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

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 […]

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 […]

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. […]

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 […]

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 […]

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 […]

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 $s$. […]

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 […]

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