Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Chowla and twin primes conjectures over $mathbb F_q[T]$
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Using geometric methods, we improve on the function field version of the Burgess bound, and show that, when restricted to certain special subspaces, the M{o}bius function over $mathbb F_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the M{o}bius function in arithmetic progressions, and resolve Chowlas $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the M{o}bius function, we obtain a level of distribution beyond $1/2$ for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.
rate research
Read More
We prove estimates for the level of distribution of the Mobius function, von Mangoldt function, and divisor functions in squarefree progressions in the ring of polynomials over a finite field. Each level of distribution converges to $1$ as $q$ goes to $infty$, and the power savings converges to square-root cancellation as $q$ goes to $infty$. These results in fact apply to a more general class of functions, the factorization functions, that includes these three. The divisor estimates have applications to the moments of $L$-functions, and the von Mangoldt estimate to one-level densities.
Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree gamma-1 where gamma is the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $ggeq 1$ defined over a finite field $F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $ggeq 2$ if $qgeq 3$ and that there always exists a non-special divisor of degree $g-1geq 1$ if $qgeq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $F_{q^n}$ of $F_q$, when $q=2^rgeq 16$.
In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers by the Jacobi symbol, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $mathbb{Z}[zeta_{12}]=mathbb{Z}[zeta_3,i]$ by using the cubic residue character on the Eisenstein integers $mathbb{Z}[zeta_3]$. Our main theorem says that the cubic spin is equidistributed along prime ideals of $mathbb{Z}[zeta_{12}]$. The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $a^2+b^6$ on the Eisenstein integers.
We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is the order of a finite field. These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one relating to column-strict tableaux and Macdonalds ``seventh variation of Schur functions, the other relating to permutation statistics and Hilbert series from the invariant theory of GL_n(F_q).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
