Tristan Freiberg

activity

20102026

most citedOn the average exponent of elliptic curves modulo p

15 citations · 83 across the 16 of their papers we have counts for

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17 papers · 1 filter

math.NT2026

Biases in the distribution of primes in short intervals

Tristan Freiberg

Assuming a suitably uniform Hardy--Littlewood prime tuples hypothesis, we obtain a second-order asymptotic for the proportion of short intervals containing a prescribed number of p…

math.NT2026

A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound

Tristan Freiberg

We give an elementary exposition of the tilted sieve introduced by GPT-5.6~Sol \cite{GPT2026}, showing that one residue class modulo each prime $p \leq x$ can cover an interval of l…

math.NT2026

Long runs of integers with small prime factors and the divisor function of $n!$

Tristan Freiberg

Let $d$ be the divisor function, and let $K (n)$ be the least positive integer $K$ for which $d ((n + K)!) \geq 2 d (n!)$ . Erdős, Graham, Ivić and Pomerance proved that, for infinitely…

math.NT2017

Sparser variance for primes in arithmetic progression

Roger Baker, Tristan Freiberg

We obtain an analog of the Montgomery-Hooley asymptotic formula for the variance of the number of primes in arithmetic progressions. In the present paper the moduli are restricted…

math.NT2015★ 14 cited

Limit points and long gaps between primes

Roger Baker, Tristan Freiberg

Let $d_{n} = p_{n + 1} - p_{n}$ , where $p_{n}$ denotes the $n$ th smallest prime, and let $R (T) = lo g T lo g_{2} T lo g_{4} T / (lo g_{3} T)^{2}$ (the "Erd{\H o}s--Rankin" function). We consider the…

math.NT2015

Short intervals with a given number of primes

Tristan Freiberg

A well-known conjecture asserts that, for any given positive real number $λ$ and nonnegative integer $m$ , the proportion of positive integers $n \leq x$ for which the interval $(n,n…

math.NT2015

Carmichael numbers and the sieve

William D. Banks, Tristan Freiberg

Using the sieve, we show that there are infinitely many Carmichael numbers whose prime factors all have the form $p = 1 + a^{2} + b^{2}$ with $a, b \in Z$ .

math.NT2014★ 4 cited

A note on square totients

Tristan Freiberg, Carl Pomerance

A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of…

math.NT2014★ 3 cited

The average of the first invariant factor for reductions of CM elliptic curves mod $p$

Tristan Freiberg, Paul Pollack

Let $E / Q$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E (F_{p}) ≅ Z / d_{p} Z \oplus Z / e_{p} Z$ , where…

math.NT2014★ 14 cited

On limit points of the sequence of normalized prime gaps

William D. Banks, Tristan Freiberg, James Maynard

Let $p_{n}$ denote the $n$ th smallest prime number, and let $L$ denote the set of limit points of the sequence ${(p_{n + 1} - p_{n}) / lo g p_{n}}_{n = 1}^{\infty}$ of normali…

math.NT2013★ 6 cited

A note on the theorem of Maynard and Tao

Tristan Freiberg

As a corollary to the recent extraordinary theorem of Maynard and Tao, we re-prove, in a stronger form, a result of Shiu concerning "strings" of consecutive, congruent primes.

math.NT2013★ 10 cited

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh

In a recent advance towards the Prime $k$ -tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$ , then for an admissible $k$ -tuple $\mathcal…

math.NT2013★ 1 cited

Mertens's theorem for splitting primes and more

Mohammad Bardestani, Tristan Freiberg

Myriad articles are devoted to Mertens's theorem. In yet another, we merely wish to draw attention to a proof by Hardy, which uses a Tauberian theorem of Landau that "leads to the…

math.NT2012★ 15 cited

On the average exponent of elliptic curves modulo p

Tristan Freiberg, Pär Kurlberg

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under…

math.NT2011

Strings of congruent primes in short intervals II

Tristan Freiberg

Let $p_{1} = 2, p_{2} = 3, ...$ be the sequence of all primes. Let $ε$ be an arbitrarily small but fixed positive number, and fix a coprime pair of integers $q \geq 3$ and $a$ . We will e…

math.NT2010★ 2 cited

Products of shifted primes simultaneously taking perfect power values

Tristan Freiberg

Let $r \geq 2$ be an integer and let $A$ be a finite, nonempty set of nonzero integers. We will obtain a lower bound for the number of squarefree integers $n$ , up to $x$ , for which…

math.NT2010★ 2 cited

Strings of congruent primes in short intervals

Tristan Freiberg

Fix ε> 0, and let p_1 = 2, p_2 = 3,... be the sequence of all primes. We prove that if (q,a) = 1 then there are infinitely many pairs p_r, p_{r+1} such that p_r \equiv p_{r+1} \equ…

Biases in the distribution of primes in short intervals

A tyro's approach to the tilted sieve: beyond the Erdős--Rankin bound

Long runs of integers with small prime factors and the divisor function of n!

Sparser variance for primes in arithmetic progression

Limit points and long gaps between primes

Short intervals with a given number of primes

Carmichael numbers and the sieve

A note on square totients

The average of the first invariant factor for reductions of CM elliptic curves mod p

On limit points of the sequence of normalized prime gaps

A note on the theorem of Maynard and Tao

Consecutive primes in tuples

Mertens's theorem for splitting primes and more

On the average exponent of elliptic curves modulo p

Strings of congruent primes in short intervals II

Products of shifted primes simultaneously taking perfect power values

Strings of congruent primes in short intervals

Long runs of integers with small prime factors and the divisor function of $n!$

The average of the first invariant factor for reductions of CM elliptic curves mod $p$