A PROOF OF MERCA’S CONJECTURES ON SUMS OF ODD DIVISOR FUNCTIONS

Merca [‘Congruence identities involving sums of odd divisors function’, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci.22(2) (2021), 119–125] posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalised m-gonal numbers. Extending Merca’s work, we complete the proof of these conjectures.

Inequalities for generalized divisor functions

We offer inequalities to $\sigma_a(n)$ as a function of the real variable $a$: Monotonicity and convexity properties to this and related functions are proved, too. Extensions and improvements of known results are provided.

On the sum of positive divisors functions

Properties of divisor functions $$\sigma _k(n)$$ σ k ( n ) , defined as sums of k-th powers of all divisors of n, are studied through the analysis of Ramanujan’s differential equations. This system of three differential equations is singular at $$x=0$$ x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions.

A lower bound for the variance of generalized divisor functions in arithmetic progressions

We prove that for a large class of multiplicative functions, referred to as generalized divisor functions, it is possible to find a lower bound for the corresponding variance in arithmetic progressions. As a main corollary, we deduce such a result for any $$\alpha $$ α -fold divisor function, for any complex number $$\alpha \not \in \{1\}\cup -\mathbb {N}$$ α ∉ { 1 } ∪ - N , even when considering a sequence of parameters $$\alpha $$ α close in a proper way to 1. Our work builds on that of Harper and Soundararajan, who handled the particular case of k-fold divisor functions $$d_k(n)$$ d k ( n ) , with $$k\in \mathbb {N}_{\ge 2}$$ k ∈ N ≥ 2 .

Some result for binomial convolution sums of restricted divisor functions

Besge presented the result about the convolution sum of divisor functions. Since then Liouville obtained the generalized version of Besge's formula, which is the binomial convolution sum of divisor functions. In 2004, Hahn obtained the results about the convolution sums of ?d|n(-1)d-1d and ?d|n (-1)n=d-1d. In this paper, we present the results for the binomial con- voltion sums, generalized convolution sums of Hahn, of these divisor functions.

Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.