Moments of averages of generalized Ramanujan sums

Let \(\beta \) be a positive integer. A generalization of the Ramanujan sum due to Cohen is given by

$$\begin{aligned} c_{q,\beta }(n) := \sum \limits _{{{(h,{q^\beta })}_\beta } = 1} {{e^{2\pi inh/{q^\beta }}}}, \end{aligned}$$

where h ranges over the non-negative integers less than \(q^{\beta }\) such that h and \(q^{\beta }\) have no common \(\beta \)-th power divisors other than 1. The distribution of the average value of the Ramanujan sum is a subject of extensive research. In this paper, we study the distribution of the average value of \(c_{q,\beta }(n)\) by computing the k-th moments of the average value of \(c_{q,\beta }(n)\). In particular we have provided the first and second moments with improved error terms. We give more accurate results for the main terms than our predecessors. We also provide an asymptotic result for an extension of a divisor problem and for an extension of Ramanujan’s formula.

     

Distribution of averages of Ramanujan sums

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.     

Ramanujan sums as supercharacters

The theory of supercharacters, recently developed by Diaconis-Isaacs and André, is used to derive the fundamental algebraic properties of Ramanujan sums. This machinery frequently yields one-line proofs of difficult identities and provides many novel formulas. In addition to exhibiting a new application of supercharacter theory, this article also serves as a blueprint for future work since some of the abstract results we develop are applicable in much greater generality.     

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.     

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.     

On an optimality property of Ramanujan sums

We evaluate , where the is taken over sequences satisfying . In particular we show that it is attained by taking for all , which reduces the summation over to a Ramanujan sum .

     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

On Ramanujan sums over the Gaussian integers

This note deals with Ramanujan sums c _m (n) over the ring ?[i], in particular with asymptotics for sums of c _m (n) taken over both variables m, n.     

A generalization of regular convolutions and Ramanujan sums

The Ramanujan Journal - Regular convolutions of arithmetical functions were first defined by Narkiewicz (Colloq Math 10:81–94, 1963). Useful identities regarding generalizations of the...     

On exponential sums involving the Ramanujan function

Let τ(n) be the arithmetical function of Ramanujan, α any real number, and x≥2. The uniform estimate $$\mathop \Sigma \limits_{n \leqslant x} \tau (n)e(n\alpha ) \ll x^6 \log x$$ is a classical result of J R Wilton. It is well known that the best possible bound would be ?x ⁶. The validity of this hypothesis is proved.     

Divisor weighted sums

Estimates for sums of divisor functions are required in many arguments of analytic number theory. Linnik was among the first to show how to handle such questions. We provide bounds that are particularly suitable for sparse sequences. Bibliography: 5 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 212–219.     

The Manin—Drinfeld theorem and Ramanujan sums

The Manin—Drinfeld theorem asserts the finiteness of the cuspidal divisor class group of a modular curve corresponding to a congruence subgroup. The purpose of the note is to draw attention to the connection between this theorem and Ramanujan sums, and to the question of what happens for non-congruence subgroups. Research partially supported by an NSERC grant. Alfred P Sloan Fellow. Research partially supported by the NSF.     

On the polynomial Ramanujan sums over finite fields

The polynomial Ramanujan sum was first introduced by Carlitz (Duke Math J 14:1105–1120, 1947), and a generalized version by Cohen (Duke Math J 16:85–90, 1949). In this paper, we study the arithmetical and analytic properties of these sums, deriving various fundamental identities, such as Hölder formula, reciprocity formula, orthogonality relation, and Davenport–Hasse type formula. In particular, we show that the special Dirichlet series involving the polynomial Ramanujan sums are, indeed, the entire functions on the whole complex plane, and we also give a square mean values estimation. The main results of this paper are new appearance to us, which indicate the particularity of the polynomial Ramanujan sums.     

Convergence of weighted averages of noncommutative martingales

Let x = (x _n ) _n?1 be a martingale on a noncommutative probability space ( $\mathcal{M}$ , τ) and (w _n ) _n?1 a sequence of positive numbers such that $W_n = \sum\nolimits_{k = 1}^n {w_k \to \infty } $ as n → ∞. We prove that x = (x _n ) _n?1 converges bilaterally almost uniformly (b.a.u.) if and only if the weighted average (σ _n (x)) _n?1 of x converges b.a.u. to the same limit under some condition, where σ _n (x) is given by $\sigma _n (x) = \frac{1} {{W_n }}\sum\limits_{k = 1}^n {w_k x_k } ,n = 1,2,... $ Furthermore, we prove that x = (x _n ) _n?1 converges in L _p ( $\mathcal{M}$ ) if and only if (σ _n (x)) _n?1 converges in L _p ( $\mathcal{M}$ ), where 1 ? p < ∞. We also get a criterion of uniform integrability for a family in L ₁( $\mathcal{M}$ ).     

On Ramanujan sums of a real variable and a new Ramanujan expansion for the divisor function

The Ramanujan Journal - We show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization, which is obtained by replacing the...