The Ramanujan Journal最新文献

Chan and Cooper proved that if the integers ({c(n) (n=0,1,2,ldots )}) are given by

then

We prove many other results of this type and apply them to the determination of congruence properties of the coefficients.

Let (S_k(N)) denote the space of cusp forms of even integer weight k and level N. We prove an asymptotic for the Petersson trace formula for (S_k(N)) under an appropriate condition. Using the non-vanishing of a Kloosterman sum involved in the asymptotic, we give a lower bound for discrepancy in the Sato–Tate distribution for levels not divisible by 8. This generalizes a result of Jung and Sardari (Math Ann 378(1–2):513–557, 2020, Theorem 1.6) for squarefree levels. An analogue of the Sato-Tate distribution was obtained by Omar and Mazhouda (Ramanujan J 20(1):81–89, 2009, Theorem 3) for the distribution of eigenvalues (lambda _{p^2}(f)) where f is a Hecke eigenform and p is a prime number. As an application of the above-mentioned asymptotic, we obtain a sequence of weights (k_n) such that discrepancy in the analogue distribution obtained in Omar and Mazhouda (Ramanujan J 20(1):81–89, 2009) has a lower bound.

A comprehensive review of diamonds, in the sense of Scholze, is presented. The diamond formulations of the Fargues-Fontaine curve and (Bun_G) are reviewed. Principal results centered on the diamond formalism in the global Langlands correspondence and the geometrization of the local Langlands correspondence are given. We conclude with a discussion of future geometrizations.

Let (mathbb {N}) be the set of all nonnegative integers. For (Ssubseteq mathbb {N}) and (nin mathbb {N}), let the representation function (R_{S}(n)) denote the number of solutions of the equation (n=s+s') with (s, s'in S) and (s<s'). In this paper, we determine the structure of (C, Dsubseteq mathbb {N}) with (Ccup D=[0, m]), (Ccap D={r_{1}, r_{2}}), (r_{1}<r_{2}) and (2not mid r_{1}) such that (R_{C}(n)=R_{D}(n)) for any nonnegative integer n.

We prove a number of new Rogers–Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the literature. This is achieved by direct summation or the constant term method. We also obtain some new single-sum identities as consequences.

Let (kappa ) be any positive real number and (min mathbb {N}cup {infty }) be given. Let (p_{kappa , m}(n)) denote the number of partitions of n into the parts from the Segal–Piatestki–Shapiro sequence ((lfloor ell ^{kappa }rfloor )_{ell in mathbb {N}}) with at most m possible repetitions. In this paper, we establish some asymptotic formulas of Hardy–Ramanujan type for (p_{kappa , m}(n)). As a necessary step in the proof, we prove that the Dirichlet series (zeta _kappa (s)=sum _{nge 1}lfloor n^{kappa }rfloor ^{-s}) can be continued analytically beyond the imaginary axis except for simple poles at (s=1/kappa -j, ~(0le j< 1/kappa , jin mathbb {Z})).

We study Diophantine equations of the form ( f(n)=m^2 pm a,; n, min mathbb {N}, )where (ain mathbb {N}^*) and (f: mathbb {N} rightarrow mathbb {N}) tends to (+infty . ) Necessary and sufficient conditions for the set of solutions to be finite are formulated in terms of asymptotic properties and the repartition of the digits of the fractional part of (sqrt{f(n)})

In the “Lost” Notebook Ramanujan defined two mock theta functions (which we denote by (phi _{{}_R}) and (xi _{{}_R})) and gave relations connecting them to some of the sixth-order mock theta functions. It turns out that these mock theta functions are related to each other by modular transformation. Their transformation formulas are similar to the transformation formulas for the second-order mock theta functions, and involve the same Mordell integrals. We prove several relations involving (phi _{{}_R}) and (xi _{{}_R}), and some relations connecting them to some of the second-order mock theta functions. Some alternate formulas for the second-order mock theta function (mu ) are given.

For (n in mathbb {N}) let (Pi [n]) denote the set of partitions of n, i.e., the set of positive integer tuples ((x_1,x_2,ldots ,x_k)) such that (x_1 ge x_2 ge ldots ge x_k) and (x_1 + x_2 + cdots + x_k = n). Fixing (f:mathbb {N}rightarrow {0,pm 1}), for (pi = (x_1,x_2,ldots ,x_k) in Pi [n]) let (f(pi ) := f(x_1)f(x_2)cdots f(x_k)). In this way we define the signed partition numbers

Following work of Vaughan and Gafni on partitions into primes and prime powers, we derive asymptotic formulae for (p(n,mu )) and (p(n,lambda )), where (mu ) and (lambda ) denote the Möbius and Liouville functions from prime number theory, respectively. In addition we discuss how quantities p(n, f) generalize the classical notion of restricted partitions.