# Questions tagged [analytic-number-theory]

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

2,436 questions
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### Hurwitz zeta function for $s=0$ $\zeta(0,1/2)$

I'm studying the Casimir Effect with perfect spherical boundary which involves the use of the Hurwitz zeta function. I've been staring for a while at this equation: \begin{align} \sum_{l=1}^{\...
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### Prove $F^* = \mu * F$

Let $f: \mathbb{Q} \cap [0,1] \to K$ and set $F(n) = \sum_{k = 1}^n f(\frac k n)$, $F^*(n) = \sum_{k = 1, (k,n) = 1}^n f(\frac k n)$. Show that $F^* = \mu * F$ where $*$ is the Dirichlet product....
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### application of distribution of primes in arithmetic progressions

I try to understand an application of distribution of primes in arithmetic progressions Let $$f(x) = \sum_{p \leq x p\equiv 3 \bmod 10} 1$$ So computing $f(40) = 3$ i.e. the primes: 3, 13, and 23 ...
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### On the proof that $\phi(n)/n$ has a limit law

In this question, $\mathbb{N}$ denotes the set of positive integers. Also, $\overline{\mathrm{d}}$, and $\mathrm{d}$ means upper natural density, and natural densitiy respectively. (They are the ...
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### Why aren't holomorphic modular forms bounded?

Let $f$ be any non-zero integral weight (holomorphic) modular form with respect to $SL_2(\mathbb{Z})$ and of weight $k, k\geq 4$. Since it is holomorphic at infinity, for given $\epsilon > 0$, it ...
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### Prove the series $\sum n^{-1-it}$ is diverge for all real $t$.

Prove that the series $\sum_{n=1}^\infty n^{-1-it}$ diverges for all real $t$. I have shown in the previous exercise that this series is bounded for nonzero $t$, and when $t=0$, it is famous that the ...
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### Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$

Guy Robin proved that $$$$\sigma(n) < e^\gamma n \log \log n$$$$ is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). The paper where ...
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### Lower Bound on the Sum of Reciprocal of LCM

While reading online, I encountered this post which the author claims that \begin{align} S(N, 1):=\sum_{1\le i, j \le N} \frac{1}{\text{lcm}(i, j)} \geq 3H_N-2 \end{align} and $S(N, 1) \geq H_N^2$ ...
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### Partial Euler product

The Riemann Zeta function defined as $$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$$ For $\Re(s)>1$ is convergent and admits the Euler product representation $$\zeta(s) = \prod_p (1-p^{-s})^{-1}$$ For ...
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### Inferences about sign of a function from abscissa point

Suppose we have a function $g(x)$ and an integral $$F(s)=\int_1^\infty \frac{g(x)}{x^{s+1}}dx$$ and $F(s)$ converges for $s>\beta$ and diverges for $s = s<\beta.$ Assume also that $\beta$ is ...
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### Summatory function of Moebius and Euler's totient function over $y$-smooth numbers

Let $y \geq 1$. We say that a positive integer $n$ is $y$-smooth if $n$ has no prime factors larger than $y$. Let $x \geq y$. Let $\mu$ and $\varphi$ be the Moebius and Euler's totient function ...
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### To show a function is bounded by a function when x is large

I want to show $\sum_{p^a\leq x}\log p = O(\sqrt{x}\log^2 x)$,where sum runs over $a\geq2$. I only know that $\sum_{\sqrt{x}<p \leq x}\log p \leq 2x\log x$. I tried using above property but I am ...
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### Problem trying to show the following:$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$

So I have to show the following: $$\sum_{n\leq x} (\omega(n)-\ln\ln x)^2=O(x\ln\ln x)$$ But the problem is in finding a suitable bound for: $$\sum_{n\leq x} (\omega(n))^2$$ I have tried the ...
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### If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $\exp_m(ab)=\exp_m(a)\exp_m(b).$

If $(a,m)=(b,m)=1$ and if $(\exp_m(a),\exp_m(b))=1$, prove that $$\exp_m(ab)=\exp_m(a)\exp_m(b).$$ The notation $\exp_m(a)$ is denote the smallest positive integer $n$ such that $a^n\equiv 1\pmod m$. ...
### Proof Verification: Let $f(\chi)$ be the conductor of $\chi$, proof that $f(\chi)=f(\chi_1)\cdots f(\chi_r)$.
This is a detailed problem, let me write down the problem and the process I have done: Assume that $k=k_1k_2\cdots k_r$ where $k_i$ and $k_j$ are relatively prime for $i\neq j$. Let $\chi$ be a ...