# 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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### sign unchanged for Dirichlet polynomials?

Let $\chi_{q}$ be a primitive Dirichlet character with modulus $q$ (see definition at wikipedia ). For example for $q=5$ we have \begin{aligned} \chi_{5,1}&=(1, 1, 1, 1, 0),\\ ...
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### Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r)$?

Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that $$\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0$$ My experimental data for $n \le 1.5 \times 10^5$...
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### Are these equalities wrong $\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x)$?

I simply found an asymptotic relation for $\sum_{n\le x}\mu(n)o(\frac xn)$ like below: $$\sum_{n\le x}\mu(n)o(\frac xn)=o(x\sum_{n\le x}\frac 1n)=o (x\log x)$$ But Basil Gordon, an American ...
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### Weyl's equidistribution theorem in the case of rational numbers

Let θ be a non-zero real number and let N be a positive integer. Consider the fractional parts of nθ for 0 ≤ n < N. This creates a maximum of N possible intervals. Show that there as only three ...
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### On Property of Ramanujan Tau function

I have been self studying number theory from Apostol Dirichlet series and modular forms and I could not solve this question from chapter 4 Pg92. There is a solution of this question on stack exchange ...
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### On proving a result on Jacobi theta function

I have been self studying Apostol Dirichlet series and Modular forms and I have could not solve this problem from Chapter 4 - Prove that $\theta(-1/t) = \sqrt{-it}\ \theta(t)$ , t belongs to upper ...
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### Reference requests: Jitsuro Nagura

I spent some time today looking for any biographical information on Jitsuro Nagura and came up empty-handed. Any suggestions welcome. Also, the Wiki note on the Chebyshev $\psi$ function says that ...
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### Density of Numbers with Exactly One Prime Factor of Multiplicity 1

Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
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### Interlacing with arithmetic progressions?

Given integers $0<a_1<\dots<a_t$ and $0<b_1<\dots<b_t$ with $a_t<b_t<M$ can we find integers $m,n,m',n'\in\mathbb Z$ such that $$ma_i+n<m'b_i+n'<ma_{i+1}+n$$ holds at ...
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### Show that $\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|}$

We must show that $$\int_1^{\infty}e^{-\pi x}(x^{\sigma/2} + x^{(1-\sigma)/2}) x^{-1} dx \ll (1 + |\sigma|)^{\pi |\sigma|}.$$ Here is my attempt, however I wondered if there is a one-trick wonder ...
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