scholarly journals Evaluation of four convolution sums and representation of integers by certain quadratic forms in twelve variables

2022 ◽  
Keyword(s):  
Quadratic Forms ◽  
Convolution Sums ◽  
Representation Of Integers

Evaluation of some convolution sums and representation of integers by certain quadratic forms in 12 variables

2017 ◽  
Vol 13 (03) ◽  
pp. 775-799
Author(s):  
Bülent Köklüce ◽  
Hasan Eser
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Representation Of Integers

In this paper, the convolution sums [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] are evaluated for all [Formula: see text] and their evaluations are used to determine the number of representation of a positive integer [Formula: see text] by the forms [Formula: see text] and [Formula: see text]

FOURTEEN OCTONARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 37-50 ◽  
Author(s):  
AYŞE ALACA ◽  
ŞABAN ALACA ◽  
KENNETH S. WILLIAMS
Keyword(s):  
Positive Integer ◽  
Quadratic Forms ◽  
Convolution Sums ◽  
Sum Of Divisors ◽  
Recent Evaluation

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 REPRESENTATION NUMBERS OF THREE OCTONARY QUADRATIC FORMS

2012 ◽  
Vol 09 (02) ◽  
pp. 505-516 ◽  
Author(s):  
BÜLENT KÖKLÜCE
Keyword(s):  
Quadratic Forms ◽  
Convolution Sums

In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms [Formula: see text], [Formula: see text] and [Formula: see text].

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

2017 ◽  
Vol 15 (1) ◽  
pp. 446-458 ◽  
Author(s):  
Ebénézer Ntienjem
Keyword(s):  
Positive Integer ◽  
Modular Forms ◽  
Quadratic Forms ◽  
Modular Space ◽  
Natural Numbers ◽  
Convolution Sums ◽  
Sum Of Divisors

Abstract The convolution sum, $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ 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 the octonary quadratic forms $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

scholarly journals REPRESENTATIONS OF INTEGERS BY TERNARY QUADRATIC FORMS

2010 ◽  
Vol 06 (01) ◽  
pp. 127-158 ◽  
Author(s):  
BEN KANE
Keyword(s):  
Quadratic Forms ◽  
Theta Series ◽  
Implied Constant ◽  
Cusp Forms ◽  
Local Conditions ◽  
Full List ◽  
Ternary Quadratic Forms ◽  
Representation Of Integers ◽  
Representations Of Integers

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space [Formula: see text], where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.

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