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It was shown by Heath-Brown that a suitable form of the pair correlation conjecture of Montgomery, in conjunction with RH, could improve Cramer's bound $p_{n+1}-p_n \ll p_n^{1/2} \log p_n$ slightly to …

If you are willing to violate multiplicativity $f(nm)=f(n)f(m)$ about $\varepsilon$ of the time, then one can make $f$ more or less arbitrary on an interval of the form $[(1-\varepsilon) x, x]$, and t …

No, such a result is not known unconditionally, even if one restricts to the low-lying case $t=O(1)$ and excludes real zeroes, and it would be a great breakthrough if one could do this. If one had su …

Mobius randomness heuristics suggest that $\sum_n \mu(n) (r e^{i\theta})^n$ does not converge to a limit as $r \to 1^-$ for any (or at least almost any) $\theta$. If it did converge for some $\theta$ …

$$ \sum_{1 \leq i,j \leq X} \frac{1}{\mathrm{lcm}(i,j)} = \sum_{1 \leq i,j \leq X} \frac{\mathrm{gcd}(i,j)}{ij} $$ $$ = \sum_{1 \leq i,j \leq X} \frac{\sum_{d|i,j} \phi(d)}{ij}$$ $$ = \sum_{d \leq X} …

The recent paper Matomäki, Kaisa; Shao, Xuancheng, When the sieve works. II, ZBL07207214. gives a fairly satisfactory answer to this question in the setting of arbitrary $S$. General sieve theory gi …

One can establish the claim using a logarithmic version of the moment method. If one factors $N = \prod_{p|N} p^{\operatorname{ord}_p(N)}$, then $\log \operatorname{gcd}(t,N) = \sum_{p|N} \log \opera …

If the even number $2n$ in question has to be chosen in a manner that does not already construct an explicit prime representation $2n = p_1 + p_2$ as a byproduct of the construction of $n$, then I thi …

Perhaps surprisingly, I don't use Poisson summation per se all that often, but I do repeatedly use the more general principle that Poisson summation exemplifies, namely that the Fourier transform inte …

This identity is true, though somewhat tricky to prove and the infinite series here might only converge conditionally rather than absolutely. The key lemma is Lemma 1 (Fourier representation of avera …

Firstly, a general comment: as understanding of a mathematical problem deepens, it is common (and even expected) for the most mathematically natural formulation of a given problem (or class of problem …

Generally speaking, Euler product type formulae involving logarithms can be derived (formally, at least) from Euler product formulae without logarithms via differentiation in the $s$ parameter. In th …

As observed by Hensley and Richards in Douglas Hensley and Ian Richards, Primes in intervals, Acta Arith. 25 (1973-74), 375--391, if the prime tuples conjecture is true, then $\limsup_{n \to \infty …

Formally (ignoring issues of integrability), one has (writing $e(\theta) := e^{2\pi i\theta}$) $$ \sigma'_\infty = \int_{{\bf R}^2} \int_{[0,1]^n} e( F_1({\bf x}) \alpha_1 + g({\bf x}) F_2({\bf x}) \ …

Ultimately, it is because of the fundamental theorem of arithmetic, which expresses each natural number $n$ as a product of the primes dividing it: $$ n = p_1^{a_1} \dots p_k^{a_k}.$$ Taking logarithm …