Prescribing the binary digits of primes,II

We obtain the expected asymptotic formula for the number of primes p < N = 2 ⁿ with r prescribed (arbitrary placed) binary digits, provided r < cn for a suitable constant c > 0. This result improves on our earlier result where r was assumed to satisfy \(r < c{\left( {\frac{n}{{\log n}}} \right)^{4/7}}\).     

The sum of digits of Gaussian primes

Let q=−a±i and denote by s _q the complex sum-of-digits function. We show that the sequence (αs _q (p)) running over all Gaussian primes lying in a circular sector is uniformly distributed modulo 1 if and only if α is irrational. Moreover, we prove that the sum-of-digits function of primes is well distributed in arithmetic progressions. This work generalizes a theorem of Mauduit and Rivat that was the solution of a long-standing conjecture by Gelfond concerning the usual q-ary sum-of-digits function. It improves also a result of Drmota, Rivat, and Stoll, who could only deal with sufficiently large prime bases q=−a±i and the full disc.     

On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.     

Construction of random variables according to the probabilities of their binary digits

One states special properties of binary expansions of random variables, having a truncated geometric (discrete uniform) and exponential (uniform) distributions, which simplify in an essential manner the construction of random variables according to the probabilities of their binary digits.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 160–163, 1988.     

Threefield identities and simultaneous representations of primes by binary quadratic forms

Kaplansky [2003] proved a theorem on the simultaneous representation of a prime p by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink?s list. We also show how such theorems are related to results of Muskat on binary quadratic forms.     

An Artin character and representations of primes by binary quadratic forms

We show how the decomposition of primes in certain dihedral extensions L of the rationals enables us to obtain results concerning representations of powers of primes by binary quadratic forms and treat here in detail the case of , where m is a square free positive integer such that the norm of the fundamental unit _m of is –1. Other cases will be treated in subsequent papers.Research supported by Natural Sciences and Engineering Research Council Canada Grant N^o A-7233     

The sum of digits of primes in $${\mathbb{Z}}$$[i]

We study the distribution of the complex sum-of-digits function s _q with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αs _q (p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.     

An Artin character and representations of primes by binary quadratic forms II

For any squarefree positive m there exists exactly one solvable antipellian equation, which can be used to construct a certain dihedral extension L/Q, cyclic of degree 4 above k=Q(–m). We calculate the conductor of L/k and the value of the Artin character of L/k on the corresponding congruence ideal classes of order 2 of k. From this, we deduce results for the representations of powers of primes by binary quadratic forms, in the case where the norm of the fundamental unit of Q(m) is +1.Research supported by Natural Sciences and Engineering Research Council Canada Grant No. A-7233     

An inequality for sums of binary digits, with application to Takagi functions

Let ?(x)=2inf{|x−n|:n∈Z}, and define for α>0 the function

     

Random sequences of binary digits in which missing values can almost certainly be restored

Infinite sequences of random (nearly independent) binary digits can be generated having (almost certainly) a remarkable property. Observation of any nonzero fraction such as sequence makes it possible to calculate the values of all the unobserved digits.     

On the sum of distinct primes or squares of primes

In 1965 Erd?s introduced $f_2(s)$: $f_2(s)$ is the smallest integer such that every $l>f_2(s)$ is the sum of s distinct primes or squares of primes where a prime and its square are not both used. We prove that for all sufficiently large s, $f_2(s)?p_2+p_3+?+p_{s+1}+3106$, and the set of s with the equality has the density 1.     

Piatetski-Shapiro primes from almost primes

Let \(\left\lfloor \cdot \right\rfloor \) be the floor function. In this paper, we show that for any fixed \(c\in \left( 1,\frac{77}{76}\right) \) there are infinitely many primes of the form \(p=\left\lfloor n^c\right\rfloor \) , where \(n\) is a natural number with at most eight prime factors (counted with multiplicity).     

Many associated primes of powers of primes

We construct families of prime ideals in polynomial rings for which the number of associated primes of the second power (or higher powers) is exponential in the number of variables in the ring.