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 11h comment Why are very large prime numbers important in cryptography? If you have a question the site functions best if you post it as a question rather than an answer. Feb 7 comment Using the Brun Sieve to show very weak approximation to twin prime conjecture Halberstam and Richert in Sieve Methods (Dover, 2011) prove using Brun's sieve that there are infinitely many p such that p+2 has at most 8 prime factors. Including some introductory material the exposition takes 67 pages. The key is their definition of the characteristic function on p. 58. The basic idea is simple enough but doesn't look like it lends itself to anything one could describe as a "straightforward exercise." Jan 18 comment Proof of inequality involving multiplicative function? Terms in the binomial expressions on the left with m factors are all covered by terms in mth powers of the expression on the right. Once we see the LHS can be written as a product of binomials we can compare the two sides. Your hint prompted me to look at the LHS again. The key (which I forgot or didn't know) is that $n=p_k\#,~\binom{k}{m}$ is the number of squarefree divisors of n having $\nu(d)=m.$ Jan 14 comment Proof of inequality involving multiplicative function? @user1952009: edited to reflect that. Jan 10 comment English wording for “first level of asymptotic expansion” You can say $f$ is asymptotically equivalent to $g.$ Jan 9 comment “The PNT obtained by statistical methods” @ErickWong: In that case I will vote to reopen. Jan 8 comment “The PNT obtained by statistical methods” Maybe asking about Erdos-Kac theorem? Jan 1 comment Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$ This result was not relegated to exercises in Apostol and OP say s/he is new to number theory. So I wonder if this is enough. Dec 31 comment Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$ This is proved at pp. 66-67 of Apostol (Intro. to Analytic Number Thy.) Different approach. Dec 24 comment Equidistribution theorem of Weyl Perhaps OP is asking if equidistribution of a sequence $a\cdot n$ can be used to show that $a$ is irrational? I don't think Weyl works in that direction but at least it's a question. Vote to reopen. Dec 12 comment What is the effective lower bound on gaps between zeta zeros? jstor.org/stable/pdf/2372402.pdf?seq=1#page_scan_tab_contents. See paragraph below formula (2). There are a lot of questions here on lim inf with good answers BTW. Nov 25 comment Zeros of the prime zeta function @mixedmath: Minimally, is it possible to show $P(s) =\zeta(s)$ implies $s$ is a zero of $C(s)$ without direct reference to $C(s).$ The argument begins, "Suppose the two integrals are equal for some value of s," and concludes, for example, "s is thus a zero of this series on the right, which with some work is seen to be $C(s).$" Nov 15 comment What is the proportion of primes that can be written as $a^2 + b^2$? See Ingham, The Distribution of Prime Numbers, pp. 106-107. Nov 8 comment What is your idea about this conjecture? @Dylan: Using Eric N's reformulation my claim is $j(2\cdot 3\cdot...\cdot13)\leq 30.$ In the table in the paper $n$ is the index of the largest prime. If $n=6$ then $h(n)$ is 22. According to this paper my claim is true, but the paper gives a stronger result. Nov 7 comment What is your idea about this conjecture? @SimonS: Good question either way, but the numerical work in this case might be misleading. Suppose it is true for some but not all n? Oct 28 comment Proof of Prime Number Theorem The Prime Number Theorem by Jameson is also good. Oct 19 comment Which progressions and sequences are guaranteed to contain infinitely many primes? You can construct infinitely many such sequences. OEIS contains some of the interesting ones. Sep 23 comment What are all of the possible fractional forms an offspring's genetic makeup? Maybe you should post this at the Bio SE site. Sep 10 comment Approximate zeros of a (hypothetical) analog of $\zeta(s)$ @draks: I looked at that question and the very nice answers there (and long ago upvoted). The r.h.s. of (1) in Ray M's answer is quite different from (1) above. Sep 5 comment A (possibly) easier version of Bertrand's Postulate Bertrand implies a prime on p, 2p. Choose p(n) max less than a non-prime n. Then there is a prime on n, 2p(n) which implies a prime on n,2n (Bertrand). So I think the two are equivalent. You don't need case 1, for the reason you give.