Even partition functions and $2$-adic analysis

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index and the same period are isomorphic. We also investigate structure of finite monogenic ternary semigroups and we prove that any finite monogenic ternary semigroup is isomorphic to a quotient ternary semigroup.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p p can be a proper divisor of N p = ( p p − 1 ) / ( p − 1 ) N_p = (p^p-1)/(p-1) . It is known that the period always divides N p N_p . The period is shown to equal N p N_p for most primes p p below 180. The investigation leads to interesting new results about the possible prime factors of N p N_p . For example, we show that if p p is an odd positive integer and m m is a positive integer and q = 4 m 2 p + 1 q=4m^2 p+1 is prime, then q q divides p m 2 p − 1 p^{m^2p}-1 . Then we explain how this theorem influences the probability that q q divides N p N_p .

Two-colorings of a dense subgroup of [formula omitted]n that forbid many distances

Let IN be the set of positive integers, $${\cal B}$$ = {b1 < ⋅s < bh}⊂IN, N∊IN and N≥ bh. $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is the set (introduced by J.-L. Nicolas, I.Z. Ruzsa and A. Sarkozy) such that $${\cal A}$$ {1,..., N} = $${\cal B}$$ and p( $${\cal A}$$ ,n)≡ 0 (mod 2) for n∊IN and n > N, where p( $${\cal A}$$ ,n) denotes the number of partitions of n with parts in $${\cal A}$$ . Let us denote by σ ( $${\cal A}$$ ,n) the sum of the divisors of n belonging to $${\cal A}$$ . In a paper jointly written with J.-L. Nicolas, we have recently proved that, for all k≥ 0, the sequence (σ( $${\cal A}$$ ,2k n))n≥ 1mod 2k+1 is periodic with an odd period qk. In this paper, we will characterize for any fixed odd positive integer q, the sets $${\cal B}$$ and the integers N such that q0 = q, and those for which qk = q for all k≥ 0. Moreover, a set $${\cal A}$$ = $${\cal A}$$ 0( $${\cal B}$$ ,N) is constructed with the property that its period, i.e. the period of (σ( $${\cal A}$$ ,n))n≥ 1mod 2, is 217, and for which the counting function is asymptotically equal to that of $${\cal A}$$ 0({1,2,3,4,5},5) which is a set of period 31.

There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.

The L2 geometry of spaces of harmonic maps S2→ S2 and [formula omitted

For p a prime ≡ 1 (mod n), where n is an odd positive integer, let k(p, n) denote the least integer k such that the numbers xn and (-x)n, where x = 1,2,…, k, yield all the non-zero n-th power residues (mod p) (possibly with repetitions). Clearly

Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi reduction theory of quadratic forms.

Let p p be an odd prime and q = p m q=p^m . Let l l be an odd positive integer. Let p ≡ − 1 ( mod l ) p\equiv -1~(\textrm {mod}~l) or p ≡ 1 ( mod l ) p\equiv 1~(\textrm {mod}~l) and l ∣ m l\mid m . By employing the integer sequence a n = ∑ t = 1 l − 1 2 ( 2 cos ⁡ π ( 2 t − 1 ) l ) n \displaystyle {a_n=\sum _{t=1}^{\frac {l-1}{2}} {\left (2\cos {\frac {\pi (2t-1)}{l}}\right )}^n} , which can be considered as a generalized Lucas sequence, we construct all the permutation binomials P ( x ) = x r + x u P(x)=x^r+x^u of the finite field F q \mathbb {F}_q .

Our study included a class of cyclic codes named δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>−cyclic codes over the ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> = F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>m</i></sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i>F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>m</i></sub> + <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>F<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>m</i></sub>, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> is an odd positive integer with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> = 1. These codes are modules over the ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i>[<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i>, α, δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>] with automorphism α and α-derivation δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>(where ζ ∈ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i>). The structure of ring <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i>[<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i>, α, δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>] is presented to develop linear codes over the ring. The center of the ring is characterized for all inner α-derivations δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub> where ζ is taken as an arbitrary element of the ring fixed by α. In addition, an algorithm was developed and implemented in MAGMA to perform division in this ring. The generator and parity-check matrices for δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>−cyclic codes are proposed, and examples of new optimal δ<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α,ζ</sub>−cyclic codes with better minimum Hamming distance than those already existing in the Markus Grassl code tables are provided.

Type 2 constacyclic codes over [formula omitted] of oddly even length

An infinite class of relations between modular forms is constructed that generalizes evaluations of the Dirichlet beta function at odd positive integers. The work is motivated by a base case appearing in Ramanujan’s Notebooks and a parallel construction for the Riemann zeta function. The identities are shown to be strong q-analogues by virtue of their reduction to the classical beta evaluations as [Formula: see text] and explicit evaluations at CM points for [Formula: see text]. We also determine asymptotic formulas for the Fourier coefficients of the associated modular forms.

Letx kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R n(xkn)=f(xkn),R n (j)(xkn)=0,j=1,2,4,k=0,1…,n−1. We setw 2(t,f) as the second modulus of continuity off(x). Then we prove that |R n(x)-f(x)|=0(nw2(1/nf)). We also examine the question of lower estimate of ‖R n-f‖. This generalizes an earlier work of the author.