# Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### Amount of numbers that won't satisfy a list of residue classes

I've been working on functions of the type Px+a where P is a prime number, x is a positive integer and a is an integer in the residue class of P. Now consider a list of primes (2....P) with different ...
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### Why is a theory of newforms in half-weight difficult? Why are $U$-operators used instead of $V$-operators?

Integer weight case: Modular forms in $S_{2k}(N) := S_{2k}(\Gamma_{0}(N))$ can come from lower levels, and we'd like to know when that happens. This is where we call a modular form $f$ "old" ...
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### Why is Abel's Identity (Apostol Theorem 4.2) valid for complex functions?

Apostol uses the Abel Identity developed early in his book as Theorem 4.2 (image below) $$\sum_{y<n\leq x}= A(x)f(x) - A(y)f(y) - \int_{y}^{x}A(t)f'(t) dt$$ to prove a result about complex ...
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### Simple proof for horizontal zero density estimate for the Riemann-Zeta-function

Let $N_c(T)$ denote the cardinality of the set $\{\rho | \zeta(\rho)=0 \text{ and } \Re \rho > c \text{ and } 0 \leq \Im \rho \leq T\}$. I am trying to find a prove for the fact $$N_c(T) = o(T)$$ ...
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### Examples of Theorems of the form “If a set could contain a prime, it will contain a prime”.

One of my favourite theorems in mathematics is Dirichlet's theorem on arithmetic progressions, which states that every set of arithmetic progressions $\{a, a+d, a+2d, \dots \}$ will contain a prime as ...
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### Explicit computation of the factor in the interpolation formula for p-adic Rankin-Selberg L functions

I'm currently trying to understand the appendix of this article by David Loeffler https://arxiv.org/pdf/1704.04049.pdf, which consists in an explicit computation of the factor allowing the ...
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### Approximation for the number of unordered $k$-factorizations of positive integer $n$

I am trying to find an approximation formula for the number of multiplicative partitions of $n$ with $k$ parts. I found that an approximation formula for the number of multiplicative partitions with ...
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### $F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ s.t $F(s_k)=G(s_k)$ show that $f(n)=g(n)$

Let $F(s)=\sum_{n=1}^\infty \frac{f(n)}{n^s}$ and $G(s)=\sum_{n=1}^\infty \frac{g(n)}{n^s}$ be two Dirichlet series which are absolutely convergent for $\Re(s)>a$ for some $a\in \Bbb R$. If there ...
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### Average Number of Small Divisors

I'm working on a pet project of mine and I've come across a seemingly simple problem that I can neither solve nor find any reference to in the literature. The problem is this: Given $x$ sufficiently ...
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### Density of primes is $1/\log x$

The prime number theorem states that $\lim_{x \rightarrow \infty} \pi(x)/Li(x) = 1$, if we denote $\pi(x) := \sum_{p \le x} 1$ for the prime counting function and $Li(x) = \int_2^x \frac{ dt}{\log t}$ ...
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### Can sufficient high degree polynomial sequences contain infinitely many primes?

I have a conjecture: For any integer $N$, there exist an positive integer $n > N$ such that there exist a degree-$n$ polynomial $P(x)$ satisfying:the sequence $\left\{P(n) \right\}$ contains ...
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### Conformal-onto vs Onto map.

In Balazard, Saias and Yor paper they assume that f is in the Hardy space $H^p\mathbb{(D)}$ where $\mathbb{D}=\{z\in \mathbb{C}\mid |z|<1\}$. Let, $f^*$ denote the function defined almost ...
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### A curious question about holomorphicity

Holomorphic functions don't necessarily have antiderivatives/primitives. For example, $\frac{1}{z}$ is holomorphic on the punctured plane $V:= \mathbb{C}- \{ 0\},$ but it does not have any ...
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### Two conjectures about the prime counting function : $\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$

Let $x\geq 100$ then we have as conjecture : $$\frac{\pi{(x)}}{\pi{(\pi{(x)}})}<\ln(x)$$ I have tested at $x=100$ to $x=5000000000$ without any counter-example. The first fact : It seems that the ...
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### Lower bound for the multiplicative order of a fixed integer $a$ modulo $n$, as $n$ grows large

Let $a \geq 2$ be fixed. Is there any good lower bound known for the multiplicative order $\text{ord}_n(a)$ of $a$ modulo $n$ (with $n,a$ relatively prime) as $n$ grows large? Clearly $\text{ord}_n(a)$...
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