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Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Ebénézer Ntienjem (2017)

Open Mathematics

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), ∑ ( l , m ) ∈ ℕ 0 2 α l + β m = n σ ( l ) σ ( m ) , where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by...

Expressing a number as the sum of two coprime squares.

Warren Dicks, Joan Porti (1998)

Collectanea Mathematica

We use hyperbolic geometry to study the limiting behavior of the average number of ways of expressing a number as the sum of two coprime squares. An alternative viewpoint using analytic number theory is also given.

Families of modular forms

Kevin Buzzard (2001)

Journal de théorie des nombres de Bordeaux

We give a down-to-earth introduction to the theory of families of modular forms, and discuss elementary proofs of results suggesting that modular forms come in families.

Galois representations, embedding problems and modular forms.

Teresa Crespo (1997)

Collectanea Mathematica

To an odd irreducible 2-dimensional complex linear representation of the absolute Galois group of the field Q of rational numbers, a modular form of weight 1 is associated (modulo Artin's conjecture on the L-series of the representation in the icosahedral case). In addition, linear liftings of 2-dimensional projective Galois representations are related to solutions of certain Galois embedding problems. In this paper we present some recent results on the existence of liftings of projective representations...

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