scholarly journals Proof of a conjecture of Ramanujan

1967 ◽  
Vol 8 (1) ◽  
pp. 14-32 ◽  
Author(s):  
A. O. L. Atkin
Keyword(s):  
Fourier Coefficient ◽  
Modular Equations ◽  
Genus Zero ◽  
Modular Invariant ◽  
Image Position

We writeandso that p(n) is the number of unrestricted partitions of n. Ramanujan [1] conjectured in 1919 that if q = 5, 7, or 11, and 24m ≡ 1 (mod qn), then p(m) ≡ 0 (mod qn). He proved his conecture for n = 1 and 2†, but it was not until 1938 that Watson [4] proved the conjecture for q = 5 and all n, and a suitably modified form for q = 7 and all n. (Chowla [5] had previously observed that the conjecture failed for q = 7 and n = 3.) Watson's method of modular equations, while theoretically available for the case q = 11, does not seem to be so in practice even with the help of present-day computers. Lehner [6, 7] has developed an essentially different method, which, while not as powerful as Watson's in the cases where Γ0(q) has genus zero, is applicable in principle to all primes q without prohibitive calculation. In particular he proved the conjecture for q = 11 and n = 3 in [7]. Here I shall prove the conjecture for q = 11 and all n, following Lehner's approach rather than Watson's. I also prove the analogous and essentially simpler result for c(m), the Fourier coefficient‡ of Klein's modular invariant j (τ) as

SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENT OF HECKE–MAASS FORMS

2015 ◽  
Vol 92 (2) ◽  
pp. 195-204 ◽  
Author(s):  
HENGCAI TANG
Keyword(s):  
Fourier Coefficient ◽  
Circle Method ◽  
Summation Formula ◽  
Cusp Forms ◽  
Maass Forms ◽  
Average Value ◽  
Laplace Eigenvalue ◽  
Image Position ◽  
Voronoi Summation ◽  
Shifted Convolution Sum

Let $\{{\it\phi}_{j}(z):j\geq 1\}$ be an orthonormal basis of Hecke–Maass cusp forms with Laplace eigenvalue $1/4+t_{j}^{2}$. Let ${\it\lambda}_{j}(n)$ be the $n$th Fourier coefficient of ${\it\phi}_{j}$ and $d_{3}(n)$ the divisor function of order three. In this paper, by the circle method and the Voronoi summation formula, the average value of the shifted convolution sum for $d_{3}(n)$ and ${\it\lambda}_{j}(n)$ is considered, leading to the estimate $$\begin{eqnarray}\displaystyle \mathop{\sum }_{n\leq X}d_{3}(n){\it\lambda}_{j}(n-1)\ll X^{29/30+{\it\varepsilon}}, & & \displaystyle \nonumber\end{eqnarray}$$ where the implied constant depends only on $t_{j}$ and ${\it\varepsilon}$.

The Fourier Coefficients of the Modular Function λ(τ)

1952 ◽  
Vol 4 ◽  
pp. 67-80 ◽  
Author(s):  
William H. Simons
Keyword(s):  
Fourier Coefficients ◽  
Modular Function ◽  
Convergent Series ◽  
Modular Invariant ◽  
Image Position

In [3], H. Rademacher obtained a convergent series for the Fourier coefficients of the modular invariant J(τ). He found that in the expansion

Modular equations and the genus zero property of moonshine functions

1997 ◽  
Vol 129 (3) ◽  
pp. 413-443 ◽  
Author(s):  
C.J. Cummins ◽  
T. Gannon
Keyword(s):  
Modular Equations ◽  
Genus Zero

On a theorem of S. Bernstein

1952 ◽  
Vol 48 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Shih-Hsun Chang
Keyword(s):  
Entire Function ◽  
Real Numbers ◽  
Genus Zero ◽  
Positive Real ◽  
Image Position

S. Bernstein (see(l), pp. 198–204) has proved that ifis an entire function of genus zero, then the seriesis convergent. By considering the most unfavourable case, when all the (βn) are positive real numbers, and writingwhich is convergent by hypothesis, he proved that.

scholarly journals Schwarzian differential equations and Hecke eigenforms on Shimura curves

2012 ◽  
Vol 149 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Yifan Yang
Keyword(s):  
Differential Equations ◽  
Automorphic Forms ◽  
Hecke Operators ◽  
Shimura Curves ◽  
Genus Zero ◽  
Shimura Curve ◽  
Hecke Eigenforms ◽  
Image Position

AbstractLet X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

scholarly journals DIFFERENTIAL EQUATIONS, DUALITY AND MODULAR INVARIANCE

2005 ◽  
Vol 07 (05) ◽  
pp. 649-706 ◽  
Author(s):  
YI-ZHI HUANG
Keyword(s):  
Differential Equations ◽  
Vertex Operator ◽  
Operator Algebra ◽  
Correlation Functions ◽  
Vertex Operator Algebra ◽  
Modular Invariance ◽  
Genus Zero ◽  
Modular Invariant ◽  
The One ◽  
Genus One

We solve the problem of constructing all chiral genus-one correlation functions from chiral genus-zero correlation functions associated to a vertex operator algebra satisfying the following conditions: (i) V(n) = 0 for n < 0 and V(0) = ℂ1, (ii) every ℕ-gradable weak V-module is completely reducible and (iii) V is C2-cofinite. We establish the fundamental properties of these functions, including suitably formulated commutativity, associativity and modular invariance. The method we develop and use here is completely different from the one previously used by Zhu and others. In particular, we show that the q-traces of products of certain geometrically-modified intertwining operators satisfy modular invariant systems of differential equations which, for any fixed modular parameter, reduce to doubly-periodic systems with only regular singular points. Together with the results obtained by the author in the genus-zero case, the results of the present paper solves essentially the problem of constructing chiral genus-one weakly conformal field theories from the representations of a vertex operator algebra satisfying the conditions above.

Modular Equations and Discrete, Genus-Zero Subgroups of SL(2, ℝ) Containing Γ(N)

2002 ◽  
Vol 45 (1) ◽  
pp. 36-45 ◽  
Author(s):  
C. J. Cummins
Keyword(s):  
Discrete Subgroup ◽  
Modular Equations ◽  
Genus Zero ◽  
Modular Polynomials ◽  
Automorphic Functions

AbstractLet G be a discrete subgroup of SL(2, ℝ) which contains Γ(N) for some N. If the genus of X(G) is zero, then there is a unique normalised generator of the field of G-automorphic functions which is known as a normalised Hauptmodul. This paper gives a characterisation of normalised Hauptmoduls as formal q series using modular polynomials.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

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