# 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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### Understanding the Hurwitz-Kronecker Class Number Formula

The Hurwitz-Kronecker Class Number is given by the formula $H(d)=\sum_{Q\in Q_d/(\Gamma=PSL_2(\mathbb{Z}))}\frac{1}{w_Q}$ where $w_Q=card(stab(\alpha_Q))$ with $\alpha_Q$ being the unique zero ...
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### Don't know what to do with little-oh!

Let $f:\mathbb{N} \to \mathbb{C}$ be a function for which there exists a positive constant $A$ such that \begin{equation} \lim_{x\to \infty} \frac{1}{x}\sum_{n\leq x} f(n) = A. \end{equation} Prove ...
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### A fun fact relating to Goldbach Conjecture

I have noticed a fact when verifying the Goldbach Conjecture. Let $n$ be an even number larger than 6, we can easily write $n=i+j$, where $i$ and $j$ are both prime numbers. Now let $i\le j$, and ...
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### A combinatorial sum and identity involving Stirling numbers of the second kind

Let $n, k \geq 1$. Let $a(j),\, 1\leq j \leq k$, be a sequence of real numbers. Consider the sum $$\sum_{j=1}^k j! S(k, j) {n \choose j} a(j),$$ where $S(k,j)$ are Stirling numbers of the second ...
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### 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 ...
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### If $(a,k)=(b,m)=1$, prove that $(ab,km)=(a,m)(b,k)$.

I'm reading the proof of the multiplicative property of $$s_k(n)=\sum_{d|(n,k)}f(d)g\bigg( \frac kd\bigg)$$ The book wrote that in order to understand the proof, we need to know if $a,b,k,m$ are ...
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### Does the first Hardy-Littlewood conjecture imply $\sum\limits_{r=1}^{n-3}\Lambda(n-r)\Lambda(n+r)\sim 2C_{2}n$?

Hardy-Littlewood conjecture predicts that the number of Goldbach decompositions $p+q=2n$ should be asymptotically equal to $K\frac{n}{\log^2 n}\prod\limits_{p>2,p\mid n}\frac{p-1}{p-2}$ for a ...
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### Prove that $\sum\limits_{n=1}^{N} \frac{1}{n} \le \text{exp}\left(2\sum\limits_{n=1}^{N} p_k^{-1}\right)$

I want to proof that $$\sum_{n=1}^{N} \frac{1}{n} \le \text{exp}(2\sum_{n=1}^{N} p_k^{-1})$$ where the sequence $(p_n)_n$ denotes the prime sequence. I tried to do this by induction over N, ...
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### The inverse of a Dirichlet product is the Dirichlet product of the inverses of each function

Let $f,g$ be arithmetic functions. According to Wikipedia, $(f*g)^{-1} = f^{-1} * g^{-1}$ if $f(1) \neq 0$ and $g(1) \neq 0$. However, it is not clear to me why this is true, and the statement does ...
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### Prove the equivalent form of the Selberg's Formula $\psi(x)\log x+\sum_{p\leq x}\psi\bigg(\dfrac xp\bigg)\log p=2x\log x+\mathcal O(x).$

This is the Selberg's Formula: $$\psi(x)\log x+\sum_{n\leq x}\Lambda(n)\psi\bigg(\dfrac xn\bigg)=2x\log x+\mathcal O(x).$$ In the book Apostol Analytic Number Theory, the exercise tells me to use this ...
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### Modular transformations of $\eta(\tau)$

Under a modular transformation the Dedekind $\eta$ function transforms as $$\eta(-1/\tau) = \sqrt{-i \tau}\eta(\tau) \, .\tag*{(*)}$$Siegel gives a proof in this paper here that uses complex ...
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### Portion of prime congruent to $a$ mod $m$ in the factorial of $n$

I'm learning analytic number theory and find this interesting question from one textbook: It is well known in analytic number theory that: \begin{equation} \sum_{\substack{p\leq x \\ p\equiv a \...
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### Existence of Dirichlet Inverse.

How do we show the existence of the Dirichlet Inverse of a function $f$? I know using induction that we can find a recursive form of $f^{-1}$. But I can't seem to show the existence. Any ideas? Most ...
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### Calculate the limit of special sequence.

There is given a sequence $A_{0},A_{1},...$ wich are all ordered zeros of Mertens fuction $M(n)$. How to find the following limit ?: $$\lim_{n\to\infty}\frac{A_{n+1}-A_{n}}{A_{n}^{0.75}}$$ Does ...
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### Why is it impossible to invert the analytic continuation of a Dirichlet series?

By Mathematica (and the truncated Euler MacLaurin formula) I know that: $$\zeta(s)=\lim_{k\to \infty } \, \left(\sum _{n=1}^k \frac{1}{n^s}+\frac{1}{(s-1) k^{s-1}}\right) \tag{1}$$ when the real part ...
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### What is sum of squared elements of Farey sequence?

Let's consider a Farey sequence: $$F_{n}=\{a_{1},...,a_{k}\}$$; Where given elements satisfy definition of $n$-th Farey sequence. My problem: Find the formula for the following sum: \sum_{l=1}^{k}...
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### Dirichlet convolution of multiplicative functions

Are there two nonzero arithmetic functions,say $f,g$, which are not multiplicative but their Dirichlet convolution is multiplicative?
Is the inverse of a completely multiplicative function $f(n)$ with respect to Dirichlet convolution again completely multiplicative? I know that for multiplicative functions its true(Apostol's ...
There is a equality in a proof in Apostol's Analytic Number Theory as follows: $O(x^{\alpha} \zeta(\alpha)) = O(x^{\alpha})$ for arbitrary real number $\alpha \ge 0$. How do we say that? Does ...