The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{array}{c}\sum _{\genfrac{}{}{0pt}{}{(l,m)\in {\mathbb{N}}_0^2}{\alpha l+\beta m=n}}\sigma \left(l\right)\sigma \left(m\right),\end{array}$ 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 convolution sum $$\sum _{\begin{array}{c}m=1\\ m\equiv a(mod4)\end{array}}^{n-1}\sigma \left(m\right)\sigma (n-m)$$ is evaluated for $a\in \{0,1,2,3\}$ and all $n\in \mathbb{N}$. This completes the partial evaluation given in the paper of J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams.

We prove that $|{\sum }_{d\le x,(d,q)=1}\mu \left(d\right)/d|\le 2.4(q/\phi \left(q\right))/log(x/q)$ for every x > q ≥ 1, and similar estimates for the Liouville function. We also give better constants when x/q is large.,

Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $\left(f(x_i,x_i)\right)$ having f evaluated at the greatest common divisor $(x_i,x_i)$ of $x_i$ and $x_i$ as its...

The structure of the group ${(\mathbb{Z}/n\mathbb{Z})}^{☆}$ and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group ${𝒢}_n:=\{a+b\mathrm{i}\in \mathbb{Z}\left[\mathrm{i}\right]/n\mathbb{Z}\left[\mathrm{i}\right]:a^2+b^2\equiv 1(modn)\}$. In particular, we characterize Gaussian Carmichael numbers...

We show that if m > 1 is a Fibonacci number such that ϕ(m) | m-1, where ϕ is the Euler function, then m is prime