# Tagged Questions

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### Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
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### Is there a von Koch-type theorem for the generalized Riemann hypothesis?

Helge von Koch proved in 1901 that the Riemann hypothesis is equivalent to the error term in the prime number theorem having the bound $$\mid\pi(x)-\textrm{li}(x)\mid=O(\sqrt{x} \log x).$$ Q1: ...
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Suppose that $S\subseteq\mathbb{Z}[i]$ has the following properties: For convenience, let $A_n = \{z : z\in\mathbb{Z}[i], \text{Nm}(z)\le n\}$ \limsup_{n\rightarrow\infty} \frac{|S\cap ... 3answers 172 views ### Expression for the derivative of Eisenstein series G_2 I am new to number theory, so I am guessing this is a standard formula. I would be grateful for a reference: We know that the Eisenstein series G_2 is quasimodular of level SL_2(\mathbb Z), so ... 1answer 313 views ### Squarefree numbers n such that 432n+1 is also squarefree This is a second attempt (see Primes p such that 432 p +1 is prime) Is the set of squarefree numbers n such that n(432 n+1) is also squarefree known to be infinite? Fact: the number of such ... 1answer 299 views ### Primes p such that 432 p +1 is prime [closed] Is the set of prime numbers p such that 432 p + 1 is also prime infinite? It doesn't follow from Dirichlet's theorem as far as I can tell. 2answers 588 views ### The shortest interval for which the prime number theorem holds [closed] It is well known that the prime number theorem on the form \begin{align*} \pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)} \end{align*} breaks down for short enough intervals, e.g. taking y=(\log ... 1answer 651 views ### Probability that a positive integer is the euler phi function of another positive integer Define f(n) = |\{m : m\le n, \exists k \text{ s.t. }\phi(k) = m\}|. Clearly, f(n)\le \left\lfloor \frac{n}{2}\right\rfloor + 1 since \phi(n) is even for all n > 2. Is ... 2answers 113 views ### Asymptotics and error terms for an arithmetic function built upon \omega and \Omega functions For any real number x, let's define Om_{k}(x) as the number of positive integers m below x such that \Omega(m)-\omega(m)=k, where \omega(n) is the number of distinct primes dividing n, ... 2answers 413 views ### Incomplete Kloosterman sum I am interested in an upper bound on the following incomplete Kloosterman sum \sum_{\substack{x=1 \\ x+_{_{\bf Z}}x^{-1}>p}}^{p-1}e\left(\frac{x+x^{-1}}{p}\right).$$Using the Weil's bound it ... 0answers 151 views ### Kloosterman-like sum with inverse to different moduli In some recent work, the following strange-looking exponential sum arose:$$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$Here e(x) = e^{2\pi i x} as ... 0answers 144 views ### Arguments for the second Hardyâ€“Littlewood conjecture being false? Assume that x,y > 2, and that x<y. Then the second Hardyâ€“Littlewood conjecture states that$$\pi(x + y) - \pi(y) \leq \pi(x).$$We can easily justify this heuristically, since$$ ...
Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...