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If you state the Density Hypothesis as $$ N(\sigma,T) \ll T^{ 2(1-\sigma)+\epsilon}$$ the factor $1-\sigma$ seems natural because the exponent $2(1-\sigma)$ is 1 when $\sigma=1/2$, where there are $ \ …

Here is an answer that is similar in spirit to Frank and Peter's answers, but possibly simpler. Summing by parts, we see that $$ \sum_{p\le x} \frac{\log^k p}{p} = (\log x)^{k-1} \sum_{p\leq x} \fra …

In many applications, you can use the fact that $$ e^{-n/x}=1 + O\Big(\frac{n}{x}\Big)$$ uniformly for $n\le x$ and therefore $$ \sum_{n\le x} a_n e^{-n/x} = \sum_{n\le x} a_n+ O\left( \frac{1}{x} \s …

I heard it said by a number of mathematicians that the thing that sets the Riemann Hypothesis apart from (almost all) other famous unsolved problems is that no one has ever suggested a reasonable firs …

Let $P_n$ be independent variables which are 1 with probability $1/\log n$ and $0$ with probability $1-1/\log n$ and let $$ \Pi(x) = \sum_{n\leq x} P_n.$$ Then Cram\'{e}r showed that, almost surely, …

It may be useful to read Section 10 of Chapter V of Ingham's "The Distribution of Prime Numbers." Let $\Pi(x)=\pi(x)+\frac{1}{2}\pi(x^{1/2})+...$, then Moebius proved that $$ \pi(x) = \sum_{n=1}^\in …

In this paper: http://arxiv.org/abs/0902.4352, the authors discuss the analytic properties of the related Dirichlet series $$ D_{2r}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)\Lambda(n+2r)}{n^s}$$ wher …

Expanding on Stopple's comment above, I believe the following argument based on Landau's explicit formula answers the question. Here is a generalization of Landau's explicit formula for the zeros of …

Let $q_n$ denote the $n^{\text{th}}$ number that is a product of exactly two distinct primes. It is known that $$\liminf_{n\to \infty} \ (q_{n+1}-q_n) \le 6.$$ This is a result of Goldston, Graham, P …

The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T \ }$ zeros in the strip with $0 < t < T$. See section 5.3 of Iwaniec and Kowalski's "Analytic Number Theory," in particular Theorem 5.8 …

Assuming the Riemann Hypothesis, I believe the best known result is due to Cramer (I cannot figure out how to add the accent of the e) and it says the following: There is a constant $C > 0$ such that …

There are infinitely many roots of $\zeta(s)-a=0$ for every complex number $a$. When $a\ne 0$, these are called "$a$-values" and there is a whole chapter discussing their distribution in Titchmarsh's …

To answer your modified question, according to Mathematica: $$ \lim_{s\to 0} \left(\frac{\hat{\zeta}'}{\hat{\zeta}}(s)+\frac{1}{s}\right) = -\frac{\gamma}{2} + \tfrac{1}{2}\log(4\pi).$$ This implies …

Edit: I endorse Juan's answer to the original question. The sum $\displaystyle{\sum_{\rho} \tfrac{1}{|\rho|}}$, running over the non-trivial zeros $\rho$ of $\zeta(s)$, is known to diverge, so at best …

The reference for this (given in the 3rd edition of Davenport's Multiplicative Number Theory) is: Grosswald, Émile. "Sur l'ordre de grandeur des différences $\psi(x)-x$ et $\pi(x)-\ell i(x)$." (Frenc …