Which integer polynomials represent fewer primes, in terms of order of magnitude, when shifted by a constant integer? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:21:44Z http://mathoverflow.net/feeds/question/115253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115253/which-integer-polynomials-represent-fewer-primes-in-terms-of-order-of-magnitude Which integer polynomials represent fewer primes, in terms of order of magnitude, when shifted by a constant integer? Timothy Foo 2012-12-03T06:41:50Z 2012-12-07T00:06:08Z <p>Let $f(x_1,\dots,x_n) \in \mathbb{Z}[x_1,\dots,x_n]$ be a polynomial for which the set of integers not represented by it is infinite. I'm curious about cases in which there exists an integer $c$ such that the density of primes represented by $f$ and those represented by $f+c$ differ in their orders of magnitude.</p> <p>[EDIT: It seems that the previous condition on the set of integers not represented by $f$ needs to be improved to: not $o(N/\log N)$. The example of $f(x,y,z)=2x^2+xy+3y^2+z^3-z$ in this interesting <a href="http://mathoverflow.net/questions/12486/integers-not-represented-by-2-x2-x-y-3-y2-z3-z" rel="nofollow">question</a>, Integers not represented by $2x^2+xy+3y^2+z^3-z$, makes one aware of this. Thank you for the hint!]</p> <p>[EDIT: A slightly more specific and maybe better way to phrase the question is: For which $f\in \mathbb{Z}[x_1,\dots,x_n]$ do we have $f$, $f+1$ both representing an infinite amount of primes and</p> <p><code>$$\sum_{\substack{p \leq N\\p-1 = f(x_1,\dots,x_n)}} 1 = o\left(\sum_{\substack{p \leq N\\p = f(x_1,\dots,x_n)}} 1\right)?$$</code></p> <p>Thanks to js who points out that if one believes the Bateman-Horn conjecture, then this phenomenon is specific to multivariate polynomials.]</p> <p>Motivation: These two questions, <a href="http://mathoverflow.net/questions/115193/integers-represented-by-the-polynomial-a2b3c6" rel="nofollow">Integers represented by the polynomial $a^2+b^3+c^6$</a> and <a href="http://mathoverflow.net/questions/92053/primes-1-x2-y2" rel="nofollow">Primes $1+x^2+y^2$</a>, made me wonder. In the second linked question, the integers less than $N$ represented by $x^2+y^2+a^2$ for any $a\in\mathbb{Z}$ are asymptotically the same, namely</p> <p><code>$$\frac{0.7642\dots N}{\sqrt{\log N}}.$$</code> </p> <p>Yet the primes less than $N$ representable by $x^2+y^2$ are asymptotically</p> <p><code>$$\frac{N}{2\log N}$$</code></p> <p>while those less than $N$ representable by $x^2+y^2+1$ have order of magnitude </p> <p><code>$$\asymp \frac{N}{(\log N)^{3/2}}.$$</code></p> <p>Question: Is there a really good way to characterize which polynomials $f$ behave somewhat like this?</p> <p>Intuitively, it seems that the above difference is due to the phenomenon that the more factors $x^2+y^2=m$ has, then the more "collisions" as well - many $(x,y)$ map to the same $m$. As in <a href="http://mathworld.wolfram.com/SumofSquaresFunction.html" rel="nofollow">here</a>. And $p-1$ is always composite, except $p=2,3$. But I don't know if this is the right way to think of the question, or whether other $f$ which are like this, behave like this for the same reason.</p> <p>Note: Let's exclude the cases where the polynomial becomes reducible over $\mathbb{Z}[x_1,\dots,x_n]$ after being shifted. I.e., let's stick to cases where both $f$ and $f+c$ still represent an infinite amount of primes.</p> <p>Thank you.</p> <p>EDIT: I'm not really sure about this, but perhaps one possible way is to consider norm forms for algebraic number fields, generalizing the case of $f(x,y)=x^2+y^2$ corresponding to $\mathbb{Q}[i]$. If $f$ is the norm of elements in the ring of integers of some number field, then the primes less than $N$ represented by it will have order of magnitude</p> <p><code>$$\asymp\frac{N}{\log N}.$$</code></p> <p>The exact constant depends on the degree and class number of the field. Then the task is to understand how often it takes values in $\mathbb{P}-1$. In the case of $f(x,y)=x^2+y^2$, it seems we had</p> <p><code>$$\sum_{\substack{p\leq N\\p-1=x^2+y^2}}1 = o(N/\log N)$$</code></p> <p>because the integers of the form $x^2+y^2$ were characterised by being divisible by even powers of primes $q\equiv 3 \bmod 4$ with no condition on the primes $q \equiv 1 \bmod 4$. This is in turn was due to the properties of $f$ coming from a number field, in this case, $\mathbb{Q}[i]$. We then had to estimate the number of primes $p\leq N$ for which the factorization of $p-1$ has only even powers of $q \equiv 3 \bmod 4$ occuring. Found something helpful on binary quadratic forms in the whole of section 4 of these <a href="http://www.math.uga.edu/~pete/thuelemmav6.pdf" rel="nofollow">notes</a> by Pete Clark and Proposition 4.1 of these course <a href="http://www.dms.umontreal.ca/~andrew/Courses/Chapter4.pdf" rel="nofollow">notes</a> of Andrew Granville. They give the analogue of this phenomena for binary quadratic forms. This motivates:</p> <p>Question 2: For any form $f$ that is the norm of elements in the ring of integers of a number field, is there a general statement that will characterise the set of integers represented by it in such a way that will allow us to conclude that</p> <p><code>$$\sum_{\substack{p\leq N\\p-1=f(x_1,\dots,x_n)}}1 = o(N/\log N)?$$</code></p> <p>Thank you.</p> <p>EDIT: It seems that Question 2 is nicely thought of in the following light: Motivated by the comment by Greg Martin in this interesting question, <a href="http://mathoverflow.net/questions/115452/average-orders-of-multiplicative-functions" rel="nofollow">Average orders of multiplicative functions</a>, one is led to consider Theorem 1 of this paper, <a href="http://arxiv.org/abs/1110.0708" rel="nofollow">Counting numbers in multiplicative sets: Landau versus Ramanujan</a> of Moree on multiplicatively-closed sets, and also this MO question, <a href="http://mathoverflow.net/questions/94543/density-of-a-set-of-integers" rel="nofollow">Density of a set of integers</a>. These indicate that the primes in such multiplicatively-closed sets occur more frequently than what would be expected from applying the heuristics on the likelihood of an integer being prime derived from the Prime Number Theorem to a "random" set which is the same size as the multiplicatively-closed set in question. The integers represented by the function $f$ in Question 2 are a multiplicatively-closed set, but shifting them by a constant integer removes this property. Guess that accounts for Question 2. Now, this interesting question, <a href="http://mathoverflow.net/questions/115430/what-is-the-geometry-of-an-undecidable-diophantine-equation" rel="nofollow">What is the geometry of an undecidable diophantine equation?</a>, shows that the general case is really not straightforward! </p>