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Let $\Delta= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $\Delta$. If $p$ is an odd prime, then the Hecke operator $T_p$ transforms $f$ in a modular form $T_p(f)$ which is a polynomial in $\Delta$ whose degree is smaller than the degree of $f$, so that $T_p$ is nilpotent. The order of nilpotence of $f$ is defined as the smallest integer $g=g(f)$ such that, for every family of $g$ odd primes $p_1,p_2,\ldots,p_g$, the relation $T_{p_1}T_{p_2}\ldots T_{p_g}(f)=0$ holds. We show how one can compute explicitly $g(f)$; if $f$ is a polynomial of degree $d\geqslant 1$ in $\Delta$, one finds that $g(f) < \frac 32 \sqrt d$., Comment: 39 pages, in French language

The Landau function $g(n)$ is the maximal order of an element of the symmetric group of degree $n$; it is also the largest product of powers of primes whose sum is $\le n$. The main result of this article is that the property " For all $n > 0$ , $log g(n) < li^{-1} (n))$ " (where $li^{-1}(n)$ denotes the inverse function of the logarithmic integral) is equivalent to the Riemann hypothesis.

Let $S_n$ be the symmetric group of $n$ letters; Landau considered the function $g(n)$ defined as the maximal order of an element of $S_n$. This function is non-decreasing. Let us define the sequence $n_1=1, n_2=2, n_3=3, n_4=4,n_5=5,n_6=7, ...,n_k$ such that $g(n_k) > g(n_k -1)$. It is known that $lim sup n_{k+1}-n_k =infinity$. Here it is shown that $lim inf n_{k+1}-n_k is finite.

We provide an historical account of equivalent conditions for the Riemann Hypothesis arising from the work of Ramanujan and, later, Guy Robin on generalized highly composite numbers. The first part of the paper is on the mathematical background of our subject. The second part is on its history, which includes several surprises., Comment: Modified the title and abstract, 11 pages, 20 images

If n is a positive integer, let h(n) denote the maximal value of the product of distinct primes whose sum does not exceed n. We give some properties of this function h and describe an algorithm able to compute h(n) for large values of n.

Modular forms mod 2 : structure of the Hecke ring We show that the Hecke algebra for modular forms mod 2 of level 1 is isomorphic to the power series ring F2[[x, y]], where x = T3 and y = T5., Comment: http://dx.doi.org/10.1016/j.crma.2012.03.019

The nilpotence order of the mod 2 Hecke operators. Let $\Delta=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form f modulo 2 of level 1 is a polynomial in $\Delta$. If p is an odd prime, then the Hecke operator Tp transforms f in a modular form Tp(f) which is a polynomial in $\Delta$ whose degree is smaller than the degree of f, so that Tp is nilpotent. The order of nilpotence of f is defined as the smallest integer g = g(f) such that, for every family of g odd primes p1, p2, ..., pg, the relation Tp1Tp2... Tpg (f) = 0 holds. We show how one can compute explicitly g(f); if f is a polynomial of degree d in $\Delta$, one finds that g(f) << d^(1/2)., Comment: C. R. Acad. Sci. Paris, Ser. I 350 (2012); http://dx.doi.org/10.1016/j.crma.2012.03.013

Let $\vfi$ be Euler's function, $\ga$ be Euler's constant and $N_k$ be the product of the first $k$ primes. In this article, we consider the function $c(n) =(n/\vfi(n)-e^\ga\log\log n)\sqrt{\log n}$. Under Riemann's hypothesis, it is proved that $c(N_k)$ is bounded and explicit bounds are given while, if Riemann's hypothesis fails, $c(N_k)$ is not bounded above or below.

Gronwall's function $G$ is defined for $n>1$ by $G(n)=\frac{\sigma(n)}{n \log\log n}$ where $\sigma(n)$ is the sum of the divisors of $n$. We call an integer $N>1$ a \emph{GA1 number} if $N$ is composite and $G(N) \ge G(N/p)$ for all prime factors $p$ of $N$. We say that $N$ is a \emph{GA2 number} if $G(N) \ge G(aN)$ for all multiples $aN$ of $N$. In arXiv 1110.5078, we used Robin's and Gronwall's theorems on $G$ to prove that the Riemann Hypothesis (RH) is true if and only if 4 is the only number that is both GA1 and GA2. Here, we study GA1 numbers and GA2 numbers separately. We compare them with superabundant (SA) and colossally abundant (CA) numbers (first studied by Ramanujan). We give algorithms for computing GA1 numbers; the smallest one with more than two prime factors is 183783600, while the smallest odd one is 1058462574572984015114271643676625. We find nineteen GA2 numbers $\le 5040$, and prove that a GA2 number $N>5040$ exists if and only if RH is false, in which case $N$ is even and $>10^{8576}$., Comment: 29 pages, 2 tables, to appear in The Ramanujan Journal; added that when Robin published his criterion for RH he was unaware of Ramanujan's theorem

For n>1, let G(n)=\sigma(n)/(n log log n), where \sigma(n) is the sum of the divisors of n. We prove that the Riemann Hypothesis is true if and only if 4 is the only composite number N satisfying G(N) \ge \max(G(N/p),G(aN)), for all prime factors p of N and all multiples aN of N. The proof uses Robin's and Gronwall's theorems on G(n). An alternate proof of one step depends on two properties of superabundant numbers proved using Alaoglu and Erd\H{o}s's results., Comment: 11 pages, 1 table, clarified Proposition 4, added reference 4

Let q be an odd positive integer and P \in F2[z] be of order q and such that P(0) = 1. We denote by A = A(P) the unique set of positive integers satisfying \sum_{n=0}^\infty p(A, n) z^n \equiv P(z) (mod 2), where p(A,n) is the number of partitions of n with parts in A. In [5], it is proved that if A(P, x) is the counting function of the set A(P) then A(P, x) << x(log x)^{-r/\phi(q)}, where r is the order of 2 modulo q and \phi is Euler's function. In this paper, we improve on the constant c=c(q) for which A(P,x) << x(log x)^{-c}.

After Landau, we define g(n) as the maximal order of a permutation of the symmetric group S(n) on n letters. We give several estimates of the largest prime divisor of g(n).

Improving on some results of J.-L. Nicolas \cite {Ndeb}, the elements of the set ${\cal A}={\cal A}(1+z+z^3+z^4+z^5)$, for which the partition function $p({\cal A},n)$ (i.e. the number of partitions of $n$ with parts in ${\cal A}$) is even for all $n\geq 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak S}_n$. If the standard factorization of $M$ into primes is $M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}$, we define $\ell(M)$ to be $q_1^{\al_1}+q_2^{\al_2}+... +q_k^{\al_k}$; one century ago, E. Landau proved that $g(n)=\max_{\ell(M)\le n} M$ and that, when $n$ goes to infinity, $\log g(n) \sim \sqrt{n\log(n)}$. There exists a basic algorithm to compute $g(n)$ for $1 \le n \le N$; its running time is $\co(N^{3/2}/\sqrt{\log N})$ and the needed memory is $\co(N)$; it allows computing $g(n)$ up to, say, one million. We describe an algorithm to calculate $g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\it $\ell$-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function $\tau(n)=\sum_{d\dv n} 1$.

Let us denote by $\tau(n)$ and $\si(n)$ the number and the sum of the divisors of $n$ and by $\vfi$ Euler's function. We give effective upper bounds for $\frac{n}{\vfi(n)}$ in terms of $\vfi(n)$, and for $\frac{\si(n)}{n}$ in terms of $\tau(n)$.