'S CLASS INVARIANT λn AND A NEW CLASS OF SERIES FOR 1/π". Journal of the London Mathematical Society 64, № 1 (2001): 93–106. http://dx.doi.org/10.1017/s0024610701002241.

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On page 212 of his lost notebook, Ramanujan defined a new class invariant λn and constructed a table of values for λn. The paper constructs a new class of series for 1/π associated with λn. The new method also yields a new proof of the Borweins' general series for 1/π belonging to Ramanujan's ‘theory of q2’.

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7

Bressoud, David M. "Book Review: Ramanujan's lost notebook, Part I." Bulletin of the American Mathematical Society 43, no. 04 (2006): 585–92. http://dx.doi.org/10.1090/s0273-0979-06-01110-4.

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8

Andrews, George E. "Simplicity and Surprise in Ramanujan's "Lost" Notebook." American Mathematical Monthly 104, no. 10 (1997): 918. http://dx.doi.org/10.2307/2974472.

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9

Andrews, George E. "Ramanujan's “Lost” Notebook V: Euler's partition identity." Advances in Mathematics 61, no. 2 (1986): 156–64. http://dx.doi.org/10.1016/0001-8708(86)90072-1.

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10

Berndt, Bruce C., Heng Huat Chan, Song Heng Chan, and Wen-Chin Liaw. "Cranks and dissections in Ramanujan's lost notebook." Journal of Combinatorial Theory, Series A 109, no. 1 (2005): 91–120. http://dx.doi.org/10.1016/j.jcta.2004.06.013.

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11

Andrews, George E. "Simplicity and Surprise in Ramanujan's “Lost” Notebook." American Mathematical Monthly 104, no. 10 (1997): 918–25. http://dx.doi.org/10.1080/00029890.1997.11990740.

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12

Berndt, Bruce C., Soon-Yi Kang, and Jaebum Sohn. "Finite and infinite Rogers–Ramanujan continued fractions in Ramanujan's lost notebook." Journal of Number Theory 148 (March 2015): 112–20. http://dx.doi.org/10.1016/j.jnt.2014.09.019.

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13

Choi, Youn-Seo. "Generalization of two identities in Ramanujan's lost notebook." Acta Arithmetica 114, no. 4 (2004): 369–89. http://dx.doi.org/10.4064/aa114-4-6.

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14

Son, Seung Hwan. "Septic theta function identities in Ramanujan's lost notebook." Acta Arithmetica 98, no. 4 (2001): 361–74. http://dx.doi.org/10.4064/aa98-4-3.

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15

Berndt, Bruce C., and Daniel Schultz. "On sums of powers in Ramanujan's lost notebook." Applicable Analysis 90, no. 3-4 (2011): 725–30. http://dx.doi.org/10.1080/00036810903437812.

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16

Andrews, George E., and F. G. Garvan. "Ramanujan's “Lost” Notebook VI: The mock theta conjectures." Advances in Mathematics 73, no. 2 (1989): 242–55. http://dx.doi.org/10.1016/0001-8708(89)90070-4.

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17

Son, Seung Hwan. "Circular Summations of Theta Functions in Ramanujan's Lost Notebook." Ramanujan Journal 8, no. 2 (2004): 235–72. http://dx.doi.org/10.1023/b:rama.0000040483.55191.d1.

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18

Berndt, Bruce C., Junxian Li, and Alexandru Zaharescu. "The final problem: an identity from Ramanujan's lost notebook." Journal of the London Mathematical Society 100, no. 2 (2019): 568–91. http://dx.doi.org/10.1112/jlms.12228.

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19

McLaughlin, James, and Andrew V. Sills. "On a Pair of Identities from Ramanujan's Lost Notebook." Annals of Combinatorics 16, no. 3 (2012): 591–607. http://dx.doi.org/10.1007/s00026-012-0148-3.

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20

Choi, Youn-Seo. "Tenth order mock theta functions in Ramanujan's Lost Notebook." Inventiones Mathematicae 136, no. 3 (1999): 497–569. http://dx.doi.org/10.1007/s002220050318.

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21

Chan, Heng Huat, Zhi-Guo Liu, and Say Tiong Ng. "Circular summation of theta functions in Ramanujan's Lost Notebook." Journal of Mathematical Analysis and Applications 316, no. 2 (2006): 628–41. http://dx.doi.org/10.1016/j.jmaa.2005.05.015.

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22

Choie, YoungJu, and Rahul Kumar. "Period function of Maass forms from Ramanujan's lost notebook." Advances in Mathematics 474 (July 2025): 110317. https://doi.org/10.1016/j.aim.2025.110317.

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23

Chandankumar, S., H. S. Sumanth Bharadwaj, and Vijay Yadav. "ON A NEW PARAMETER INVOLVING RAMANUJAN'S THETA-FUNCTIONS." South East Asian J. of Mathematics and Mathematical Sciences 19, no. 02 (2023): 35–52. http://dx.doi.org/10.56827/seajmms.2023.1902.3.

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Srinivasa Ramanujan recorded explicit evaluations of certain quotients of theta functions in his lost notebook. Motivated by the works of Ramanujan, Jinhee Yi systematically studied the analogues of explicit evaluation of quotients of theta functions by defining parameters. In this work, we define a new parameter involving theta-functions and establish some modular relations to explicitly evaluate the parameter.

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24

Choi, Youn-Seo. "Tenth order mock theta functions in Ramanujan's lost notebook III." Proceedings of the London Mathematical Society 94, no. 1 (2006): 26–52. http://dx.doi.org/10.1112/plms/pdl006.

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25

Berndt, Bruce C., and Wenchang Chu. "Two entries on bilateral hypergeometric series in Ramanujan's lost notebook." Proceedings of the American Mathematical Society 135, no. 01 (2006): 129–34. http://dx.doi.org/10.1090/s0002-9939-06-08553-4.

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26

BERNDT, BRUCE C., and JAEBUM SOHN. "ASYMPTOTIC FORMULAS FOR TWO CONTINUED FRACTIONS IN RAMANUJAN'S LOST NOTEBOOK." Journal of the London Mathematical Society 65, no. 02 (2002): 271–84. http://dx.doi.org/10.1112/s0024610701002952.

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27

Berndt, Bruce C., Ae Ja Yee, and Jinhee Yi. "Theorems on partitions from a page in Ramanujan's lost notebook." Journal of Computational and Applied Mathematics 160, no. 1-2 (2003): 53–68. http://dx.doi.org/10.1016/s0377-0427(03)00613-7.

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28

Andrews, George E., and Dean Hickerson. "Ramanujan's “lost” notebook VII: The sixth order mock theta functions." Advances in Mathematics 89, no. 1 (1991): 60–105. http://dx.doi.org/10.1016/0001-8708(91)90083-j.

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29

Kang, Soon-Yi, Sung-Geun Lim, and Jaebum Sohn. "The continuous symmetric Hahn polynomials found in Ramanujan's lost notebook." Journal of Mathematical Analysis and Applications 307, no. 1 (2005): 153–66. http://dx.doi.org/10.1016/j.jmaa.2004.12.054.

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30

Choi, Youn-Seo. "Tenth Order Mock Theta Functions in Ramanujan's Lost Notebook II." Advances in Mathematics 156, no. 2 (2000): 180–285. http://dx.doi.org/10.1006/aima.2000.1948.

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31

McIntosh, Richard J. "The H and K Family of Mock Theta Functions." Canadian Journal of Mathematics 64, no. 4 (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 Rama

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32

ON, GENERALIGATIONS OF PARTITION FUNCTIONS, and Das Sabuj. "ON GENERALIGATIONS OF PARTITION FUNCTIONS." International Journal of Research – Granthaalayah 3, no. 10 (2017): 1–29. https://doi.org/10.5281/zenodo.851756.

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In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for X(n),Y(n) . In 2014, Sabuj developed the generating functions for . In 2005, George E. Andrews found the generating functions for In 1916, Ramanujan showed the generating functions for , , and . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan’s Lost Notebook. In 1967, G

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33

Andrews, George E. "Ramanujan's “Lost” notebook IX: the partial theta function as an entire function." Advances in Mathematics 191, no. 2 (2005): 408–22. http://dx.doi.org/10.1016/j.aim.2004.03.013.

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34

Mortenson, Eric. "Ramanujan's Radial Limits and Mixed Mock Modular Bilateralq-Hypergeometric Series." Proceedings of the Edinburgh Mathematical Society 59, no. 3 (2015): 787–99. http://dx.doi.org/10.1017/s0013091515000425.

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AbstractUsing results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsomet al.for mock theta functions. Here we see that Mortenson's previous work on the dual nature of Appell–Lerch sums and partial theta functions and on constructing bilateralq-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 bilateralq-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function of Gordon a

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35

Yesilyurt, Hamza. "Four identities related to third order mock theta functions in Ramanujan's lost notebook." Advances in Mathematics 190, no. 2 (2005): 278–99. http://dx.doi.org/10.1016/j.aim.2003.12.007.

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36

Dixit, Atul, and Bibekananda Maji. "Generalized Lambert series and arithmetic nature of odd zeta values." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 2 (2019): 741–69. http://dx.doi.org/10.1017/prm.2018.146.

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AbstractIt is pointed out that the generalized Lambert series $\sum\nolimits_{n = 1}^\infty {[(n^{N-2h})/(e^{n^Nx}-1)]} $ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page 332 of Ramanujan's Lost Notebook in a slightly more general form. We extend an important transformation of this series obtained by Kanemitsu, Tanigawa and Yoshimoto by removing restrictions on the parameters N and h that they impose. From our extension we deduce a beautiful new generalization of Ramanujan's famous formula for odd zeta values which, for N odd and m > 0, gives a relation between ζ(2m + 1) an

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37

Berndt, Bruce C., Byungchan Kim, and Ae Ja Yee. "Ramanujan's lost notebook: Combinatorial proofs of identities associated with Heine's transformation or partial theta functions." Journal of Combinatorial Theory, Series A 117, no. 7 (2010): 957–73. http://dx.doi.org/10.1016/j.jcta.2009.07.005.

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38

Bernd, Bruce C., and Heng Huat Chan. "Some Values for the Rogers-Ramanujan Continued Fraction." Canadian Journal of Mathematics 47, no. 5 (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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39

ANDREWS, GEORGE E., BRUCE C. BERNDT, SONG HENG CHAN, SUN KIM, and AMITA MALIK. "FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS." Nagoya Mathematical Journal 239 (September 21, 2018): 173–204. http://dx.doi.org/10.1017/nmj.2018.35.

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In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3) 4 (1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190 (2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions ar

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40

Bhattacharjee, Nil Ratan, and Sabuj Das. "GENERALIZATIONS OF RAMANUJAN’S RANK FUNCTIONS COLLECTED FROM RAMANUJAN’S LOST NOTEBOOK." International Journal of Research -GRANTHAALAYAH 4, no. 3 (2016): 1–20. http://dx.doi.org/10.29121/granthaalayah.v4.i3.2016.2780.

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In1916, Srinivasa Ramanujan defined the Mock Theta functions in his lost notebook and unpublished papers. We prove the Mock Theta Conjectures with the help of Dyson’s rank and S. Ramanujan’s Mock Theta functions. These functions were quoted in Ramanujan’s lost notebook and unpublished papers. In1916, Ramanujan stated the theta series in x like A(x), B(x), C(x), D(x). We discuss the Ramanujan’s functions with the help of Dyson’s rank symbols. These functions are useful to prove the Mock Theta Conjectures. Now first Mock Theta Conjecture is “The number of partitions of 5n with rank congruent to

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41

Berndt, Bruce C., Sen-Shan Huang, Jaebum Sohn, and Seung Hwan Son. "Some Theorems on the Rogers–Ramanujan Continued Fraction in Ramanujan’s Lost Notebook." Transactions of the American Mathematical Society 352, no. 5 (2000): 2157–77. http://dx.doi.org/10.1090/s0002-9947-00-02337-0.

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42

Bhattacharjee, Nil Ratan, and Sabuj Das. "ON GENERALIGATIONS OF PARTITION FUNCTIONS." International Journal of Research -GRANTHAALAYAH 3, no. 10 (2015): 1–29. http://dx.doi.org/10.29121/granthaalayah.v3.i10.2015.2928.

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In 1742, Leonhard Euler invented the generating function for P(n). Godfrey Harold Hardy said Srinivasa Ramanujan was the first, and up to now the only, Mathematician to discover any such properties of P(n). In 1916, Ramanujan defined the generating functions for X(n),Y(n) . In 2014, Sabuj developed the generating functions for . In 2005, George E. Andrews found the generating functions for In 1916, Ramanujan showed the generating functions for , , and . This article shows how to prove the Theorems with the help of various auxiliary functions collected from Ramanujan’s Lost Notebook. In 1967, G

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43

Kostov, V. "A domain free of the zeros of the partial theta function." Matematychni Studii 58, no. 2 (2023): 142–58. http://dx.doi.org/10.30970/ms.58.2.142-158.

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The partial theta function is the sum of the series \medskip\centerline{$\displaystyle\theta (q,x):=\sum\nolimits _{j=0}^{\infty}q^{j(j+1)/2}x^j$,}\medskip\noi where $q$is a real or complex parameter ($|q|<1$). Its name is due to similaritieswith the formula for the Jacobi theta function$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$. The function $\theta$ has been considered in Ramanujan's lost notebook. Itfinds applicationsin several domains, such as Ramanujan type$q$-series, the theory of (mock) modular forms, asymptotic analysis, statistical physics, combinatorics and most recently

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44

Berndt, Bruce C. "Some integrals in Ramanujan’s lost notebook." Proceedings of the American Mathematical Society 132, no. 10 (2004): 2983–88. http://dx.doi.org/10.1090/s0002-9939-04-07430-1.

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45

Hirschhorn, Michael D., and Vasile Sinescu. "Elementary Algebra in Ramanujan’s Lost Notebook." Fibonacci Quarterly 51, no. 2 (2013): 123–29. http://dx.doi.org/10.1080/00150517.2013.12427953.

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46

Hirschhorn, Michael D. "“Algebraical oddities” in Ramanujan’s lost notebook." Ramanujan Journal 29, no. 1-3 (2012): 69–77. http://dx.doi.org/10.1007/s11139-012-9395-5.

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47

Hirschhorn, Michael D. "Elementary analysis in Ramanujan’s lost notebook." Ramanujan Journal 37, no. 3 (2014): 641–51. http://dx.doi.org/10.1007/s11139-013-9551-6.

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48

Vijaya Shankar, A. I. "Eisenstein series of level 6 and level 10 with their applications to theta function identities of Ramanujan." Notes on Number Theory and Discrete Mathematics 28, no. 3 (2022): 581–88. http://dx.doi.org/10.7546/nntdm.2022.28.3.581-588.

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S. Ramanujan recorded theta function identities of different levels in the unorganized pages of his second notebook and the lost notebook. In this paper, we prove level 6 and level 10 theta function identities by using Eisenstein series identities.

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49

Lee, Jongsil, and Jaebum Sohn. "Some Continued Fractions in Ramanujan’s Lost Notebook." Monatshefte für Mathematik 146, no. 1 (2005): 37–48. http://dx.doi.org/10.1007/s00605-005-0304-5.

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50

Berndt, Bruce C., and Geumlan Choi. "A continued fraction from Ramanujan’s lost notebook." Aequationes mathematicae 69, no. 3 (2005): 257–62. http://dx.doi.org/10.1007/s00010-004-2738-6.

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