Let $R_s\left(n\right)$ denote the number of representations of the positive number n as the sum of two squares and s biquadrates. When $s=3$ or 4, it is established that the anticipated asymptotic formula for $R_s\left(n\right)$ holds for all $n\le X$ with at most $O\left(X^{(9-2s)/8+\epsilon }\right)$ exceptions.

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n\ge 1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u={uₙ}_{n\ge 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

We prove that the error term ${\sum }_{n\le x}\Lambda \left(n\right)/n-logx+\gamma$ differs from (ψ(x)-x)/x by a well controlled function. We deduce very precise numerical results from the formula obtained.

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.,

On établit les majorations $\left|M\left(x\right)\right|\le \frac{0.002969x}{{(logx)}^{1/2}}$, valable pour $x\ge 142194,\left|M\left(x\right)\right|\le \frac{0.6437752x}{logx}$ qui est la meilleure majoration possible en $\frac{x}{logx}$ valable pour tout $x\>1(\left|M\left(5\right)\right|=2=\frac{0.6437752\times 5}{log5})$, et d’autres analogues. On montre enfin comment trouver des majorations effectives $\left|M\left(x\right)\right|\>\frac{c_{k}x{(loglogx)}^{2k}}{{(logx)}^k}$ pour tout $k$.