Ramanujan journal

2015Panaitopol, L.: A formula for \(\pi (x)\) applied to a result of Koninck-Ivić. Nieuw Arch. Wiskd. 5(1), 55–56 (2000)MathSciNet Google Scholar Ramanujan, S.: A proof of Bertrand’s postulate. J. Indian Math. Soc. 11, 181–182 (1919) Google Scholar Rosser, J.B.: The \(n\)-th prime is greater than \(n \log n\). Proc. Lond. Math. Soc. 45, 21–44 (1939)Article Google Scholar Rosser, J.B., Schoenfeld, L.: Approximate formulas for some functions of prime numbers. Ill. J. Math. 6, 64–94 (1962)MATH MathSciNet Google Scholar Shevelev, V.: Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), Article 12.1.1Sondow, J.: Ramanujan primes and Bertrand’s Postulate. Am. Math. Monthly 116, 630–635 (2009)Article MATH MathSciNet Google Scholar Sondow, J.: Sequence A104272. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015Sondow, J.: Sequence A233739. The on-line encyclopedia of integer sequences. Accessed 13 Apr 2015Sondow, J., Nicholson, J.W., Noe, T.D.: Ramanujan primes: Bounds, Runs, Twins, and Gaps. J. Integer Seq. 14 (2011), Article 11.6.2Srinivasan, A.: An upper bound for Ramanujan primes. Integers 19 (2014), #A19Tchebychev, P.: Mémoire sur les nombres premiers. Mémoires des savants étrangers de l’Acad. Sci. St.Pétersbourg 7 (1850), 17–33 [Also, Journal de mathématiques pures et appliques 17 (1852), 366–390]Trost, E.: Primzahlen. Birkhäuser, Basel/Stuttgart (1953)MATH Google Scholar Download referencesAcknowledgmentsI would like to thank Benjamin Klopsch for the helpful conversations. Also I would like to thank Elena Klimenko and Anitha Thillaisundaram for their careful reading of the paper.Author informationAuthors and AffiliationsMathematisches Institut, Heinrich-Heine-Universität Düsseldorf, 40225, Düsseldorf, GermanyChristian AxlerAuthorsChristian AxlerYou can also search for this author in PubMed Google ScholarCorresponding authorCorrespondence to Christian Axler.Rights and permissionsAbout this articleCite this articleAxler, C. On generalized Ramanujan primes. Ramanujan J 39, 1–30 (2016). citationReceived: 07 March 2014Accepted: 01 April 2015Published: 30 April 2015Issue Date: January 2016DOI: Subject Classification

Few explicit methods for constructing Ramanujan graphs. Many of the currently known techniques are reprised in Sect. 4.3 Two new families of unbalanced Ramanujan bigraphsThe existence of Ramanujan bigraphs of specified degrees and sizes has been well studied in recent years. A “road map” for constructing Ramanujan bigraphs is given in [4], and some abstract constructions of Ramanujan bigraphs are given in [5, 14]. These bigraphs have degrees \((p+1,p^3+1)\) for various values of p, such as \(p \equiv 5 ~\mathrm{mod}~12\), \(p \equiv 11 ~\mathrm{mod}~12\) [5], and \(p \equiv 3 ~\mathrm{mod}~4\) [14].Footnote 3 For each suitable choice of p, these papers lead to an infinite family of Ramanujan bigraphs. At present, these constructions are not explicit in terms of resulting in a biadjacency matrix of 0 s and 1 s. There is a paper under preparation by the authors of these papers to make these constructions explicit.The celebrated results of [23,24,25] show that there exist bipartite Ramanujan graphs of all degrees and all sizes. However, these results do not imply the existence of Ramanujan graphs of all sizes and degrees. To understand the approach of [24], we recall the notion of a Ramanujan covering as follows: A covering of a graph G refers to a graph \(G'\) and a surjection \(f:\,\mathcal {V}(G') \rightarrow \mathcal {V}(G)\) such that for any vertex \(v \in \mathcal {V}(G')\), the neighborhood of any vertex in \(G'\) [that is, the set of all vertices in \(G'\) which are joined to v by an edge] is mapped bijectively to the neighborhood of f(v) in G. Moreover, for a positive integer s, a covering \(G'\) of G is called an s-covering if every vertex in G has exactly s preimages in \(G'\). Finally, if an s-covering \(G'\) of a Ramanujan bigraph G satisfies the Ramanujan property in (3), it is said to be a Ramanujan s-covering of G.A special case of one of the key results in [24] implies the existence of a Ramanujan 2-covering of any Ramanujan bigraph G. Subsequently, this was generalized to the existence of a Ramanujan s-covering of a Ramanujan bigraph G for \(s=3,\,4\) in

Their adjacency or biadjacency matrices. We also focus on the two earlier-known constructions of Ramanujan bipartite graphs due to Lubotzky–Phillips–Sarnak [22] and Gunnells [19] and show that each can be converted into a nonbipartite graph. Note that every graph can be associated with a bipartite graph, but the converse is not true in general. Also, the research community prefers nonbipartite graphs over bipartite graphs. Thus, our proof that the LPS and Gunnells constructions can be converted to nonbipartite graphs is of some interest. All of this is addressed in Sect. 4. 3. [Construction of Ramanujan graphs with prohibited edges]: The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? The approach that we follow to answer this question is to start with an existing Ramanujan bigraph, and then to perturb its edge set so as to eliminate the prohibited edges and replace them by other edges that are nonprohibited. This procedure retains the biregularity of the graph. We then show that our replacement procedure also retains the Ramanujan nature of the bigraph, provided the gap between the second largest singular value of the biadjacency matrix and the “Ramanujan bound” is larger than twice the maximum number of prohibited edges at each vertex. These questions are studied in Section 7. 1.1 Organization of paperThis article is organized as follows. In Sect. 2, we present a brief review of Ramanujan graphs and bigraphs.In Sect. 3, we address the first goal of this article, and present the first explicit construction of an infinite family of unbalanced Ramanujan bigraphs.In Sect. 4 (Sects 4.1–4.4), we address the second goal of this article. We review many of the known methods for constructing Ramanujan graphs, based on the original publications.In Sect. 5, we shed further light on the two constructions of Ramanujan bipartite graphs due to Lubotzky–Phillips–Sarnak [22] and Gunnells [19] and show how each can be converted into a nonbipartite graph.In Sect. 6,

Of Ramanujan graphs and balanced bigraphs, and no explicit constructions of an unbalanced Ramanujan bigraph. In this paper, we presented for the first time an infinite family of unbalanced Ramanujan bigraphs with explicitly constructed biadjacency matrices. In addition, we have also shown how to construct the adjacency matrices for the currently available families of Ramanujan graphs. These explicit constructions, as well as forthcoming ones based on [5, 14], are available for only a few combinations of degree and size. In contrast, it is known from [24, 25] that Ramanujan graphs are known to exist for all degrees and all sizes. The main limiting factor is that these are only existence proofs and do not lead to explicit constructions. A supposedly polynomial-time algorithm for constructing Ramanujan graphs of all degrees and sizes is proposed in [11]. But it is still a conceptual algorithm and no code has been made available. Therefore it is imperative to develop efficient implementations of the ideas proposed in [25], and/or to develop other methods to construct Ramanujan graphs of most degrees and sizes. We should also note that the work of [24, 25] shows the existence of balanced, bipartite graphs of all degrees and sizes. Therefore, in Sect. 5 of this article, we have also looked at how the constructions of [22] and Gunnells [19] can be further analyzed to derive nonbipartite Ramanujan graphs.It is worth pointing out that efficient solutions of the matrix completion problem do not really require the existence of Ramanujan graphs of all sizes and degrees. It is enough if the “gaps” in the permissible values for the degrees and the sizes are very small. If this extra freedom leads to substantial simplification in the construction procedures, then it would be a worthwhile tradeoff. However, research on this problem is still at a nascent stage.Finally, in Sect. 7, we address another issue in the matrix completion problem, namely the “missing measurements” problem. This leads to the problem of the construction of a Ramanujan bigraph (not necessarily balanced) in which a certain set of edges is prohibited. In a typical real-life application, the

Observe too that not all authors make this distinction. Definition 2 A \((d_r,d_c)\)-biregular bipartite graph is said to be a Ramanujan bigraph if the second largest singular value of its biadjacency matrix, call it \(\sigma _2\), satisfies$$\begin{aligned} \sigma _2 \le \sqrt{d_r-1} + \sqrt{d_c-1} . \end{aligned}$$ (3) It is easy to see that Definition 2 contains the second case of Definition 1 as a special case when \(d_r = d_c = d\). A Ramanujan bigraph with \(d_r \ne d_c\) is called an unbalanced Ramanujan bigraph.The rationale behind the bounds in these definitions is given the following results. In the interests of brevity, the results are paraphrased and the reader should consult the original sources for precise statements. Theorem 1 (Alon–Boppana bound; see [1]) Fix d and let \(n \rightarrow \infty \) in a d-regular graph with n vertices. Then$$\begin{aligned} \liminf _{n \rightarrow \infty } | \lambda _2 | \ge 2 \sqrt{d-1} . \end{aligned}$$ (4) Theorem 2 (see [16]) Fix \(d_r,d_c\) and let \(n_r, n_c\) approach infinity subject to \(d_r n_r = d_c n_c\). Then$$\begin{aligned} \liminf _{n_r \rightarrow \infty , n_c \rightarrow \infty } \sigma _2 \ge \sqrt{d_r-1} + \sqrt{d_c-1} . \end{aligned}$$ (5) Given that a d-regular graph has d as its largest eigenvalue \(\lambda _1\), a Ramanujan graph is one for which the ratio \(\lambda _2/\lambda _1\) is as small as possible, in view of the Alon–Boppana bound of Theorem 1. Similarly, given that a \((d_r,d_c)\)-regular bipartite graph has \(\sigma _1 = \sqrt{d_r d_c}\), a Ramanujan bigraph is one for which the ratio \(\sigma _2/\sigma _1\) is as small as possible, in view of Theorem 2.In a certain sense, Ramanujan graphs and Ramanujan bigraphs are pervasive. To be precise, if d is kept fixed and \(n \rightarrow \infty \), then the fraction of d-regular, n-vertex graphs that satisfy the Ramanujan property approaches one; see [17, 18]. Similarly, if \(d_r,d_c\) are kept fixed and \(n_r, n_c \rightarrow \infty \) (subject of course to the condition that \(d_r n_r = d_c n_c\)), then the fraction of \((d_r,d_c)\)-biregular graphs that are Ramanujan bigraphs approaches one; see [7]. However, despite their prevalence, there are relatively

Thinking. It was very difficult for great mathematicians to understand how it happened. A problem which would take about six hours for an eminent mathematician to solve – and then too he was not sure about being right – Ramanujan solved instantaneously, unerringly.It proved that Ramanujan was not replying through the medium of the mind. He was not very learned, he had actually failed in matriculation; there was no other sign of intellectual ability, but in connection with mathematics he was superhuman. Something happened that was beyond the human mind. He died when he was thirty-six because of tuberculosis.When he was in hospital, Hardy, along with two or three other mathematician friends, went to see him. As it happened, he parked his car in such a place so that Ramanujan could see its number plate. When Hardy went into Ramanujan’s room, he told Hardy that his number plate was unique: it had four special aspects to it. After that, Ramanujan died. Hardy took six months to understand what Ramanujan meant, but he could only discover three of the four aspects. On his death he left a will that research work on that number should continue, to find out the fourth aspect. Because Ramanujan had said there was a fourth, there had to be. Twenty-two years after Hardy’s death, the fourth was discovered. Ramanujan was right.Whenever he began to look into any mathematical problem something began to happen in the middle space between his two eyebrows. Both his eyeballs turned upwards,