NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/18570

Internet Message Format | 1993-01-21 | 2.4 KB

Path: sparky!uunet!olivea!mintaka.lcs.mit.edu!zurich.ai.mit.edu!ara
From: [e-mail] (Allan Adler)
Newsgroups: sci.math
Subject: Re: Distribution of primes mod 4
Message-ID: <[e-mail]>
Date: 21 Jan 93 13:12:39 GMT
References: <[e-mail]> <[e-mail]>
Sender: [e-mail]
Organization: M.I.T. Artificial Intelligence Lab.
Lines: 35
In-Reply-To: [e-mail]'s message of 21 Jan 93 10:09:48 GMT


How would one formulate and prove analogous theorems for primes in
arithmetic progressions other than 4n+/-1, 6n+/-1? One could take
any two congruence classes of unit modulo b, say, and compare which
the number of primes in one congruence class minus the number of
primes in the other congruence class, but one would have to do that
for all pairs of units, which seems like it might be inefficient
in some sense. If one works with a prime modulus b, say, one could
get around this by asking about the number of primes which are
quadratic residues modulo b minus the of primes which are not
quadratic residues modulo b. But one can work more generally, I think.
For example, let X be a multiplicative character modulo b, where b is
any modulus, and for each unit u mod b let a(u,n) denote the number
of primes not exceeding n and congruent to u mod b. Then one can
consider the sum S(X,n) = sum over units u mod b of a(u,n).X(u).
Now how does one generalize this business of the sign changing
infinitely often?  Since this expression will in general be complex,
it seems like there are two approaches. First, one can let
r denote the order of X and divide the plane into r sectors by rays
through the r 2r-th roots of unity which are not r-th roots of unity,
and try to show that S(X,n) lies in each sector infinitely often.
Alternatively, one can consider ANY region of the plane and ask
whether that region is hit infinitely often, what is the density of
the set of n for which  it is hit, etc. One could also consider an
analogous question for number fields or even a nonabelian version
by considering the values of characters of irreducible representations
of Galois groups of number fields (or algebraic function fields in one
variable) on automorphisms arising as Artin symbols (or characteristic
polynomials instead of traces).

In view of all this potential generality, is 4n+/-1, 6n+/-1 really
the state of the art?

Allan Adler
[e-mail]