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 obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.

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.,

Given a multivariate polynomial $h\left(X_1,\cdots ,X_n\right)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h\left(\mathbf{0}\right)=1$), we determine the maximal domain of meromorphy of the Euler product ${\prod }_{p\mathrm{prime}}h\left(p^{-s_1},\cdots ,p^{-s_n}\right)$ and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.

The history of the construction, organisation and publication of factor tables from 1657 to 1817, in itself a fascinating story, also touches upon many topics of general interest for the history of mathematics. The considerable labour involved in constructing and correcting these tables has pushed mathematicians and calculators to organise themselves in networks. Around 1660 J. Pell was the first to motivate others to calculate a large factor table, for which he saw many applications, from Diophantine...