Ramanujan's Formula Research Articles

The Bauer–Ramanujan formula: historical analyses and perspectives

The formula 2 π = 1 − 5 ⋅ ( 1 2 ) 3 + 9 ⋅ ( 1 ⋅ 3 2 ⋅ 4 ) 3 − ⋯ was famously included as a discovery in Ramanujan's first letter to Hardy in 1913, and has been referred to as the Bauer–Ramanujan formula, in view of Bauer's 1859 proof of the above formula. There is a rich history associated with this formula and its many and dramatically different proofs, including a computer-based proof due to Zeilberger that may be seen as groundbreaking in the history of computer-assisted proofs. In addition to a complete survey we provide of all known proofs of the Bauer–Ramanujan formula, we introduce historical analyses based on these proofs, by arguing that the history of the Bauer–Ramanujan formula and our account of this history may be seen as being representative of much broader trends in the history of mathematics. In this regard, the earlier proofs tend to rely on one of the oldest and most basic tools in classical analysis, namely, interchanging the order of limiting operations. In contrast, the more modern proofs tend to rely on computer-related approaches toward summation problems, as in with Zeilberger-type and Gosper-type telescoping arguments.

Recent developments pertaining to Ramanujan’s formula for odd zeta values

In this expository article, we discuss the contributions made by several mathematicians with regard to a famous formula of Ramanujan for odd zeta values. The goal is to complement the excellent survey by Berndt and Straub (2017) with some of the recent developments that have taken place in the area in the last decade or so.

Some Fourier transforms involving confluent hypergeometric functions

ABSTRACT In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas.

Remarks on a formula of Ramanujan

Assuming an averaged form of Mertens’ conjecture and that the ordinates of the non-trivial zeros of the Riemann zeta function are linearly independent over the rationals, we analyse the finer structure of the terms in a well-known formula of Ramanujan.

On a higher-order version of a formula due to Ramanujan

A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1 + 1 5 ( 1 2 ) 2 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 2 + ⋯ = Γ 4 ( 1 4 ) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1 + 1 5 ( 1 2 ) 4 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 4 + ⋯ = Γ 8 ( 1 4 ) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.

Optimizing the coefficients of the Ramanujan expansion

In this paper, we provide the coefficients of the Ramanujan expansion proposed by Wang [23] with a different method. Then, we present the rates of convergence of the expansion for the harmonic numbers, the values of the parameter h of the coefficients and some relevant inequalities. To demonstrate the superiority of our expansion with new coefficients over the classic Ramanujan formula, some numerical computations are also given.

Ramanujan-type series for , revisited

AbstractIn this note, we revisit Ramanujan-type series for $\frac {1}{\pi }$ and show how they arise from genus zero subgroups of $\mathrm {SL}_{2}(\mathbb {R})$ that are commensurable with $\mathrm {SL}_{2}(\mathbb {Z})$ . As illustrations, we reproduce a striking formula of Ramanujan for $\frac {1}{\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for $\frac {1}{\pi }$ . As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.

Hurwitz zeta functions and Ramanujan's identity for odd zeta values

Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing that associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.

A new Ramanujan-type identity for $$L(2k+1, \chi _1)$$

One of the celebrated formulas of Ramanujan is about odd zeta values, which has been studied by many mathematicians over the years. A notable extension was given by Grosswald in 1972. Following Ramanujan’s idea, we rediscovered a Ramanujan-type identity for \(\zeta (2k+1)\) that was first established by Malurkar and later by Berndt using different techniques. In the current paper, we extend the aforementioned identity of Malurkar and Berndt to derive a new Ramanujan-type identity for \(L(2k+1, \chi _1)\), where \(\chi _1\) is the principal character modulo prime p. In the process, we encounter a new family of Ramanujan-type polynomials. Furthermore, we establish a character analogue of Grosswald’s identity.

Extended higher Herglotz function II

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.

Exact limit theorems for restricted integer partitions

For a set of positive integers A, let pA(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy–Ramanujan formula for p(n) by showing that A has density α if and only if log⁡pA(n)∼log⁡p(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if log⁡pA(n)/log⁡p(αn) has lower limit 1. We answer this question negatively by constructing, for every α>0, a set of integers A of lower density α, satisfyinglim infn→∞log⁡pA(n)log⁡p(αn)≥(6π−oα(1))log⁡(1/α). We further show that the above bound is best possible (up to the oα(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.

A simple evaluation of a theta value, the Kronecker limit formula and a formula of Ramanujan

We evaluate the classic sum sum _{nin {mathbb {Z}}} e^{-pi n^2}. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuation. Even the Gamma function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula. Finally, employing the evaluation of the sum and some other ideas, we also obtain an undemanding proof of one of the most emblematic formulas of Ramanujan.

Parametrization of Kloosterman sets and SL3-Kloosterman sums

We stratify the SL3 big cell Kloosterman sets using the reduced word decomposition of the Weyl group element, inspired by the Bott-Samelson factorization. Thus the SL3 long word Kloosterman sum is decomposed into finer parts, and we write it as a finite sum of a product of two classical Kloosterman sums. The fine Kloosterman sums end up being the correct pieces to consider in the Bruggeman-Kuznetsov trace formula on the congruence subgroup Γ0(N)⊆SL3(Z). Another application is a new explicit formula, expressing the triple divisor sum function in terms of a double Dirichlet series of exponential sums, generalizing Ramanujan's formula.

Koshliakov zeta functions I: Modular relations

We examine an unstudied manuscript of N.S. Koshliakov over 150 pages long and containing the theory of two interesting generalizations ζp(s) and ηp(s) of the Riemann zeta function ζ(s), which we call Koshliakov zeta functions. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each p∈R+. The first one is a generalization of Ramanujan's famous formula for ζ(2m+1) and the second is an elegant extension of a modular relation on page 220 of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for ζ(4m+3).

On Ramanujan's formula for ζ(1/2) and ζ(2m + 1)

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for ζ(1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series,1rexp⁡(1sx)−1+2rexp⁡(2sx)−1+3rexp⁡(3sx)−1+⋯, where s is a positive integer and r−s is any even integer. Unfortunately, Ramanujan doesn't give any formula for it. This series was rediscovered by Kanemitsu, Tanigawa, and Yoshimoto, although they studied it only when r−s is a negative even integer. Recently, Dixit and the second author generalized the work of Kanemitsu et al. and obtained a transformation formula for the aforementioned series with r−s is any even integer. While extending the work of Kanemitsu et al., Dixit and the second author obtained a beautiful generalization of Ramanujan's formula for odd zeta values. In the current paper, we investigate transformation formulas for an infinite series, and interestingly, we derive Ramanujan's formula for ζ(1/2), Wigert's formula for ζ(1/k) as well as Ramanujan's formula for ζ(2m+1). Furthermore, we obtain a new identity for ζ(−1/2) in the spirit of Ramanujan.

Several transformation formulas for basic hypergeometric series

ABSTRACT In 1981, Andrews gave a four-variable generalization of Ramanujan's summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two series and Bailey's transformation formula for three series. Then, it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two summation formulas in terms of two limiting relations and Bailey's another transformation formula for three series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo–Schlosser transformation formula and other two transformation formulas.

On Euler–Ramanujan formula, Dirichlet series and minimal surfaces

In this paper, we rewrite two forms of an Euler–Ramanujan identity in terms of certain Dirichlet series and derive functional equation of the latter. We also use the Weierstrass–Enneper representation of minimal surfaces to obtain some identities involving these Dirichlet series and one complex parameter.

Finding Modular Functions for Ramanujan-Type Identities

This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for a(mn + t). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for p(11n + 6) with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions \( \mathit(\overline{p}) \)(5n + 2) and \( \mathit(\overline{p}) \)(5n + 3) and Andrews–Paule’s broken 2-diamond partition functions \( \triangle_{2} \)(25n+14) and \( \triangle_{2} \)(25n+24). It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews’ singular overpartition functions \( \mathit(\overline{Q}_{3,1}) \)(9n + 3) and \( \mathit(\overline{Q}_{3,1}) \)(9n + 6) due to Shen, the 2-dissection formulas of Ramanujan, and the 8-dissection formulas due to Hirschhorn.

On squares of odd zeta values and analogues of Eisenstein series

A Ramanujan-type formula involving the squares of odd zeta values is obtained. The crucial part in obtaining such a result is to conceive the correct analogue of the Eisenstein series involved in Ramanujan's formula for ζ(2m+1). The formula for ζ2(2m+1) is then generalized in two different directions, one, by considering the generalized divisor function σz(n), and the other, by studying a more general analogue of the aforementioned Eisenstein series, consisting of one more parameter N. A number of important special cases are derived from the first generalization. For example, we obtain a series representation for ζ(1+ω)ζ(−1−ω), where ω is a non-trivial zero of ζ(z). We also evaluate a series involving the modified Bessel function of the second kind in the form of a rational linear combination of ζ(4k−1) and ζ(4k+1) for k∈N.

A q-microscope for supercongruences

By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a ‘q-microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formula∑n=0∞(4n2n)(2nn)228n32n(8n+1)=23π, of the two supercongruencesS(p−1)≡p(−3p)(modp3)andS(p−12)≡p(−3p)(modp3), valid for all primes p>3, where S(N) denotes the truncation of the infinite sum at the N-th place and (−3⋅) stands for the quadratic character modulo 3.