Şaban Alaca

FOURTEEN OCTONARY QUADRATIC FORMS

We use the recent evaluation of certain convolution sums involving the sum of divisors function to determine the number of representations of a positive integer by certain diagonal octonary quadratic forms whose coefficients are 1, 2 or 4.

THE NUMBER OF REPRESENTATIONS OF A POSITIVE INTEGER BY CERTAIN QUATERNARY QUADRATIC FORMS

Some theta function identities are proved and used to give formulae for the number of representations of a positive integer by certain quaternary forms x2 + ey2 + fz2 + gt2 with e, f, g ∈ {1, 2, 4, 8}.

REPRESENTATIONS BY CERTAIN OCTONARY QUADRATIC FORMS WHOSE COEFFICIENTS ARE 1, 2, 3 AND 6

We determine formulae for the numbers of representations of a positive integer by certain octonary quadratic forms whose coefficients are 1, 2, 3 and 6.

Evaluation of the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m)

We determine the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m) for all positive integers n. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of n by the octonary quadratic forms x12 + x 1x2 + x22 + x 32 + x 3x4 + x42 + 9(x 52 + x 5x6 + x62 + x 72 + x 7x8 + x82) and x12 + x 22 + x 32 + x 42 + 8(x 52 + x 62 + x 72 + x 82). A modular form approach is used.

Fourier coefficients of a class of eta quotients of weight 2

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that where and are Jacobi–Kronecker symbols. We prove our results using the theory of modular forms.

Congruences for Brewer sums

We prove congruences modulo 4 and modulo 8 for certain polynomial character sums, and use these congruences to give conditions for the nonvanishing of Brewer sums.

SEXTENARY QUADRATIC FORMS AND AN IDENTITY OF KLEIN AND FRICKE

Formulae, originally conjectured by Liouville, are proved for the number of representations of a positive integer n by each of the eight sextenary quadratic forms

On the number of representations of a positive integer as a sum of two binary quadratic forms

x_1^2 +x_2^2+x_3^2 +x_4^2+x_5^2+4x_6^2

Infinite products with coefficients which vanish on certain arithmetic progressions

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Character values of the Sidelnikov-Lempel-Cohn-Eastman sequences

x_1^2 +x_2^2+x_3^2+x_4^2+4x_5^2+4x_6^2