Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

The Lambert series factorization theorem

A factorization for partial sums of Lambert series is introduced in this paper. As corollaries, we derive some connections between partitions and divisors. These results can be easily used to discover and prove new combinatorial identities involving important functions from number theory: the Möbius function \(\mu (n)\), Euler’s totient \(\varphi (n)\), Jordan’s totient \(J_k(n)\), Liouville’s function \(\lambda (n)\), the von Mangoldt function \(\Lambda (n)\), and the divisor function \(\sigma _x(n)\). The fascinating feature of these identities is their common nature.     

A generalization of the Laurent series

A system of functions, biorthogonal to the D-basis introduced by M. K. Fage for a linear differential operator D with analytic coefficients, is constructed. Series in the combination of the D-basis with this system generalize Laurent series in the same way as series in the D-basis generalize Taylor series.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 521–531, May, 1971.     

On the generalization of a retrocirculant

We study the properties of matrices of the form P(σ)A where σ is induced by an automorphism of an abelian group G and A is a group matrix. P(σ)A is a generalization of a retrocirculant. We also determine the eigenvalues of P(σ)A.     

A generalization of the Gauss hypergeometric series

A generalization of the Gauss hypergeometric function to t variables is given, and the Euler identity is shown to hold for this generalized function. The corresponding generalization of the Saalschütz theorem is also obtained.     

Factorization theorems for generalized Lambert series and applications

The Ramanujan Journal - We prove new variants of the Lambert series factorization theorems studied by the authors which correspond to a more general class of Lambert series expansions of the form...     

On a generalization of the Gallai-Sylvester theorem

It is shown that ifS ^d , affS=aff ^d , and every hyperplane spanned by (a subset of)S misses fewer thank points ofS(k2), then (a) #Skm ifd=2m–1 is odd and (b) #Skm+1 ifd=2m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For oddd the proofs are later modified to purely algebraic ones, and carry over to , where is an arbitrary field. For evend, (b) is generally not true when , but we prove some weaker inequalities that do hold over arbitrary fields.This is part of a Ph.D. thesis, supervised by Professor Micha A. Perles at the Hebrew University of Jerusalem. This research was supported in part by the Landau Center for Mathematical Research.     

On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψk(x;q,a) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression . This theorem not only gives the asymptotic formula for ψk(x;q,a) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.     

On a generalization of the Kalman-Yakubovic lemma

In this paper we generalize the Kalman-Yakubovic lemma to infinite dimensions—or, more precisely, to semigroups of operators over a Hilbert space. The proof differs substantially from the finite-dimensional version and is based on the Paley-Wiener-Helson-Lowdenslager factorization theorem.     

On a generalization of the auslander-bridger transpose

Let λ be an artin algebra and _λω λ a faithfully balanced self-orthogonal bimodule. We generalize the notion of the Auslander-Bridger transpose to that of the transpose with respect to _λω _λand obtain some properties about dual modules with respect to_λω _λFurther, we characterize cotilting bimodules and give criteria for computing generalized Goren-stein dimension.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.     

On a generalization of the logistic distribution

Summary Expressions for the moment generating functions, cumulants and coefficients of kurtosis of a generalization of the logistic distribution are derived and used to show that any symmetric version of this distribution can be closely and simply approximated by a Studentt distribution. Related approximation of the distribution of the logistic sample median is discussed.     

On a generalization of the bessel inequality

Translated from Matematicheskie Zametki, Vol. 51, No. 5, pp. 149–150, May, 1992.