Math 105

Analytic Number Theory

Instructor: Carl Pomerance (

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Textbooks (Davenport, 3rd edition) are in!

Homework due February 8:

(1) Do exercise 1 in section 2.3 of Montgomery--Vaughan.

(2) Do exercise 2 in section 2.3 of Montgomery--Vaughan.

(3) Using that the sum of (omega(n)-loglog x)^2 for n in [1,x] is O(x loglog x),
prove that for each epsilon >0, the asymptotic density of the set of integers n with
|omega(n)-loglog n| < epsilon loglog n
is 1.

(4) Let chi be a primitive character mod q, and let q1 be a divisor of q with 1 < q1 < q.
Show that chi is not constant on those integers c with (c,q)=1 and c = 1 mod q1.


This course will discuss some of the key ideas and ways of thinking in analytic number theory. The role of cominatorial methods will be discussed, as well as some applications.


Choate House, Emmy's
Lectures: Tuesday & Thursday 10:00am--11:50am (10A hour)
X-hour: Wednesday 3:00 pm -- 4:00 pm


Carl Pomerance -- 102 Choate House / Tel. 6-2635
Office hours: Tuesday, Wednesday, Thursday 9:00am--10:00am and by arrangement at other times.


Harold Davenport
Multiplicative number theory, Third edition


Grading will be based entirely on written assignments and occasional in-class presentations.

Past assignments

Homework due February 1:

(1) Show that the number of integers n in [1,x] divisible by a square > log n is O(x/(log x)^{1/2}).

(2) Let S be a set of primes with an infinite reciprocal sum. Show that the set of integers not divisible by any member of S has asymptotic density 0.
(That is, show that the number of n in [1,x], with n not divisible by any member of S, is o(x).)

(3) Show that the set of integers which are expressible as the sum of 2 squares has asymptotic density 0.
(You might first do this for numbers that are the sum of 2 coprime squares.)

(4) Given an arbitrary positive integer k, show that the set of integers n with phi(n) divisible by k has asymptotic density 1.

Homework due Tuesday, January 24:

(1) Prove the assertion in Davenport that the Euler-Mascheroni constant is equal to the negative of the integral of (log u)/e^u for u running from 0 to infinity.

(2) Justify the interchange of integration and summation on the center of page 61 of Davenport.

(3) Numbers 12 and 11 from Chapter 2 of the Montgomery-Vaughan book. You might try the first part of 12 by using partial summation and the result from class about phi(n)/n.

Homework due Tuesday, January 18:

(1) Using only first principles (that is, without citing Stirling's formula), prove that
log(n!) = n log n - n +O(log n).

(2) Assuming that the sum of (log p)/p for primes p up to x is log x +O(1), prove (using partial summation) that
the sum of 1/p for primes p up to x is loglog x + A + O(1/log x) for an appropriate constant A.

(3) Show that psi(x) (the sum of the von Mangoldt values for n in [1,x]) is the logarithm of the least common multiple of the integers in [1,x].

(4) With theta(x) the sum of log p for p in [1,x], show the following are equivalent:
(a) pi(x) = (1+o(1))x/log x
(b) theta(x) = (1+o(1))x
(c) psi(x) = (1+o(1))x

(5) Show that if pi(x) = (c+o(1))x/log x for some positive number c, then c = 1.

Homework due Tuesday, January 11:
(1) Give a simple proof that there are infinitely many primes that are congruent to 1 modulo 3.
(2) Show the sum of 1/(mp^m), over all primes p and all integers m at least 2, is convergent.

Honor Code

Students are encouraged to work together to do homework problems. What is important is a student's eventual understanding of homework problems, and not how that is achieved. The honor principle applies to homework in the following way. What a student turns in as a homework solution is to be his or her own understanding of how to do the problem. Students must state what sources they have consulted, with whom they have collaborated, and from whom they have received help. The solutions you submit must be written by you alone. Any copying (electronic or otherwise) of another person's solutions, in whole or in part, is a violation of the Honor Code.

If you have any questions as to whether some action would be acceptable under the Academic Honor Code, please speak to me.


I encourage any students with disabilities, including "invisible" disabilities such as chronic diseases and learning disabilities, to discuss appropriate accommodations with me, which might help you with this class, either after class or during office hours. Dartmouth College has an active program to help students with disabilities, and I am happy to do whatever I can to help out, as appropriate.

The Student Disabilities Coordinator, Nancy Pompian, can be reached at 6-2014 if you have any questions. Any student with a documented disability requiring academic adjustments or accommodations is requested to speak with me by the end of the second week of the term. All discussions will remain confidential, although the Academic Skills Center may be consulted to verify the documentation of the disability and advise on an appropriate response to the need. It is important, however, that you talk to me soon, so that I can make whatever arrangements might be needed in a timely fashion.