A note on the number of divisors of forms $$m^2+Nn^2$$m2+Nn2

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

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Two classical lattice point problems

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100035830 ◽

1961 ◽

Vol 57 (4) ◽

pp. 699-721 ◽

Cited By ~ 13

Author(s):

K. S. Gangadharan ◽

A. E. Ingham

Keyword(s):

Lattice Point ◽

Error Term ◽

Divisor Problem ◽

Lattice Points ◽

Classical Lattice ◽

Rectangular Hyperbola ◽

Number Of Divisors ◽

Image Position ◽

Euler's Constant ◽

Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

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On the number of divisors which are values of a polynomial

The Ramanujan Journal ◽

10.1007/s11139-006-9010-8 ◽

2007 ◽

Vol 17 (3) ◽

pp. 387-403

Author(s):

Katalin Gyarmati

Keyword(s):

Number Of Divisors

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De Jonquières’ formula for a family of nodal curves

International Journal of Algebra and Computation ◽

10.1142/s0218196716500661 ◽

2016 ◽

Vol 26 (08) ◽

pp. 1503-1528

Author(s):

Kwangwoo Lee

Keyword(s):

Linear System ◽

Parameter Family ◽

Nodal Curves ◽

Number Of Divisors ◽

Single Curve

For a given linear system on a curve, the number of divisors of a certain type contained in this system is known as the formula of de Jonquières. In this paper, we give an algorithm for getting a general de Jonquières formula for a family of nodal curves. Using this algorithm we obtain one of such formulas for a particular case; on a single partition [Formula: see text] of [Formula: see text] for a one parameter family of nodal curves. Moreover we retrieve the classical de Jonquières’ formula for a single curve using the method developed in this paper.

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On the number of divisors of x2 + x + A

Journal of Number Theory ◽

10.1016/0022-314x(74)90041-9 ◽

1974 ◽

Vol 6 (6) ◽

pp. 434-442 ◽

Cited By ~ 5

Author(s):

George Szekeres

Keyword(s):

Number Of Divisors

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Another generalization of Menon’s identity

International Journal of Number Theory ◽

10.1142/s1793042117501299 ◽

2017 ◽

Vol 13 (09) ◽

pp. 2373-2379 ◽

Cited By ~ 10

Author(s):

Xiao-Peng Zhao ◽

Zhen-Fu Cao

Keyword(s):

Euler’S Totient Function ◽

Number Of Divisors ◽

Dirichlet Characters

Let [Formula: see text] be the Euler’s totient function and [Formula: see text] be the number of divisors of [Formula: see text]. Menon’s beautiful identity states that [Formula: see text] Here we extend this identity to Dirichlet characters mod [Formula: see text].

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