# Questions tagged [goldbach-type-problems]

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89
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### Possible implications of the bound $\sum_{n\leq x}\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)=O\left(x\right)$

Let $\lambda(n)$ be the Liouville function and consider the Goldbach-type problem $\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$. Assume the Riemann hypothesis, the ...

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### Bateman-Horn-type generalization of the Goldbach conjecture

The Bateman-Horn conjecture is a generalization of the twin prime conjecture that roughly states that given a set $S=\{f_1, \dots, f_m\}$ of irreducible polynomials with integer coefficients, there ...

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### Effective bound for odd numbers expressed as sums of three primes

I am interested in the representation of odd numbers greater than five as sums of three primes, inspired by Harald Helfgott's seminal proof of the ternary Goldbach conjecture and the nuanced findings ...

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### Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture

I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight ...

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### Goldbach conjecture and the difference of two primes

The Goldbach conjecure is not yet proved. But, when an even number is represented as a sum of two primes, is there any knwon result about the difference of the two primes?
That is, if $2n$ is a sum of ...

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### Is the Goldbach conjecture easier if we allow 1 as a prime?

I hope this is the right site for the question.
Is the Goldbach conjecture easier if we allow 1 as a prime? (12=1+11 would be allowed as Goldbach sum for 12)
IOW: if we can prove Goldbach for the case ...

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### On Buchstab et al's "forgotten" sieve and the Goldbach conjecture for certain integers

There is a somewhat forgotten sieve-theoretic approach to the Goldbach conjecture, due to Buchstab et al, see e.g. pp.247-248 of R.D. James.
On p.247, James defines some function $F$ such that for any ...

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### Another Goldbach variation for odd numbers?

Lemoine's conjecture (also called Levy's conjecture according to Professor Wikipedia) states that every odd integer larger than $5$ is the sum of a prime and of twice a prime.
Dabbling in the dark art ...

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### Reducing the number of terms in Waring-Goldbach problem by allowing exponents to vary

Assuming the Waring-Goldbach problem (see https://en.m.wikipedia.org/wiki/Waring%E2%80%93Goldbach_problem) has a positive solution, can we reduce the number of terms $t$ to some value $t'$ by allowing ...

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### Inversion shift of a Galois radius

Say a non negative $r$ is a Galois radius of $n$ of type $(a,b)$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime and positive $a$ and $b$. If $a\neq b$, say $r$ is "unbalanced" and say $s$ ...

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### Lower bounding the number of Galois radii of an integer

Recall that I call $r>0$ a Galois radius of an integer $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ primes and positive $a$ and $b$ and a primality radius of $n$ if $a=b=1$.
Does it suffice to ...

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### Does asymptotic Goldbach imply GRH?

It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...

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### Selberg's 1943 result on primes in short intervals and primality radius

This preprint: https://arxiv.org/abs/2207.05038 states in the last paragraph of the first page that a result of Selberg (1943) implies that under RH, almost all intervals of the form $(x,x+\left(\log ...

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### Why does this convolution of the prime counting function $\pi$ look like a parabola?

In this previous question it is shown that the convolution of the prime counting function $\pi$ with itself, is related to the Goldbach conjecture:
$$\pi^*(n):=\sum_{k=0}^n \pi(k) \pi(n-k)$$
The ...

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### Prime powers gap of type $(a,b)$

For $n$ a given positive integer, say $r$ is a Galois radius of $n$ of type $(a,b)$, level $l=ab$ and rank $\rho=a+b$ if $n-r=p^a$ and $n+r=q^b$ with both $p$ and $q$ prime.
Denote by $PPG_{a,b}(m)$ ...

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### Reference request Re Vinogradov's ternary Goldbach proof

I believe that I.M. Vinogradov's proof of the ternary Goldbach conjecture used the observation that the number of ways $n$ can be written as a sum of three primes equals
$$
\int_0^1 \sum_{p , q , r \...

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### Is the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ provable for some values of $a$, $b$ and $d$?

Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ ...

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### A relation of the prime counting function $\pi$ to counting the ordered ways of a number $n$ as a sum of two primes and two questions?

The definitions are from these two questions:
https://math.stackexchange.com/questions/3164216/a-series-related-to-prime-numbers
https://math.stackexchange.com/questions/4349186/trying-to-understand-...

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### $l$-th power radioprimal conjecture

I would like to know if some widely believed conjecture, be it GRH, Hardy-Littlewood conjecture, or any other would imply the following statement for some $l>1$:
$l$-th power radioprimal growth ...

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### Is there a link between Elliott-Halberstam and weak Hardy-Littlewood-Goldbach conjectures?

Let $\theta$ be such that $EH(\theta)$ holds, where $EH$ stands for Elliott-Halberstam. Can one get an explicit lower bound $\delta_{\theta}$ for the quantity $\delta$ appearing in the weak Hardy-...

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### Can a lower bound for this weakening of Goldbach's conjecture be reached?

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a Galois radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, where $\...

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### $g$-gap radius of an integer

For $n$ a large enough composite integer, define the $g$-gap radius of $n$, if it exists, for positive even $g$ as the smallest positive integer $\rho_{g}(n)$ such that both $n-\rho_{g}(n)$ and $n+\...

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### Consequences of Goldbach's conjecture

In a letter of 1742 to Euler, Goldbach expressed the belief that ‘Every integer $N>5$ is the sum of three primes’. Euler replied that this is easily seen to be equivalent to the following statement:...

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### Robin's criterion, Goldbach's conjecture and upper bound for $r_{0}(n)$

This question is a follow-up to both About Goldbach's conjecture and Question in Proof of Hardy Ramanujan theorem about $\omega(n) =\sum_{p|n} 1$.
Can one derive from Robin's criterion for RH an ...

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### Weak Hardy-Littlewood-Goldbach conjecture

Assuming the weak Hardy-Littlewood-Goldbach conjecture as stated in this paper,
does the density $d(\delta,\varepsilon)$ of integers $m$ below $n$ such that
$$
\left\vert\frac{G(m)}{{\frak{S}}(m)m}-\...

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### Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?

This question is a follow-up to About Goldbach's conjecture and as such deals with the notion of primality radius of a composite integer $n$, that is, a positive integer $r$ such that both $n-r$ ...

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### Weak Goldbach conjecture with distinct primes for odd integers between $4\times 10^{18}$ and $10^{27}$

This is related to the conjecture that all odd integers greater than $17$ can be written as the sum of 3 distinct primes.
Schinzel showed that the Goldbach conjecture implied this in 1959 and as the ...

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### Upper bound for the number of $k$-central numbers in a prime gap

Let $I_{n}:=]p_{n},p_{n+1}[$ be the open interval between the $n$-th and $(n+1)$-th prime. Under Goldbach's conjecture, denote by $r_{0}(m)$ the smallest positive integer $r$ such that both $m-r$ and $...

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### Staircase numbers

Assuming Goldbach's conjecture, denote as usual by $r_{0}(n)$ for any large enough positive integer $n$ the smallest positive integer $r$ such that both $n-r$ and $n+r$ are prime.
Let's define the ...

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### A number theoretical identity of exponential sum

I try to understand a number theoretical identity used by
Jan-Christoph Schlage-Puchta in this answer.
He defined the function
$$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$
where $\Lambda(n)$ is ...

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1
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### Does one have $2r_{0}(n)\lesssim k_{0}(n)(\log n)^{1+1/k_{0}(n)}$?

Under Goldbach's conjecture, I'm trying to find an upper bound for $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ that would generalize Cramer's conjecture.
Denoting by $k_{0}(n)$ the quantity ...

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### Symmetry in Hardy-Littlewood k-tuple conjecture

Assuming Hardy-Littlewood $k$-tuple conjecture, do the "dual" prime constellations $(0,h_1, h_2,\cdots, h_i,\cdots, h_{k-1}=d)$ and $(0, h_{k-1}-h_{k-2}, h_{k-1}-h_{k-3},\cdots,h'_i=h_{k-1}-...

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### Goldbach conjecture and the representation number

Let $g(2n)$ be the number of representations of $2n=p+q$ with primes $p$ and $q$. Many people have asked whether $g(2n) \ge 2$ when $2n = p+q$ for some primes $p$ and $q$. That is, does $g(2n) \ge 1$ ...

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### Prime gap distribution in residue classes and Goldbach-type conjectures

Update on 7/20/2020: It appears that conjecture A is not correct, you need more conditions for it to be true. See here (an answer to a previous MO question).
The general problem that I try to solve is ...

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### A nice pattern about Goldbach conjecture in French Wikipedia

In the following link: https://fr.m.wikipedia.org/wiki/Conjecture_de_Goldbach, one can see a nice pattern of pink and blue lines coming from each prime number, the intersection points thereof are ...

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### Congruential equidistribution, prime numbers, and Goldbach conjecture

Let $S$ be an infinite set of positive integers, $N_S(z)$ be the number of elements of $S$ less than or equal to $z$, and let
$$D_S(z, n, p)= \sum_{k\in S,k\leq z}\chi(k\equiv p\bmod{n}).$$
Here $\chi$...

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### Curious inversion formula in additive combinatorics

Let $S$ be an infinite set of positive integers, and $T=S+S=\{x+y, \mbox{ with } x,y\in S\}$.We definte the following functions:
$N_S(z)$ is asymptotic continuous version of the function counting the ...

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### Paradox in additive combinatorics

Let $S$ be an infinite set of positive integers. Let us define the following quantities:
$N_S(z)$ is the number of elements of $S$, less or equal to $z$
$r_S(z)$ if the number of positive integer ...

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### General asymptotic result in additive combinatorics (sums of sets)

Let $S_1,\cdots,S_k$ be $k$ infinite sets of positive integers. Let $N_i(z)$ be the numbers of elements in $S_i$ that are less or equal to $z$. Let us further assume that
$$N_i(S) \sim \frac{a_i z^{...

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### $t$-balanced numbers

Disclaimer: throughout this question, we'll assume the truth of Goldbach's conjecture.
For $n$ a large enough composite positive integer, write $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$, $...

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### Is this conjecture equivalent to Polignac's conjecture?

Under Goldbach's conjecture denote by $r_{0}(n)$ for $n$ a large enough composite integer the quantity $\inf\{r\geq 0,(n-r,n+r)\in\mathbb{P}^{2}\}$, by $k_{0}(n)$ the quantity $\pi(n+r_{0}(n))-\pi(n-...

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### Need help in understanding meaning of a notation and theorem used in research paper due to a reference being in German Language

I thought of utilizing this lockdown period to study research papers in number theory by myself.
I began reading the research paper By T Estermann ->" On Goldbach Problem : Proof that Almost all ...

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### Can this number be interpreted as a fractal dimension?

Under Goldbach's conjecture, let's denote for a large enough integer $n$ by $r_{0}(n)$ the quantity $\inf\{r>0,(n-r,n+r)\in\mathbb{P}^2\}$ and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n+r_{0}(n))$.
Let's ...

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### Statement about upper density of even numbers satisfying the Goldbach condition

For $A\subseteq \mathbb{N}$, let the upper density of $A$ be defined by $$\mu^+(A) = \lim\sup_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}.$$
Let $$A = \{n\in\mathbb{N}: 2n \text{ is the sum of } 2 \...

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### Are there infinitely many karmic numbers, i.e numbers whose primality radii equal one or a prime power?

For $n$ a large enough positive composite integer, say $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime. Say $n$ is a karmic number if the following holds: $r$ is a primality radius ...

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### Equivalent Statements of Goldbach Conjecture in Terms of the Properties of Riemann Zeta Function?

Riemann Hypothesis has many equivalent statements.
Many of them are not about prime distribution, instead, are about the properties of Riemann Zeta function, such as the distribution of zeros of Zeta ...

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### Riemann hypothesis and ternary Goldbach

Is there any result of the following shape: There exists an absolute constant $\delta>0$ such that the Riemann hypothesis for some $L$-functions is equivalent to the following estimate for all ...

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### Upper bound for $\alpha_{n}$ from Mertens' third theorem

This question is a follow-up to About Goldbach's conjecture.
I would like to know if an unconditional upper bound for $\alpha_{n}$, defined as $n(N_{2}(n)-\dfrac{nN_{1}(n)}{P(n)})$ (where $N_{2}(...

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### Distribution of Goldbach's weak-conjecture's prime-triples

From Harald Helfgott's proof of Goldbach's weak conjecture,1
we know that every odd number $> 7$ is the sum of three odd primes.
If $n$ is such an odd number, say that two sums that yield $n$
are ...

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### Primality radii and Sidon sets

I learned tonight what a Sidon set is, in a book about Erdős. This notion inspires me the following question :
For $n$ a large enough composite integer, say $r>0$ is a primality radius of $n$ if ...