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Whether for a finite set $\mathcal{R}$ of roots the approximation $$ \pi(x)\approx\mathrm{Li}(x)-\frac{1}{2}\mathrm{Li}(x^{1/2})-\sum_{\rho\in\mathcal{R}}\mathrm{Li}(x^\rho) $$ is "on average" better …

The non-vanishing of $L$-series on the real line received a lot of attention, unfortunately, there is still a lot we do not know, even in the non-quadratic case. This circle of problems even has its o …

From the point of view of analytic number theory the most important specific result which is proved using algebraic geometry is Burgess' bounds for character sums. The proof relies on Wiles bound for …

The maximal domain of meromorphic continuation of a Dirichlet series can be anything. More precisely, for every connected open subset $O$ of $\mathbb{C}$, which contains the half plane $\{\Re s>1\}$, …

In the 1960's Turán wrote several papers on a function-theoretic sieve. He managed to express the number of prime twins in terms of roots of $L$-series. He began like you did by expressing $\Lambda$ a …

In "The equation $\omega(n)=\omega(n+1)$" (Mathematika 50, 99-101, 2003) it was shown that the equation $\omega(n)=\omega(n+1)$ has infinitely many solutions. Pintz has a series of results on consecut …

Wolke (On the large sieve with primes, Acta Math. Acad. Sci. Hungar. 22 (1971/72), 239–247, MathSciNet MR0291121 (45 #215)) has worked on this question. His motivation was Gallagher's approach to the …

The right context of your question is probably the area of Beurling primes. An arithmetic semigroup is a semigroup $(G,\cdot)$ together with a norm $\|\cdot\|:G\rightarrow[1,\infty)$, such that $\|gh\ …

Goldston, Graham Pintz and Yildirim ( https://arxiv.org/abs/0803.2636 ) showed that among three linear forms $\ell_1, \ell_2, \ell_3$, which satisfy the obvious local conditions, there are two, say $\ …

Studying the distribution of patterns of the Moebius function falls into an easy part, which deals with the distribution of zeroes, and a difficult part, which deals with the distribution of signs. Th …

Checking whether an integer $n$ has a divisor in a given interval is essentially equivalent to factorization. Factoring is essentially equivalent to finding the smallest prime factor. So suppose that …

The assertion is generally believed to be true. In fact, much more is conjectured: For every $\epsilon>0$ we have $\left|\zeta(\frac{1}{2}+it)-\sum_{n\leq t^\epsilon} n^{-\frac{1}{2}-it}\right|= \math …

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by $$ \mathfrak{S}(d_1, \ldots, d_k) (Ck) …

The analytic version of Vaughan's identity is $$ \frac{\zeta'}{\zeta} = F+\zeta'G-FG\zeta + \left(\frac{\zeta'}{\zeta}-F\right)(1-\zeta G). $$ Here the last factor to the right is the most complicated …

Write $f(x)=\frac{e^{-0.3\sqrt{\log t}}}{\log^2 t}$. On the interval $[2, xf(x)]$ bound the integral trvially by $xf(x)$. On $[xf(x), x]$ the integrand is close to constant. More precisely, we have $$ …