Contributions to the theory of Ramanujan&#39;s function τ(n) and similar arithmetical functions

Suppose thatis an integral modular form of dimensions −κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for Thenwhere a, b, c, d are integers such that ad − bc = 1.

Contributions to the theory of Ramanujan's function τ(n) and similar arithmetical functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100017114 ◽

1940 ◽

Vol 36 (2) ◽

pp. 150-151 ◽

Cited By ~ 11

Author(s):

R. A. Rankin

Keyword(s):

Modular Form ◽

Title I ◽

Fundamental Region ◽

Arithmetical Functions ◽

Suppose that is an integral modular form of dimensions − κ, where κ > 0, and Stufe N, which vanishes at all the rational cusps of the fundamental region, and which is absolutely convergent for The purpose of this note is to prove thatThe notation employed is that of my second paper under the same general title* I refer to this paper as II.

On the divisibility of r2(n)

10.1017/s0017089500003128 ◽

1977 ◽

Vol 18 (1) ◽

pp. 109-111 ◽

Cited By ~ 2

Author(s):

E. J. Scourfield

Keyword(s):

Modular Form ◽

Algebraic Number ◽

Modular Forms ◽

Congruence Subgroup ◽

Algebraic Number Field ◽

The Past ◽

Divisibility Property ◽

Integral Weight ◽

Image Position ◽

Positive Integers

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Letbe a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such thatas x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

On Ramanujan and Dirichlet series with Euler products

10.1017/s0017089500005620 ◽

1984 ◽

Vol 25 (2) ◽

pp. 203-206 ◽

Cited By ~ 3

Author(s):

S. Raghavan

Keyword(s):

Modular Form ◽

Dirichlet Series ◽

Modular Forms ◽

Hecke Operators ◽

Euler Product ◽

Arithmetical Functions ◽

In Trans ◽

Euler Products ◽

Unpublished Manuscripts ◽

Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

Arithmetical Functions of a Greatest Common Divisor, III. Cesàro's Divisor Problem

Proceedings of the Glasgow Mathematical Association ◽

10.1017/s2040618500034328 ◽

1961 ◽

Vol 5 (2) ◽

pp. 67-75 ◽

Cited By ~ 2

Author(s):

Eckford Cohen

Keyword(s):

Remainder Term ◽

Substantial Reduction ◽

Divisor Problem ◽

Greatest Common Divisor ◽

Arithmetical Functions ◽

Precise Statement ◽

Elementary Methods ◽

Dirichlet Divisor Problem ◽

Riemann Ζ Function ◽

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also placewhere γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

On Ramanujan's Arithmetical Function Σr, s(n)

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100013943 ◽

1929 ◽

Vol 25 (3) ◽

pp. 255-264 ◽

Cited By ~ 5

Author(s):

J. R. Wilton

Keyword(s):

Error Term ◽

Arithmetical Function ◽

Arithmetical Functions ◽

Image Position ◽

The Right ◽

Positive Integers

Let σs(n) denote the sum of the sth powers of the divisors of n,and let , where ζ(s) is the Riemann ζfunction. Ramanujan, in his paper “On certain arithmetical functions*”, proves that, ifandthen , whenever r and s are odd positive integers. He conjectures that the error term on the right of (1·1) is of the formfor every ε > 0, and that it is not of the form . He further conjectures thatfor all positive values of r and s; and this conjecture has recently been proved to be correct.

On the order of magnitude of Ramanujan's arithmetical function τ(n)

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100027122 ◽

1951 ◽

Vol 47 (4) ◽

pp. 668-678 ◽

Cited By ~ 7

Author(s):

W. B. Pennington

Keyword(s):

Asymptotic Formula ◽

Arithmetical Function ◽

Modular Functions ◽

Arithmetical Functions ◽

Order Of Magnitude ◽

Image Position ◽

Ramanujan’S Formula

1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansionThis function appears in the discussion of an asymptotic formula for the functionand also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

Lacunarity of Dedekind η-products

10.1017/s0017089500030329 ◽

1995 ◽

Vol 37 (1) ◽

pp. 1-14 ◽

Cited By ~ 11

Author(s):

Basil Gordon ◽

Sinai Robins

Keyword(s):

Modular Form ◽

Fourier Coefficients ◽

Partition Functions ◽

Multiplicative Structure ◽

Multiplier System ◽

Form Representation ◽

Upper Half Plane ◽

Group 3 ◽

Monster Group ◽

The Dedekind η-function is defined bywhere τ lies in the upper half plane ℋ = {tau;|Im(τ) > 0}, and x = e2πiτ. It is a modular form of weight ½ with a multiplier system. We define an η-product to be a function f (τ) of the formwhere rδ ε ℤ. This is a modular form of weight with a multiplier system. The Fourier coefficients of η-products are related to many well-known number-theoretic functions, including partition functions and quadratic form representation numbers. They also arise from representations of the “monster” group [3] and the Mathieu group M24 [13]. The multiplicative structure of these Fourier coefficients has been extensively studied. Recent papers include [1], [4], [5] and [6]. Here we study the connections between the density of the non-zero Fourier coefficients of f(τ) and the representability of f(τ) as a linear combination of Hecke character forms (defined in Section 4 below). We first make the following definition.

A New Method in Arithmetical Functions and Contour Integration

Canadian Mathematical Bulletin ◽

10.4153/cmb-1973-060-6 ◽

1973 ◽

Vol 16 (3) ◽

pp. 381-387 ◽

Cited By ~ 5

Author(s):

Bruce C. Berndt

Keyword(s):

Meromorphic Function ◽

Closed Form ◽

Meromorphic Functions ◽

Arithmetical Function ◽

Contour Integration ◽

Partial Answer ◽

Arithmetical Functions ◽

Classical Technique ◽

Limited Class ◽

If f is a suitable meromorphic function, then by a classical technique in the calculus of residues, one can evaluate in closed form series of the form,Suppose that a(n) is an arithmetical function. It is natural to ask whether or not one can evaluate by contour integration(1.1)where f belongs to a suitable class of meromorphic functions. We shall give here only a partial answer for a very limited class of arithmetical functions.

An analogue of a conjecture of Sato and Tate for a Hilbert modular form

10.1017/s001708950000255x ◽

1975 ◽

Vol 16 (2) ◽

pp. 69-87 ◽

Cited By ~ 1

Author(s):

H. L. Resnikoff ◽

R. L. Saldaña

Keyword(s):

Modular Form ◽

Elliptic Curve ◽

Density Function ◽

Number Field ◽

Hilbert Modular Form ◽

Hilbert Modular ◽

Complex Multiplications ◽

If k denotes a number field and εm is the product of an elliptic curve ε with itself m times over k, then for each prime π where ε has non-degenerate reduction, the zeta factor ζ(επ'S) can be expressed asWhere |π| denotes the norm of π. It is a consequence of a conjecture of Tate [16] that if ε does not have complex multiplications, then the numbers are distributed according to the density functionthat is, the density of the set of primes π such that – is

The vanishing of Poincaré series

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500003035 ◽

1980 ◽

Vol 23 (2) ◽

pp. 151-161 ◽

Cited By ~ 17

Author(s):

R. A. Rankin

Keyword(s):

Positive Integer ◽

Modular Form ◽

Linear Combination ◽

Cusp Form ◽

Modular Group ◽

Poincaré Series ◽

Complex Field ◽

Poincare Series ◽

Holomorphic Modular Form ◽

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series.Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} andThe summation is over all matriceswith different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.