The Ramanujan Journal最新文献

In this article, we are interested in modular forms with non-vanishing central critical values and linear independence of Fourier coefficients of modular forms. The main ingredient is a generalization of a theorem due to VanderKam to modular symbols of higher weights. We prove that for sufficiently large primes p, Hecke operators (T_1, T_2, ldots , T_D) act linearly independently on the winding elements inside the space of weight 2k cuspidal modular symbol (mathbb {S}_{2k}(Gamma _0(p))) with (kge 1) for (D^2ll p). This gives a bound on the number of newforms with non-vanishing arithmetic L-functions at their central critical points and linear independence on the reductions of these modular forms for prime modulo (lnot =p).

Let f and g be two distinct normalized primitive Hecke cusp forms of even integral weights (k_{1}) and (k_{2}) for the full modular group (Gamma =SL(2,{mathbb {Z}})), respectively. Denote by (lambda _{fotimes fotimes fotimes g}(n)) and (lambda _{text {sym}^{2}fotimes fotimes g}(n)) the nth normalized coefficients of the automorphic L-functions (L(fotimes fotimes fotimes g,s)) and (L(text {sym}^{2}fotimes fotimes g,s)), respectively. In this paper, we are interested in the average behavior of the coefficients (lambda _{fotimes fotimes fotimes g}(n)) and (lambda _{text {sym}^{2}fotimes fotimes g}(n)) on a primitive integral binary quadratic form with negative discriminant whose class number is 1, and we also provide the asymptotic formulae of these summatory functions. As an application, we also consider the number of sign changes of the sequences ({lambda _{fotimes fotimes fotimes g}(n)}_{ngeqslant 1}) and ({lambda _{text {sym}^{2}fotimes fotimes g}(n)}_{ngeqslant 1}) on the same binary quadratic form in short intervals.

Let (f(t)=sum _{n=0}^{+infty }frac{C_{f,n}}{n!}t^n) be an analytic function at 0, and let (C_{f, n}(x)=sum _{k=0}^{n}left( {begin{array}{c}n kend{array}}right) C_{f,k} x^{n-k}) be the sequence of Appell polynomials, referred to as C-polynomials associated to f, constructed from the sequence of coefficients (C_{f,n}). We also define (P_{f,n}(x)) as the sequence of C-polynomials associated to the function (p_{f}(t)=f(t)(e^t-1)/t), called P-polynomials associated to f. This work investigates three main topics. Firstly, we examine the properties of C-polynomials and P-polynomials and the underlying features that connect them. Secondly, drawing inspiration from the definition of P-polynomials and subject to an additional condition on f, we introduce and study the bivariate complex function (P_{f}(s,z)=sum _{k=0}^{+infty }left( {begin{array}{c}z kend{array}}right) P_{f,k}s^{z-k}), which generalizes the (s^z) function and is denoted by (s^{(z,f)}). Thirdly, the paper’s main contribution is the generalization of the Hurwitz zeta function and its fundamental properties, most notably Hurwitz’s formula, by constructing a novel class of functions defined by (L(z,f)=sum _{n=n_{f}}^{+infty }n^{(-z,f)}), which are intrinsically linked to C-polynomials and referred to as LC-functions associated to f (the constant (n_{f}) is a positive integer dependent on the choice of f).

In this note we give exact formulas (and asymptotics) for the number of rational points of bounded height on weighted projective stacks over global function fields.

We present a new proof of the transformation law of (vartheta _1) under the action of the generator of the full modular group (Gamma ) using Siegel’s method.

We consider two multiplicative statistics on the set of integer partitions: the norm of a partition, which is the product of its parts, and the supernorm of a partition, which is the product of the prime numbers (p_i) indexed by its parts i. We introduce and study new statistics that are sums of reciprocals of supernorms on three statistical ensembles of partitions, labelled by their size (|lambda |=n), their perimeter equaling n, and their largest part equaling n. We show that the cumulative statistics of the reciprocal supernorm for each of the three ensembles are asymptotic to (e^{gamma } log n) as (n rightarrow infty ).

A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number a, which we denote (frac{p}{q}), satisfies the inequality (left| a-frac{p}{q} right| < frac{C}{q^2}) with (C=frac{1}{sqrt{5}}). In this paper we give more precise information about the constant C.

In this paper, we study the zeros of polynomials obtained from the L-functions and their derivatives associated to non-cuspidal modular forms in Eisenstein spaces of prime levels as a generalization of work by Diamantis and Rolen.

For a real number k, define (pi _k(x) = sum _{ple x} p^k). When (k>0), we prove that

as (xrightarrow infty ), and we prove a similar result when (-1<k<0). This strengthens a result in a paper by Gerard and the author and it corrects a flaw in a proof in that paper. We also quantify the observation from that paper that (pi _k(x) - pi (x^{k+1})) is usually negative when (k>0) and usually positive when (-1<k<0).

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems 1 and 2). These, together with the explicit expression of the latter remainder (Theorem 3), naturally transfer to several new variants of the celebrated formulae of Euler and of Ramanujan for specific values of the Riemann zeta-function (Theorem 4 and Corollaries 4.1–4.5), and to various modular type relations for the classical Eisenstein series of any even integer weight (Corollary 4.6) as well as for Weierstraß’ elliptic and allied functions (Corollaries 4.7–4.9). Crucial roles in the proofs are played by certain Mellin-Barnes type integrals, which are manipulated with several properties Kummer’s confluent hypergeometric functions.