nt.number theory – Prime gap distribution in residue classes and Goldbach-type conjectures

The general problem that I try to solve is this: if $S$ is an infinite set of positive integers, equidistributed in a sense defined here, and large enough as defined in the same post, then all large enough integers can be written as the sum of two elements of $S$. I call this conjecture A, and the purpose of my previous question (same link) was to find whether this is a conjecture, a known fact, or not very hard to prove.

Here I try to solve what I call conjecture B. Let $p_k$ be the $k$-th prime ($p_1 = 2$) and $q_k = p_{k} + p_{k+1} = p_{k} + g_{k}$ where $g_{k} =(p_{k+1}-p_{k})/2$ is the half-gap between $p_{k}$ and $p_{k+1}$. Let $S_1$ be the set of all the $q_k$‘s, for $k=2,3,cdots$. Is $S_1$ equidistributed in the same sense, that is equidistributed in all residue classes? For this to be true, it suffices to prove that the half-gaps are equidistributed in residue classes. There is an attempt to answer that question here, but it is not clear to me if the answer is yes, no, or unsure. What is your take on this?

Assuming conjectures A and B are true, then any large enough integer is the sum of two elements of $S_1$. Another interesting result is this: let $S_2$ be the set of all $lfloor alpha p_krfloor$ where the brackets represent the floor function, $k=1,2,cdots$, and $alpha > 0$ is an irrational number. Then any large enough integer is the sum of two elements of $S_2$.

The interesting thing about $S_2$ is that it is known to be equidistributed and furthermore, you can choose $alpha=1+epsilon$ with $epsilon$ an irrational number as close to zero as you want, but NOT exactly zero. Since $lfloor(1+epsilon)p_krfloor = p_k + lfloor epsilon p_krfloor$, if conjecture A is true you have this result:

Any large enough integer $n$ can be written as $n=p + q + lfloor epsilon prfloor + lfloor epsilon qrfloor$, with $p, q$ primes and
$epsilon>0$ an irrational number as close to zero as you want (but not zero).

With $epsilon=0$, this would be equivalent to Goldbach conjecture, but of course it does not work with $epsilon=0$ since no odd integer $n$ is the sum of two primes, unless $n=p+2$ and $p$ is prime.

nt.number theory – Is there a difference between using nats and bits to express entropy?

It seems to me like questions involving decimal vs binary representations of some number are not particularly interesting: for instance $pi$ or $sqrt{2}$ are conjectured to be normal in every base, and as far as I know this is open for any particular base.

On the other hand, in calculating entropy there is again a choice of basis. Further, this gives us certain ‘distinguished’ real numbers: e.g. the entropy of the Gauss-Kuzmin distribution is $3.432527514776…$ bits, while it is $2.379246769061…$ nats.

Are the properties of these digit strings of the same number ‘similar’ in any way, or is one ‘nicer’ in some sense?

nt.number theory – Simultaneous small fractional parts of polynomials

Fix $epsilon>0$. For a finite set of arbitrary-degree polynomials with integer constant term, $p_1(x), …, p_m(x)in mathbb{R}(x)$ is it possible to find an $nin mathbb{N}$ such that $$max_{i=1,…,m}||p_i(n)||<epsilon$$ where $||cdot||$ denotes the distance to the nearest integer?

In the book Small Fractional Parts of Polynomials this is listed as an open question for degree greater than 4, but the book is from 1977. Research has continued since then on small fractional parts of polynomials. Has progress improved on this problem, or is it still open?

nt.number theory – Quantifying shrinking of solutions of a particular linear diophantine equation when target is small linear combination of coefficients?

Consider the linear diophantine equation $$cex_1+cfx_2+dex_3+dfx_4=a$$

where $a=c-d=e-f$ holds with $c,d$ coprime and $e,f$ coprime and $a,d,f$ are odd and $T<c,d,e,f<2T$ holds and $T^alpha<a<2T^alpha$ holds where $alphain(0,1)$.

We pick uniformly random $ain(T^alpha,2T^alpha)$ and uniformly random $c,ein(T+a,2T)$ and set $d,f$ accordingly.

What is the minimum of $$L=|(x_1,x_2,x_3,x_4)|_infty?$$

If the constraints $a=c-d=e-f$ were not there and $c,d,e,f$ were random and unrelated to $a$ then since $L^4>T^2L$ should hold we need $L>T^{2/3}$.

However the constraints should force it to a smaller solution. How small is it?

In two variable case the situation corresponds to $cx+dy=(c-d)$ with $T<c,d<2T$ and $c,d$ coprime. Here $x=-y=1$ suffices instead of norm $|(x,y)|_infty$ being roughly $T$ where $(c-d)$ is replaced by a random number in $(T^alpha,2T^alpha)$ at an $alphain(0,1)$. This is a huge reduction.

nt.number theory – In how many ways to generate a vector with odd/even entries

Let $bar v=(k_1, ldots k_n)$ be a vector with entries $k_i$ from ${0, ldots, K}$ with $K in N$ and such that $sum_{i=1}^n k_i=K.$

Find in how many ways we can generate vector $bar v$ so that it contains even number of odd values of $k_i$ equal to each other.

nt.number theory – Asymptotic for the probability that a number has $k$ prime factors less than $Q$

If we let $omega_Q(n)$ denote the number of distinct prime factors of $n$ less than a bound $Q$, then what asymptotic formulas exist for $Pr_{ninmathbb{N}}(omega_Q(n)=k)$ as $Qtoinfty$ if $k$ remains fixed (or perhaps very small with respect to n)?

I am asking this question since my study led me to want to bound the quantity

$$mathbf{E}_{ninmathbb{N}}left(frac{2^{omega_Q(n)}}{sqrt{omega_Q(n)}}right)$$

as $Qtoinfty$. Since

$$mathbf{E}_{ninmathbb{N}}left(frac{2^{omega_Q(n)}}{sqrt{omega_Q(n)}}right)=sum_{n=1}^{pi(Q)}left(Pr_{ninmathbb{N}}(omega_Q(n)=k)right)left(frac{2^{omega_Q(n)}}{sqrt{omega_Q(n)}}right)$$

and

$$sum_{n=1}^{pi(Q)}Pr_{ninmathbb{N}}(omega_Q(n)=k)2^{omega_Q(n)}sim_{Qtoinfty} clog(Q)$$

is well understood, good (upper) bounds on $Pr_{ninmathbb{N}}(omega_Q(n)=k)$ could help me in my effort.

For small values of $k$ computations can be done directly, like

$$Pr_{ninmathbb{N}}(omega_Q(n)=0)simfrac{c}{log(Q)}$$

and

$$Pr_{ninmathbb{N}}(omega_Q(n)=1)sim cfrac{log(log(Q))}{log(Q)}$$

The main approach I have been using is noting that $Pr_{ninmathbb{N}}(omega_Q(n)=k)$ is exactly the coefficient of $x^k$ in the polynomial

$$prod_{p<Q}left(frac{x}{p}+1-frac{1}{p}right)$$

Asymptotics of this full polynomial are easy to come by, for instance as $Qtoinfty$ we have that

$$prod_{p<Q}left(frac{x}{p}+1-frac{1}{p}right)sim c log^{x-1}(Q)$$

Heuristically this would suggest that

begin{align*}
Pr_{ninmathbb{N}}(omega_Q(n)=k)&=frac{1}{k!}left.frac{d^k}{dx^k}prod_{p<Q}left(frac{x}{p}+1-frac{1}{p}right)right|_{x=0}\
&sim frac{c}{k!}left.frac{d^k}{dx^k}log^{x-1}(Q)right|_{x=0}\
&=frac{c}{k!}frac{log^k(log(Q))}{log(Q)}
end{align*}

This argument is however by no means rigorous so I would appreciate true asymptotics.

nt.number theory – On a certain integral involving the Chebyshev psi function

Let $psi(x)=sum_{nleq x} Lambda(n)$, where $Lambda$ denotes the von Mangoldt function. Assuming the Riemann Hypothesis, it is known that $f(X)=int_{2}^{X} (psi(x)-x)^2 mathrm{d}x ll X^2$. Independently of the the RH, one would ”conjecture” that $f(X) ll X^{2Theta+1}$, where $Theta$ is the supremum of the real parts of the zeros of the Riemann zeta function. Is this conjecture true ?

nt.number theory – Is there a collection of evidence and heuristic arguments against the Riemann hypothesis?

There is undoubtedly an overwhelming collection of evidence for the Riemann hypothesis. However, is there any evidence against it ? Below is my own heuristic argument against it:

Let $M(x)=sum_{nleq x} mu(n)$, where $mu$ denotes the Mobius function and let $Theta$ be the supremum of the real parts of the zeros of the Riemann zeta function. Define $f(sigma)=int_{1}^{infty} M^{2}(x)x^{-2sigma-1} mathrm{d}x$ and $g(X)=int_{1}^{X} M^{2}(x)x^{-2}dx$. Recall that $M(x) ll_{varepsilon} x^{Theta + varepsilon}$ for every $varepsilon>0$. Consider $sigma >Theta$ and take $varepsilon in (0, sigma-Theta)$. Notice that $g(X) ll _{varepsilon}X^{2Theta-1+varepsilon}ll X^{sigma+Theta-1}$ and integrating by parts,
begin{align}
f(sigma) &= int_{1}^{infty} g'(x)x^{1-2sigma} mathrm{d}x \
& = (2sigma -1)int_{1}^{infty} g(x)x^{-2sigma} mathrm{d}x \
& ll (2sigma-1)int_{1}^{infty} x^{Theta – sigma -1} mathrm{d}x \
& = frac{2sigma – 1}{sigma-Theta}.
end{align}
Assuming that $f(sigma)$ converges whenever $sigma > Theta$ and diverges whenever $sigma<Theta$, it follows by Landau’s Theorem for integrals that $f(sigma)$ has a singularity at $sigma=Theta$, which entails that $(sigma-Theta)f(sigma)gg 1$ as $sigma rightarrow Theta^+$, contradicting the above inequality if $Theta$ were equal to $1/2$. $blacksquare$

nt.number theory – Full measure properties for Zariski open subsets in $p$-adic situation

Let $F$ be a $p$-adic field and let $X$ be a smooth integral variety over $F$ (I am chiefly interested in the case when $X$ is a connected reductive group over $F$). Let $U$ be a non-empty open subset of $X$ with complement $Z$.

We can endow $X(F)$ with the Serre-Oesterle measure (e.g. as in (1,Section 2.2) or (2, Section 7.4))–this is just the standard measure coming from a top form of $X$).

My question is then whether one knows a simple proof/reference for the following:

The subset $Z(F)$ of $X(F)$ has measure zero.

I think this is proven in (1, Lemma 2.14)–but this is concerned with a more specific context which makes it non-ideal as a reference.

Any help is appreciated!

(1) http://www.math.uni-bonn.de/people/huybrech/Magni.pdf

(2) Igusa, J.I., 2007. An introduction to the theory of local zeta functions (Vol. 14). American Mathematical Soc..

nt.number theory – Q-curves and twisting

An elliptic curve $E$ over $overline{mathbb{Q}}$ is called a $mathbb{Q}$-curve if it is isogenous (over $overline{mathbb{Q}}$) to all its Galois conjugates — see Are Q-curves now known to be modular? for example.

If I take a finite Galois extension $K / mathbb{Q}$ and an elliptic curve $E / K$ whose base-extension to $overline{mathbb{Q}}$ is a $mathbb{Q}$-curve, then all the Galois conjugates $E^{sigma}$ are also defined over $K$, but the isogenies between them might not be. Supposing $E$ to be non-CM for simplicity, then what you get instead is a $K$-isogeny from each conjugate $E^{sigma}$ to some possibly non-trivial quadratic twist of $E$. Let me say $E$ is a strong $mathbb{Q}$-curve over $K$ if it’s non-CM and it’s actually $K$-isogenous to all its Galois conjugates. (Clearly any $mathbb{Q}$-curve over $K$ becomes a strong $mathbb{Q}$-curve over some finite extension $L / K$, but I want to keep $K$ fixed here.)

It’s easy to produce examples of $mathbb{Q}$-curves which aren’t strong $mathbb{Q}$-curves, by taking a strong $mathbb{Q}$-curve and applying a quadratic twist by an element of $K^times / K^{times 2}$ that’s not stable under $Gal(K / mathbb{Q})$. However, I can’t find any examples which aren’t of this form.

Are there $mathbb{Q}$-curves which are not twists of strong $mathbb{Q}$-curves?

(I’m chiefly interested in the case when $K$ is a real quadratic field here.)