# 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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### the constant in the asymptotics of $\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}$

The following thread at math.stackexchange.com proposes a constant term for the asymptotic expansion of $$\sum_{1\le k \le n} \frac{\varphi(k)}{k^2}.$$ I am getting a different term and I would like ...
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### Convergence of an integral involving the radical of an integer versus the convergence of $\int_2^\infty\frac{dx}{x(\log x)^2}$

For positive integers $n\geq 2$, let $\operatorname{rad}(n)$ the radical of the integer $n$. For example $\operatorname{rad}(24)=6$. See this Wikipedia to know this definition. Question. I would ...
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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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### Mathematics felt by Srinivasa Ramanujan [closed]

At the moment I am reading the book Ramanujan's Papers by B.J. Venkatachala, V. Vinay and C.S. Yogananda; when clarifying some doubt with a professor, he told me that Srinivasa Ramanujan used Galois ...
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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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### Showing $\pi(ax)/\pi(bx) \sim a/b$ as $x \to \infty$

I'm having a bit of a problem with exercise 4.12 in Apostol's "Introduction to Analytic Number Theory". I don't think it's supposed to be a very hard exercise, it's the first one in its section (they'...
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### Inequality appearing in proof of Mills' Theorem

I'm reading this (very short, 1 page long) paper by W.H. Mills where he determines that there exists a real number $A$ such that $f(n) = \lfloor {A^3}^n \rfloor$ is a prime number for all positive ...
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### Does the sum of reciprocals of primes congruent to $1 \mod{4}$ diverge?

Let $P$ be the set of primes $p$ greater than $3$ such that $p\equiv1 \pmod{4}$. Does the following sum converge or diverge? $$\sum_{p\in P}\frac{1}{p}$$
I came across two remarks that I would appreciate help in making the connections. I) In Riemann's Explicit Formula: for $x > 1$ $\Pi = Li(x) - \sum_{\rho:\zeta(\rho)=0}Li (x^{\rho})- \log(2) +$ ...