Arithmetical properties of multiple Ramanujan sums

In the present paper, we introduce a multiple Ramanujan sum for arithmetic functions, which gives a multivariable extension of the generalized Ramanujan sum studied by D.R. Anderson and T.M. Apostol. We then find fundamental arithmetic properties of the multiple Ramanujan sum and study several types of Dirichlet series involving the multiple Ramanujan sum. As an application, we evaluate higher-dimensional determinants of higher-dimensional matrices, the entries of which are given by values of the multiple Ramanujan sum.     

Certain weighted averages of generalized Ramanujan sums

We derive certain identities involving various known arithmetical functions and a generalized version of Ramanujan sum. L. Tóth constructed certain weighted averages of Ramanujan sums with various arithmetic functions as weights. We choose a generalization of Ramanujan sum given by E. Cohen and derive the weighted averages corresponding to the versions of the weighted averages established by Tóth.     

Results on vanishing coefficients in infinite q-series expansions for certain arithmetic progressions mod 7

The Ramanujan Journal - Recently, Mc Laughlin proved some results on vanishing coefficients in the series expansions of certain infinite q-products for arithmetic progressions modulo 5, modulo 7...     

Digit sums and generating functions

The Ramanujan Journal - We connect a primitive operation from arithmetic—summing the digits of a base-B integer—to q-series and product generating functions analogous to those in...     

Matching coefficients in the series expansions of certain q-products and their reciprocals

The Ramanujan Journal - We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ramanujan’s...     

On the asymptotics of some partial theta functions

Using the Euler–Maclaurin sum formula, we develop an asymptotic expansion for a fairly general sum of exponentials, which when specialized includes some common partial theta functions. Some conjectured asymptotic expansions for relevant integrals are given. We give a simple proof of a theorem by Bruce Berndt and Byungchan Kim generalizing a result found in Ramanujan’s second notebook.     

On some continued fraction expansions of the Rogers–Ramanujan type

By guessing the relative quantities and proving the recursive relation, we present some continued fraction expansions of the Rogers–Ramanujan type. Meanwhile, we also give some J-fraction expansions for the q-tangent and q-cotangent functions.     

Unified treatment of several asymptotic formulas for the gamma function

We consider a class of the asymptotic expansions for the gamma function, and derive a formula for determining the coefficients of the asymptotic expansions. Thus, we give a unified treatment of several asymptotic expansions for the gamma function due to Laplace, Ramanujan–Karatsuba, Gosper, Mortici and Batir.     

Remarks on asymptotic expansions for the gamma function

We unify several asymptotic expansions for the gamma function due to Laplace, Ramanujan–Karatsuba, Gosper, Mortici, Nemes and Batir. Furthermore we present new asymptotic expansions for the gamma function.     

Congruences for arithmetic functions

The Ramanujan Journal - Motivated by congruences for partitions, we study congruences for three well known arithmetic functions: the divisor function d(n), the sum-of-divisors function $$\sigma...     

Averages of Ramanujan sums: note on two papers by E. Alkan

We give a simple proof and a multivariable generalization of an identity due to E. Alkan concerning a weighted average of the Ramanujan sums. We deduce identities for other weighted averages of the Ramanujan sums with weights concerning logarithms, values of arithmetic functions for gcd’s, the Gamma function, the Bernoulli polynomials, and binomial coefficients.     

Asymptotic formulas related to the $$M_2$$M2-rank of partitions without repeated odd parts

The Ramanujan Journal - We give asymptotic expansions for the moments of the $$M_2$$-rank generating function and for the $$M_2$$-rank generating function at roots of unity. For this we apply the...     

Eisenstein series and convolution sums

The Ramanujan Journal - We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the...     

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...     

Sums of products of Ramanujan sums

The Ramanujan sum c _n (k) is defined as the sum of k-th powers of the primitive n-th roots of unity. We investigate arithmetic functions of r variables defined as certain sums of the products \({c_{m_1}(g_1(k))\cdots c_{m_r}(g_r(k))}\), where g ₁, . . . , g _r are polynomials with integer coefficients. A modified orthogonality relation of the Ramanujan sums is also derived.     

Some new asymptotic expansions of certain partial theta functions

In (Proc. Am. Math. Soc. 139(11):3779–3788, 2011), B.C. Berndt and B. Kim established asymptotic expansions for a class of partial theta functions, generalizing a result by S. Ramanujan. In this paper, we also prove some asymptotic expansions of such functions which are different from the results in (Berndt and Kim in Proc. Am. Math. Soc. 139(11):3779–3788, 2011).     

Identities for the Hurwitz zeta function, Gamma function, and L-functions

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet L-functions. They involve a sequence of polynomials α _k (s) whose study was initiated in Rubinstein (Ramanujan J. 27(1): 29–42, 2012). The expansions given here are practical and can be used for the high precision evaluation of these functions, and for deriving formulas for special values. We also present a summation formula and use it to generalize a formula of Hasse.     

Modular equations for cubes of the Rogers–Ramanujan and Ramanujan–Göllnitz–Gordon functions and their associated continued fractions

In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions.     

On sums of coefficients of Borwein type polynomials over arithmetic progressions

The Ramanujan Journal - We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$ \prod _{j=1}^n\prod _{k=1}^{p-1}(1-q^{pj-k})^s, $$ where p...