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Counting primes by circle method $$\int_{0}^{1}\sum_{n_0=1}^{N}e^{2\pi ian_0}\sum_{n=1}^{N}\Lambda (n)e^{-2\pi ian}da$$ I tried to find the main term by looking at major arcs, but the singular series …

Is this special case known? For $\lambda(q)$ -- well-factorable function and $q|P(z)$, $\pi(x;q,a)$ $a=1$. $\displaystyle \sum_{q\leq x^{1-\epsilon}} \lambda(q) ( \pi (x;q,1)-\frac{\pi(x)}{\varphi (q) …

Is it possible to get a good upper bound for the integral $$\int_{0}^{T}\zeta ^{3}(\frac{1}{3}+it)dt$$ (unconditionally)?

Could you please provide a link to the source? $$\sum_{\chi\neq \chi_0}\int_{0}^{T}|L(1/2+it,\chi)|^4dt\ll (qT)^{1+\varepsilon},$$ where $\chi_0$ is the principal character modulo $q$, and $L(s,\chi)$ …

Is it possible to get a good upper bound for $$\sum_{1\leq |h|\leq q}\frac{c_{q}(a-h)}{h}$$ with $(a,q)=1$ and $1\leq a\leq q$.

Look at the first page of this paper --> Daniel A. Goldston, Julian Ziegler Hunts, Timothy Ngotiaoco, The Tail of the Singular Series for the Prime Pair and Goldbach Problems, Funct. Approx. Comm …

Is it possible to get an upper bound better than $\ll_\sigma T^{3/2-\sigma}$ for $$\int_{0}^{T}|\zeta (\sigma +it)|\,dt,\qquad 0<\sigma<1/2\,?$$