# 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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### $\forall n\in\mathbb N:n^x\in\mathbb Q$ implies $x\in\mathbb Z$ - elementary proof?

Consider the following two problems: Show that if for some $x\in\mathbb R$ and for each $n\in\mathbb N$ we have $n^x\in\mathbb N$, then $x\in\mathbb N$. Show that if for some $x\in\mathbb R$ ...
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### Using the Brun Sieve to show very weak approximation to twin prime conjecture

I recently stumbled across MIT OCW for analytic number theory. As a budding number theorist, my ears perked up and I looked through some of the material haphazardly. I don't really know much about ...
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### Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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### A map from zeros of $\zeta(s)$ to zeros of $C(s)?$

Let $P(s),C(s),\zeta(s)$ be the prime zeta function, the analogous composite zeta function, and the classical zeta function. I do not know whether it is known that there are infinitely many zeros of ...
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### Equidistribution of roots of prime cyclotomic polynomials to prime moduli

Here is a relevant - and longstanding, I'm told - conjecture. Let $f \in \mathbb{Z}[x]$ be irreducible and of degree > 1. Set $E_p = \{x/p \: | \: 0 \leq x < p, f(x) \equiv 0 \: (p) \}$ = { ...
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### Is there an equivalent statement of Riemann Hypothesis in term of Random Matrix or physics theory?

We all know that Riemann Hypothesis has many equivalent statements. After Montgomery’s works on pair-relationship, we now know that ZEROs of Riemann Zeta function has similar properties as ...
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### Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
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### For what $k\in \mathbb{C}$ do we have $\lim_{x\to\infty}\frac{1}{x^{k+1}}\sum_{n=1}^x \sigma_k(n)=\frac{\zeta(k+1)}{k+1}$?

Can the following claim be extended for some complex $k$. Perhaps: all $k$ with real part greater than or equal to $1$? Do the arguments below fall apart for complex $k$ for some reason? Claim. ...
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### The ordinary generating function for the square-free kernel: reference request about singularities and its phase plot

Let $n\geq 1$ an integer, in this post I denote the product of distinct prime numbers dividing dividing $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ is the famous ...
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### Functional equation of the complete $L$-function of the twisted $L$-function of a cuspidal modular form

Let $f(z)=\sum a(n)n^{(k-1)/2}q^n\in S_k(\Gamma_0(N),\chi)$ a cuspidal modular form of integral weight with nebentypus $\chi.$ I am looking for the expression of $\Lambda(\psi\otimes f,s)$ the ...
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### Large gap between two consecutive square-free numbers

Let $q_n$ denote the $n$-th square-free number. By Chinese remainder theorem (see this post), it is not difficult to show that there is arbitrarily large gap between two consecutive square-free ...
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### Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...