On the variance of squarefree integers in short intervals and arithmetic progressions

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Let $\varepsilon> 0$ be sufficiently small and let $0 < \eta < 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

Factorization Tests and Algorithms Arising from Counting Modular Forms and Automorphic Representations

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$.It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.

Converegence of a series leading to an analogue of Ramanujan's assertion on squarefree integers

Let d be a squarefree integer. We prove that(i) Pnμ(n)nd(n′) converges to zero, where n′ is the product of prime divisors of nwith ( dn ) = +1. We use the Prime Number Theorem.(ii) Q( dp )=+1(1 −1ps ) is not analytic at s=1, nor is Q( dp )=−1(1 −1ps ) .(iii) The convergence (i) leads to a proof that asymptotically half the squarefree ideals have an even number of prime ideal factors (analogue of Ramanujan’s assertion).

ON TWISTS OF THE FERMAT CUBIC x3 + y3 = 2

We consider the Fermat elliptic curve E2 : x3 + y3 = 2 and prove (using descent methods) that its quadratic twists have rank zero for a positive proportion of squarefree integers with fixed number of prime divisors. We also prove similar result for rank zero cubic twists of this curve. Then we present detailed description of rank zero quadratic and cubic twists of E2 by primes and by products of two primes. We also consider twists of Jacobians of Fermat curves x5 + y5 = m and distribution of their root numbers.

PRODUCTS OF SHIFTED PRIMES SIMULTANEOUSLY TAKING PERFECT POWER VALUES

Let r be an integer greater than 1, and let A be a finite, nonempty set of nonzero integers. We obtain a lower bound for the number of positive squarefree integers n, up to x, for which the products ∏ p∣n(p+a) (over primes p) are perfect rth powers for all the integers a in A. Also, in the cases where A={−1} and A={+1}, we will obtain a lower bound for the number of such n with exactly r distinct prime factors.