Articles of analytic number theory

Discrete fourier transform on $\mathbb{Z}/N\mathbb{Z}$ vanishing on an interval of size at least $\sqrt{N}$

Let $f : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ and let $\hat{f} : \mathbb{Z}/N\mathbb{Z} \to \mathbb{C}$ be its DFT given by $\hat{f}(m) = \sum_{j \in \mathbb{Z}/N\mathbb{Z}} f(j)e_N(jm)$, where $e_N(x) := e^{2\pi i x/N}$. If $\hat{f}(m) = 0$ for every $m$ in some interval $[M+1, M+K]$ with $K \geq \sqrt{N}$, is there anything interesting we can say about $f$ […]

How to prove that the partial Euler product of primes less than or equal x is bounded from below by log(x)?

How does one prove $\prod_{p \leq x}(1 – \frac{1}{p})^{-1} \geq \log(x)$?

Identity between absolutely convergent generalized Dirichlet series

The following identity between generalized Dirichlet series, both absolutely convergent in the whole complex plane, occurs when investigating functions of the Selberg class of degree $d=0$: \begin{equation} \sum_{n=1}^\infty a(n)\left(\frac{Q^2}{n}\right)^s=\omega Q\sum_{n=1}^\infty\overline{a(n)}n^{s-1}. \end{equation} To prove that \begin{equation} F(s)=\sum_{n=1}^\infty \frac{a(n)}{n^s} \end{equation} is actually a Dirichlet polynomial, using an uniqueness theorem for generalized Dirichlet series one derives from the […]

Some asymptotics for zeta function.

I need to use the functional equation for $\zeta(s)$ and Stirling’s formula, to show that for $s=\sigma +it$ , with $\sigma <0$: $$ |\zeta(s)| << \left(\frac{t}{2\pi}\right)^{1/2-\sigma}$$ as $t\rightarrow \infty$ (where $\sigma$ is fixed), i.e, $\,\displaystyle{\frac{|\zeta(s)|}{\left(\frac{t}{2\pi}\right)^{1/2-\sigma}}}$ is bounded by some constant that depends on $\sigma$ as $t\rightarrow \infty$. Any reference or the solution itself? I tried […]

Elementary Proof of Landau's count on number representable as sum of two squares

In Analytic Number Theory by Iwaniec and Kowalski, there is an elementary proof of Landau’s result of $\#\{n \le x: \exists a, b,\ s.t.\ n = a^2 + b^2\} \sim Cx/\sqrt{\log x}$ with an explicit constant $C$. However, it is left as exercise 4 in Chapter 1, and the hint is to use Thm 1.1. […]

connection between odd primes and a certain q-series

I posed a conjecture about odd primes and a certain q-serieshere.I thought it would be more appropriate ,if I could ask the converse of the aforementioned problem . Is $p$ an odd prime iff $$\prod_{n=0}^{\infty}\frac{1}{(1-q^{4n+1})^p}=1+\sum_{n=1}^{\infty}\phi(n)\,q^n$$ such that $$\phi(n)\equiv 0\pmod{p}$$ is true for all natural numbers $n\in\mathbb{N}$ except at multiples of $p$. I’ve experimentally verified the […]

Dirichlet hyperbola methods : estimate functions # of ordered pairs

Let $f$ be an arithmetic function defined by $$f(n) = |A_n|$$ where $A_n = \{(a, b) : n = ab^2\}$.Estimate $$\sum_{n \leq x} f(n)$$ where $x \in \mathbb{R}^+$, using Dirichlet hyperbola method. The error term should be $O(x^{1/3})$. Dirichlet hyperbola method requires the function $f$ to be written as Dirichlet convolution of two functions. The […]

Product of the logarithms of primes

I would like to know if there is a result for the product $$f(x)=\prod_{p\leq x}\log p,\quad \text{where $p$ is prime}.$$ A simple upper bound is $f(x)<(\log x)^{\pi(x)}$, where $\pi(\cdot)$ is the prime counting function. But I’m hoping to find a closer approximate to the function. The Chebyshev’s theta function $\displaystyle\theta(x)=\sum_{p\leq x}\log p$ is the closest […]

Amount of real numbers, in a given sequence, whose fractional part lies in a given closed interval

Fix an interval $[a,b] \subset [0,1]$ and let $S$ be a given sequence of real numbers. Are there any ad-hoc methods that may be used to estimate $$ T := \#\{ s \in S \ : \ \{s\} \in [a,b]\}? $$ Here $\{s\} := s – \lfloor s \rfloor$ denotes the fractional part of $s$. […]

Derive zeta values of even integers from the Euler-Maclaurin formula.

Euler showed: \begin{equation} B_{2 k} = (-1)^{k+1} \frac{2 \, (2 \, k)!}{ (2 \, \pi)^{2 k}} \zeta(2 k) \end{equation} for $k=1,2, \cdots$. We could from here find $\zeta(2k)$ in terms of the even Bernoulli coefficients $B_{2k}$. How can we derive the equivalent representation by using the Euler Maclaurin formula ? Thanks. Update Here is what […]