Let us consider the classical Kloosterman sum Ac.m; n / D

数理解析研究所講究録, 2012

The significance of Kloosterman sums to the theory of modular forms dates back a century to an astonishingly little-known work of Poincare [10]. In 1926 Kloosterman [4] published his seminal paper regarding Ramanujan's problem of representing sufficiently large integers by quatemary quadratic forms. Since then these sums have surfaced with an almost unreasonable ubiquity throughout arithmetic.

1998

Introduction The general structure of this work reeects the main problems that I studied as a PhD student at the University of Pisa (1993{1997). There are ve chapters, which deal with three diierent topics: Practical numbers (Chapters 1 and 2); Sum-free sequences (Chapter 3); Arithmetical identities related to the theory of modular forms (Chapters 4 and 5). In Chapter 1 we extensively survey the theory of practical numbers, i.e., those positive integers m such that every positive integer n < m can be represented as a sum of distinct positive divisors of m. This theory has recently received attention for some unexpected similarities with the properties of primes. We deal with both arithmetical and analytical aspects of the theory. Among other things, we prove the analogue of Goldbach's conjecture for practical numbers, showing that every even positive integer can be expressed as a sum of two practical numbers. This result gives a positive answer to a conjecture raised in 1984 ...

2015

Modular forms are complex analytic objects, but they also have many intimate connections with number theory. This paper introduces some of the basic results on modular forms, and explores some of their uses in number theory.

Analytic Number Theory, 1996

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f (z) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a( N ) ≡ 0 mod where N is any quadratic residue (resp. non-residue) modulo . For every prime > 3 we exhibit a natural holomorphic weight 2 + 1 modular form whose coefficients satisfy the congruence a( N ) ≡ 0 mod for every N satisfying`− N ´= 1. This is proved by using the fact that the Fourier coefficients of these forms are essentially the special values of real Dirichlet L−series evaluated at s = 1− 2 which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of . ¿From the works of Carlitz and Leopoldt, one can deduce that the Fourier coefficients of these forms are almost always a multiple of the denominator of a suitable Bernoulli number. Using these examples as a template, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given positive integer M. We also present two examples of half-integer weight forms, whose coefficients are determined by the special values at the center of the critical strip for the quadratic twists of two modular L−functions, possess such congruence properties. These congruences are related to the non-triviality of the −primary parts of Shafarevich-Tate groups of certain infinite families of quadratic twists of modular elliptic curves with conductors 11 and 14.

The Ramanujan Journal, 2009

We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x 2 + 5y 2 . Making use of Ramanujan's 1 ψ 1 summation formula we establish a new Lambert series identity for P ∞ n,m=−∞ q n 2 +5m 2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don't stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x 2 + 6y 2 , 2x 2 + 3y 2 , x 2 + 15y 2 , 3x 2 + 5y 2 , x 2 + 27y 2 , x 2 + 5(y 2 + z 2 + w 2 ), 5x 2 + y 2 + z 2 + w 2 . In the process, we find many new multiplicative eta-quotients and determine their coefficients.

The Annals of Mathematics, 1998

arXiv (Cornell University), 2022

Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for 1/π. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.

Journal of Number Theory, 2011

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both "naive" and "modified"), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.

Compositio Mathematica, 2004

We investigate the arithmetic and combinatorial significance of the values of the polynomials j n (x) defined by the q-expansion ∞ n=0 j n (x)q n := E 4 (z) 2 E 6 (z) ∆(z) • 1 j(z) − x. They allow us to provide an explicit description of the action of the Ramanujan Thetaoperator on modular forms. There are a substantial number of consequences for this result. We obtain recursive formulas for coefficients of modular forms, formulas for the infinite product exponents of modular forms, and new p-adic class number formulas.

Journal of the American Mathematical Society, 2007

In 1988, Hickerson proved the celebrated “mock theta conjectures” in a collection of ten identities from Ramanujan’s “lost notebook” which express certain modular forms as linear combinations of mock theta functions. In the context of Maass forms, these identities arise from the peculiar phenomenon that two different harmonic Maass forms may have the same non-holomorphic parts. Using this perspective, we construct several infinite families of modular forms which are differences of mock theta functions.