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The Elliott-Halberstam conjecture is an immediate consequence of Montgomery's (modified) conjecture, which proposes that if $\epsilon>0$, $\gcd(a,q)=1$ and $x>q^{1+\epsilon}$, then $\Big|\displaystyl …

Notice that $$\Big|\sum_{n\leq x}a_n e(n\beta)-\sum_{n=1}^{\infty}a_n e(n\beta)w_X(n)\Big|\leq \sum_{X<n\leq X+Y}|a_n|.$$ So if you have strong control of $|a_n|$ in short intervals, then you can pass …

If $|a_n|\ll 1$ and $c>1$, then $\displaystyle\sum_{n\leq x}(x-n)a_n = \frac{1}{2\pi i}\int_{c-iT}^{c+iT}\mathcal{F}(s)\frac{x^{s+1}}{s(s+1)}ds+O\Big(\frac{x^{c+1}(\log x)^2}{T^2}\Big)$. A detailed pr …

I wouldn't say that there is a single all-prevailing reason, but here are some easily discernible sources: Given a Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$$ in its abscissa of absolute …

Sieve theory is not saturated. It is alive and thriving. The ICM just awarded the Fields medal to James Maynard in no small part due to his work in sieve theory (see here and here). Because of the …

Matti Jutila (On Linnik's Constant, Math. Scand. 41 (1977), 45-62) proved that if $Q\geq 2$, $T\geq 1$, and $4/5\leq\alpha\leq 1$, and $\epsilon>0$, then $\sum_{q\leq Q}\sideset{}{'}\sum_{\chi\bmod{q …

There is no uniform, arithmetically "simple" condition (like lying in a residue class) that classifies the Artin symbol in nonabelian extensions. Roughly speaking, if you write $K=\mathbb{Q}(\alpha)$ …

I think that Bill Duke proves a "near miss" to what you seek in his Ph.D. thesis (http://matwbn.icm.edu.pl/ksiazki/aa/aa52/aa5231.pdf). The idea proceeds as follows (roughly speaking). Let $K$ be a …

Let $K/\mathbb{Q}$ be a number field, and let $\pi$ be a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_K)$, where $\mathbb{A}_K$ is the ring of adeles over $K$. Consider the standa …

In general, this product is hard to work with, and good upper/lower bounds require a bit of work. I'll also assume the simplest case of $k=12$, in which case $\lambda_f(n)$ is the famous Ramanujan ta …

The answer is addressed in Siegel-Walfisz Theorem with smooth weights. A very particular smooth weight is used, but the ideas can be adapted to other smooth weights with minor changes. Also, the ans …

Let us consider the related problem of finding a suitable $\delta>0$ such that $\displaystyle\sum_{\substack{q\leq x^{\delta-\epsilon} \\ K\cap \mathbb{Q}(e^{2\pi i/q}) = \mathbb{Q}}}\max_{(a,q)=1}\B …

It seems, essentially, that you want to estimate the $L$-function $L(s,\Delta)=\sum_{n=1}^{\infty}\tau(n) n^{-s}=\prod_p (1-\tau(p)p^{-s}+p^{11-2s})^{-1}$ as a short Euler product, in the sense that …

What follows is a refinement of Peter's answer that might be useful to you, depending on what exactly you want to pursue. Here is what is known for $\zeta(s)$ on RH: If $t$ is large and $0<h\leq \sq …

There is enough in the literature to extract a result of this form, but it might not appear explicitly. I will reference recent work of Thorner and Zaman instead of Lagarias-Odlyzko, since it gives a …