# 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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### How to use an inverse Mellin transform to get to the $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
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### Proof of Prime Number Theorem

I am looking for a detailed proof of the Prime Number Theorem using analytic methods (that is, using $\zeta(s)$). What is a good reference to read?
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### is there a Globally convergent series for Riemann Xi function?

According to Wikipedia, there is a global convergent series for Riemann Zeta function: https://en.wikipedia.org/wiki/Riemann_zeta_function#Globally_convergent_series Is there a similar global ...
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### What are some recursive properties of Merten function or Summatory Liouville function?

Both Merten function and Summatory Liouville function show some kinds of "scale invariance" properties. (Those functions also display some kind of "periodic" behavior.( Just wonder if those "scale ...
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### Convergence of a sum involving the divisor function and characters

Could someone please show me that $\sum_{n = 1}^{\infty} \sigma_{x}(n)n^{-(x + 1)/2} \chi(n)$ converges where $\chi$ is a Dirichlet character of modulus $m$?
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### Limit of the ratio of the square root of a Mersenne number to the product of its prime factors

Mersenne numbers with prime exponents are numbers of the form $M_p = 2^p-1$, where $p$ is prime. Suppose that $p$ is such that $M_p$ has exactly two prime factors, $\rho, P$. Given $\epsilon > 0$, ...
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### “Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
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### On the summatory function of $\Lambda(n)/n$

In this paper is written that the prime number theorem in the form $\psi(x) = ( 1 + o(1) ) x$ is elementary equivalent to $$\sum_{n \le x } \frac{\Lambda(n)}{n} = \log x - \gamma + o(1)$$ I started ...
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### want a better bound the expression

Consider the expression $$f_n(x)=\sum_{d|n,1<d\leq x} \Lambda(d)\left(\frac{1}{\log d}-\frac{1}{\log x}\right)$$ I've got $f_n(x)=\operatorname{O}\left(\frac{ x}{(\log x)^2}\right)$, but I've ...
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### definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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### Proof Janusz Algebraic number fields, convergence of Dirichlet Series.

The book Algebraic number fields, Janusz Please, Could you explain the proof of the part b) a little more? Thank you all.
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### Generalised sum of divisors.

Let $$F(x,k)=\sum_{a_{1}a_{2}...a_{k}\le x}1$$ Find asymptotic formula for $F(x,k)$ For example it is known. $$F(x,2)=\sum_{n\le x}\tau(x)=x\log x+(2\gamma-1)x+O(x^{\frac{1}{2}})$$ The proof of ...
Let $P$ denotes the set of all primes. Find a subset $Q$ of $P$ such that the following holds: (i) Relative density of $Q$ is zero in $P$ and (ii) $\sum_{p \in Q} \frac{1}{p} = \infty$.