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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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16 views

Bound on log integral

I am looking for an explanation of the bound $$\frac{1}{2\pi}\left(-\frac{T \log T}{1+(t-T)^2} - 2 \int_T^\infty \frac{x \log x (t-x)}{(1+(t-x)^2)^2} dx \right)\ll \left( \frac{1}{t+1} + \frac{1}{T-t+...
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17 views

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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12 views

Open problems related to modular objects

What are some open problems in the theory of modular forms, mock modular forms, L-functions, quantum modular forms and all our little modular friends ! If you have interesting research papers to share ...
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25 views

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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0answers
23 views

Step by step derivation of Robin's inequality $\sigma(n) < e^\gamma n \log \log n$

Guy Robin proved that $$\begin{equation} \sigma(n) < e^\gamma n \log \log n \end{equation}$$ 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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1answer
39 views

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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1answer
37 views

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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1answer
43 views

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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0answers
17 views

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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1answer
19 views

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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66 views

Analytic Number Theory Research [closed]

I am finishing up a masters degree and am thinking about branching into Analytic Number theory, and have a pretty strong foundation in basic elementary number theory and wanted to know what texts I ...
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13 views

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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1answer
12 views

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 $a^n\equiv 1\pmod m$. ...
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1answer
59 views

Limit of a function, given the recurrence relation

Let $f(n)$ be a function defined for $n\ge 2$ and $n\in N$ which follows the recurrence(for $n\ge 3$) $$\displaystyle f(n)=f(n-1) +\frac {4\cdot (-1)^{(n-1)} \cdot \left(\sum_{d \vert (n-1)} (\chi (d))...
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33 views

Asymptotic for the gamma function on vertical lines

On page 135 of Joerg Bruedern's "Einfuehrung in die analytische Zahlentheorie" he claims that Stirling's formula implies for fixed $\sigma <0$, any $t\geq 1$, and some constant $C$ (I assume ...
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27 views

What information do the moments of the Riemann-Zeta function give us

I have seen an explicit formula for what a moment of the Riemann-Zeta function is but I am unsure what information this give us? If we are looking at the zero's of the function then this can be ...
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1answer
25 views

Bounding sum (log factor)

I want to prove that $$ \sum_{\substack{1\leq n\leq T \\ n\neq m}}n^{-\frac{1}{2}}\left|\log \frac{m}{n}\right|^{-1}\ll T^{\frac{1}{2}}\log T $$ for any $1\leq m \leq T$. Do you have any hint how I ...
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16 views

Equidistributed sequences satisfying the central limit theorem

For a sequence (x_n) of reals that are equidistributed modulo one (such as (prime-) multiples of irrationals, or most geometric sequences) does the (properly scaled) sum of the remainders converge to ...
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22 views

The sum $\sum_{r=1}^{p-1} r(r|p)$ when $p$ is an odd prime of the form $4k+3$, $k\geq 1$.

In the book Apostol Analytic Number Theory, $(r|p)$ denotes the Legendre Symbol. The exercise tell us to prove when $p\equiv 1\pmod 4$, $$\sum_{r=1}^{p-1}r(r|p)=0.$$ I can solve this quickly, but I ...
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33 views

Solving $\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du=0$ (an extension to the Ramanujan-Soldner constant)

For $u,x>0$, let $P$ be the function given by $$P(x)=\int_0^x\int_u^{u+1}\frac1{\ln t}\,dt\,du\tag1.$$ Is there a closed form for the positive root of $P(x)$, denoted by $\nu$? Can it be ...
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1answer
18 views

Let $P=\{1,2,\cdots,p-1\}$, $P=S\cup T$, prove that $S$ is quadratic residues and $T$ is quadratic nonresidues.

Let $p$ be an odd prime. Assume that the set $\{1,2,\cdots,p-1\}$ can be expressed as the union of two nonempty subsets $S$ and $T$. $S\neq T$, such that the product (mod $p$) of any two elements in ...
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11 views

Problem understanding summation of Big-O notation i.e $F(x)=\sum_{n\leq x}O(\frac{x}{n})$

I know that $O(f(x))+O(g(x))=O(g(x))$ if $f(x)=O(g(x))$. But I cant seem to find a bound for F(x) since n changes when x increases. Any ideas. Im assuming that $F(x)=O(\sum_{n\leq x}\frac{x}{n})$ ...
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1answer
46 views

Show that $\sum_{pq\leq x}\frac{1}{pq}$ = $(\ln \ln x)^2+O(\ln \ln x)$

I know that $\sum_{pq\leq x}\frac{1}{pq}$=$\sum_{p\leq x}\frac{1}{p}\sum_{q\leq\frac{x}{p}} \frac{1}{q}$=$\sum_{p\leq x}\frac{1}{p}(\ln\ln(\frac{x}{p})+A+O(\frac{1}{\ln (\frac{x}{p})}))$. However, I'm ...
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130 views

Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
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47 views

Obscure approximate functional equation for the Riemann zeta function

The following result is supposed to follow from an approximate functional equation for the Riemann zeta function, but I've never seen anything close enough to it : For $T \leq t \leq 2T$ where $T$ is ...
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26 views

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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30 views

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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2answers
44 views

Prove that $\left|\sum\limits_{m=N+1}^M\frac{\chi(m)}{m}\right|< \frac{2}{N+1}\sqrt{k}\log k$.

This problem is from Apostol's Analytic Number Theory, Chapter $9$. The problem: Let $\chi$ be a primitive Dirichlet character mod $k$. Prove that if $N< M$ we have $$ \bigg|\sum_{m=N+1}^M\...
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1answer
21 views

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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1answer
49 views

Approximation of $\zeta$(s) at $s=1$

I am currently taking an Analytic Number Theory unit and we're working on the zeros of the zeta function. In the proof of $\zeta(1+\textit{i}t) \neq 0$ for $t \in \mathbb{R}$, we suppose that $\zeta$(...
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Asymptotic's for Fourier coefficients of $GL(3)$ Maass forms

Let $f$ be a $GL(3)$ Hecke-Maass cusp form and $A(m,n)$ denote its Fourier coefficients. (1) Are there any lower bounds known for $\sum_{p\leq x}|A(1,p)|^2$ or $\sum_{n\leq x}|A(1,n)|^2$ ? (we know ...
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1answer
93 views

Statement Equivalent to the Riemann Hypothesis

I am told that the Riemann Hypothesis is equivalent to the condition: $\psi(x) = x + O(x^{1+o(1)})$, and asked to prove this in the forward direction. (Here $\psi(x)$ is the Chebyshev Function). ...
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46 views

Estimating $\sum\limits_{n\leq x} d_3(n)$.

If $d_3(n)$ denotes the number of ways to write $n$ as a product of $3$ positive integers then how do I show that as $x\to \infty$, $\sum\limits_{n\leq x}d_3(n)=\frac{x(\log x)^2}{2}+O(x\log x)$. ...
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1answer
39 views

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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An Engineer sets out to Prove Fermat's Last Theorem …

This started off as a joke post of mine on a Facebook Group called "Bad Maths that Gives the Right Answer", in which I pulled a Fermat and claimed that the last bit of the proof was too long to post. ...
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55 views

Upper and lower bound of $\pi(n+x)-\pi(n-x)$

Let $\pi(x)$ be the prime counting function. What is the best known unconditional estimate of $\pi(n+x)-\pi(n-x)$? What is the best known conditional estimate? My brute force approach was to use ...
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1answer
29 views

A not very obvious question about $\{h+tk\}$ sequence.

Let $h$ and $k$ be positive integers such that $\gcd(h,k)=1$. Let $A(h,k)$ be the sequence $$A(h,k)=\{h+kx|x=0,1,2,3,\cdots\}.$$ Let $S$ be a infinite subset of $A(h,k)$, prove that for each positive ...
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1answer
58 views

The limit on the ratio of the Dirichlet eta functions

If we accept-- $s_o$ -- as one of the non-trivial zeros of the Riemann zeta function by $0 <Re(s_o)<1$ and $Re(s_o)$ is the real part of a complex variable, we know: $$\eta(s_o ) = \...
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31 views

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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32 views

Show $\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^s}}{\zeta(s)}=\prod_{p}(1-1/p^s+1/(p-1)^s)$.

I want to show that the following equality and that the product is absolutely convergent and uniformally convergent on compact subsets of ${s:Re(s)>1}$. $$\frac{\sum_{n=1}^{\infty}\frac{1}{\phi(n)^...
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1answer
25 views

Solutions for $n$? Use Stirling approximations if needed

$$(2n)! = a^{2n}$$ where $a \in \mathbb R$, and $n \in \mathbb N$. This is relevant because of a research question I'd asked and received an answer to by Sotiris here
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1answer
29 views

Number of integers coprime to a given integer $q$ in some range $[x, x+y]$

I am asked to show that for $1 \leq x,y$ and an integer $q$, we have: $S(x,x+y,q) = |\{x < n \leq x + y \mid n \text{ is comprime to } q\}| = \frac{\phi(q)y}{q} + O(2^{\omega(q)})$, where: $\...
6
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1answer
135 views

Density of integers $n$ with all prime factors of order $O(\log n)$?

For a rational integer $n \in \mathbb{Z}_{+}$, let $\mathfrak{p}(n)$ denote the set of (distinct) prime factors of $n$. Then for a positive constant $c$, let $$f(x) = \vert\{n\in\mathbb{Z}_{+}:\ n\...
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1answer
37 views

About absolute convergence of complex series

I know I need to incorporate uniqueness theorem of Dirichlet series to get some kind of contradiction, but don't know how to proceed?
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0answers
55 views

Convergence of $\sum\limits_p \frac{\chi(p)}{p}$ and the prime number theorem

Consider the sum $$\sum\limits_p (-1)^{\frac{p-1}{2}}\frac{1}{p}=-\frac{1}{3} + \frac{1}{5} - \frac{1}{7} - \frac{1}{11} + \frac{1}{13} + \frac{1}{17} - \cdots \tag{1}$$ of signed reciprocals of the ...
6
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0answers
77 views

Chebyshev function: variational formulation

The Wikipedia article on the Chebyshev function $\psi(x)$ states that, evaluated at $x=e^t$, it minimizes the functional $$J[f] = \int_0^\infty \dfrac{f(s)\zeta'(s+c)}{\zeta(s+c)(s+c)}ds - \int_0^\...
2
votes
1answer
84 views

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, ...
2
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0answers
28 views

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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0answers
60 views

Lower bound for number of primes up to $x$.

It is possible to prove the statement that $\pi(x)$, the number of primes up to $x$ is bigger than $\sqrt x$ for $x \ge 3$, in an elementary way? More generally can we prove that for every $0<\...
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0answers
10 views

Fundamental approximation in the circle method

I am interested in the circle method and I am currently working on Vaughan's book. Let $f$ be the generating function $f$ of the squares, that is to say the power series sum of $z^{m^2}$. One of the ...