Questions tagged [analytic-number-theory]
On the blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.
3,246 questions
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Problem understanding the proof of $L(s,\operatorname{Ind}_N^H(\rho))=L(s,\rho)$
I'm following Jurgen Neukirch's proof of $L(s,\operatorname{Ind}_H^G(\rho))=L(s,\rho)$ and I am having trouble understanding one key step.
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The set up is that we have a tower of extensions of ...
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Distribution of the sign of the real part of derivatives of zeta on the critical line
In https://arxiv.org/pdf/1604.00517 it was shown that Hardy's $Z$-function, on the critical line, is roughly positive half of the time, and negative half of the time. I am interested in a broader ...
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On the set $\{\sum_{k=1}^n p_k:\ n = 1,2,3,\ldots\}$
For any positive integer $n$, let $S(n)$ be the sum of the first $n$ primes. Then
$$S(1) = 2,\ S(2)=2+3=5,\ S(3)=2+3+5 =10,\ S(4) = 2+ 3+5+7 =17.$$
By the Prime Number Theorem,
$$S(n)\sim \frac{n^2}2\...
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Major arc approximations - partial summation and Gallagher's lemma
The motivation for my question is that I think I need to use Gallagher's lemma for exponential sums (``A large sieve density estimate near $\sigma =1$", Lemma 1, https://link.springer.com/article/...
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Are there infinitely many differences of cubed primes that are perfect squares?
So the question is formulated as:
Does equation $$ p^3-q^3=x^2 $$ admits infinitely many prime solutions $p,q$ with $x\ge1$?
Some trivial analysis: it's equivalent to $(p-q)(p^2+pq+q^2)=x^2$. Note ...
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Weighted sums of four primes
Sums of primes have been studied by number theorists for many years. Goldbach's conjecture is the most famous unsolved problem in this direction.
Here I'd like to consider weighted sums of primes. For ...
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Questions motivated by Goldbach's conjecture and the four-square theorem
Goldbach's conjecture asserts that for any integer $n>1$ we have $2n=p+q$ for some primes $p$ and $q$. A similar conjecture of Lemoine states that for any integer $n>2$ we can write $2n+1=p+2q$ ...
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Best formulation of Riemann hypothesis for a general audience
There are many equivalent ways to state the Riemann Hypothesis. I'm looking for a statement that is mathematically precise and yet at the same time as accessible as possible to a general audience. The ...
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Zero density estimates for zeta translated to Dirichlet $L$-functions
In this recent paper of Bellotti the author gives a new zero-density estimate for $\zeta(s)$ close to the boundary of the Vinogradov-Korobov region. It is natural to ask if this generalises to ...
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Sums of distinct powers of 3, 4 and 7
Following the notation introduced in the paper "Complete sequences of sets of integer powers" by Burr, Erdös, Graham and Wen-Ching Li (Acta Arith., 1996), let $\Sigma(\rm{Pow}( \{3,4,7\};1))$...
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Barriers to a fixed-width zero-free region for zeta
The classical zero-free region of the Riemann zeta function $\zeta(s)$ says there is a constant $A>0$ such that there are no zeta zeros $$\rho=\sigma+iT$$
with $\sigma>1-\frac{A}{\log T}$. ...
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Is it true that $\{p_{2^m+1}-p_{2^m}:\ m\in\mathbb Z^+\}=\{2n:\ n\in\mathbb Z^+\}$?
For $n\in\mathbb Z^+=\{1,2,3,\ldots\}$, let $p_n$ denote the $n$th prime. A well known conjecture of de Polignac states that for any $n\in\mathbb Z^+$ there are infinitely many $k\in\mathbb Z^+$ with $...
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Sign of Laurent coefficients of $-\zeta'(s)/\zeta(s)$ at $s=1$
Let $\zeta(s)$ be the Riemann zeta function. Write $-\zeta'(s)/\zeta(s)-1/(s-1) = \sum_{n=0}^\infty a_n (s-1)^n$. It would seem that this is an alternating sum: $$a_0<0, \;\;\;a_1>0, \;\;\;a_2&...
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A new twin prime pair related conjecture? [closed]
CONJECTURE =>
every twin prime pair (x,y) > (11,13) can be expressed as (x,y) = (a+c+1, b+d-1) where
(a,b) < (c,d)
are both smaller twin prime pairs themselves.
Moreover,
a = 6m-1 , b = 6m+1 ,...
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Average of $\Lambda(n)^2$
Let $\Lambda$ be the von Mangoldt function. I am interested in understanding the average $$\sum_{n=1}^x \Lambda(n)^2.$$ By partial summation and the prime number theorem one can prove that this is $$ ...
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