Proc. Legacy of Ramanujan, RMS-Lecture Notes Series No. 20, 2013, pp. 111127.
The Circle and Divisor Problems, and Ramanujans
Contributions through Bessel Function Series
Bruce C. Berndt1 , Sun Kim2 and Alexandru Zaharescu3
1 Department of Mathematics, Un
Math 531 Fall 2016
Homework # 4
Problem 1
Express each function in terms of the Riemann zeta function. Say in which half-plane the
identity is valid.
(i)
X
(n)
n=1
ns
(ii)
X
2 (n)
n=1
(iii)
ns
X
(n2 )
n=1
ns
Problem 2
P
Let r be a positive integer. Evalu
Math 531 Fall 2016
Homework # 1
Problem 1
Let f (n) = (n)/n, and let cfw_nk
k=1 be the sequence of values n at which f attains a record
low; i.e., n1 = 1 and, for k > 2, nk is defined as the smallest integer > nk1 with f (nk ) < f (n) for
all n < nk . (F
Math 531 Fall 2016
Homework # 5
Problem 1
(a) Prove that (s) = 1 + O(2 ) for = <s > 2.
X
(b) Find the value of
(m) 1). The proof must include justification of the convergence,
m=2
not just formal manipulations.
Problem 2
(i) Show that (s) < 0 for real s,
Math 531 Fall 2016
Homework # 4
Solutions
Problem 1
Express each function in terms of the Riemann zeta function. Say in which half-plane the
identity is valid.
(i)
X
(n)
n=1
ns
(ii)
X
2 (n)
n=1
(iii)
ns
X
(n2 )
n=1
ns
Solution
P
P
s
s
(i) Since = 1 id,
Math 531 Fall 2016
Homework # 7
Solutions
Problem 1
Given positive integers l and k with (l, k) = 1, express the Dirichlet series
X
(n)ns
nl (mod k)
in terms of Dirichlet L-functions.
Solution
Use the orthogonality relation to eliminate the summation cond
Math 531 Fall 2016
Homework # 2
Solutions
Problem 1
Z x This exercise outlines one way to obtain an estimate for the logarithmic integral Li(x) =
dt
, with error term O(x(log x)k ), for any given integer k > 2.
log
t
2
Rx
(a) Set Ik (x) = 1 et tk dt and o
Math 531 Fall 2016
Homework # 2
Problem 1
Z x This exercise outlines one way to obtain an estimate for the logarithmic integral Li(x) =
dt
, with error term O(x(log x)k ), for any given integer k > 2.
2 log t
Rx
(a) Set Ik (x) = 1 et tk dt and observe tha
Math 531 Fall 2016
Homework # 6
Problem 1
(20 points)
(a) For c > 0 define
1
I(y) =
2i
Z
(c)
ys
ds.
s2
Recall that (c) means the infinite vertical line with real part c, traversed upward. Show that
(
log y y > 1
I(y) =
0
0 < y 6 1.
Note: since the integra
Math 531 Fall 2016
Homework # 3
For these problems, you may use the Mertens estimates, but not the Prime Number Theorem (the
exception being Problem 4 (i).
Problem 1
Show that lim sup
n
(n)
= e , where (n) is the sum of the positive divisors of n.
n log l
Math 531 Fall 2016
Homework # 7
Problem 1
Given positive integers l and k with (l, k) = 1, express the Dirichlet series
X
(n)ns
nl (mod k)
in terms of Dirichlet L-functions.
Problem 2
Let be a non-principal character modulo k, and let t 6= 0.
(a) Prove th
Math 531 Fall 2016
Homework # 5
Solutions
Problem 1
(a) Prove that (s) = 1 + O(2 ) for = s > 2.
(b) Find the value of
X
(m) 1). The proof must include justification of the convergence,
m=2
not just formal manipulations.
Solution
(a) Since f (t) = t is dec
Math 531 Fall 2016
Homework # 1
Solutions
Problem 1
Let f (n) = (n)/n, and let cfw_nk
k=1 be the sequence of values n at which f attains a record
low; i.e., n1 = 1 and, for k > 2, nk is defined as the smallest integer > nk1 with f (nk ) < f (n) for
all n
Math 531 Fall 2016
Homework # 3
Solutions
For these problems, you may use the Mertens estimates, but not the Prime Number Theorem (the
exception being Problem 4 (i).
Problem 1
Show that lim sup
n
(n)
= e , where (n) is the sum of the positive divisors of
INFINITELY MANY ZEROS OF (s) LIE ON =
1
2
BRUCE C. BERNDT
1. T HE G ROWTH OF (s)
Theorem 1.1. Let > 0 be given. Then, for 0 1, as |t| ,
1
(s) = O(|t| 2 (1)+ ).
Proof. Recall that we previously have shown that, as |t| ,
1
< 0,
O(|t| 2 ),
3
O(|t| 2 ),
MATH 531
MID-SEMESTER EXAM
October 21, 2009
Name
Mighty are numbers, joined with art resistless.
Euripedes
SHOW ALL WORK. INDICATE ALL REASONING.
1.
2.
3.
4.
Total
1
2
Math 531, Mid-semester exam
1. Liouvilles function (n) is dened by
(n) =
1,
if n = 1,
a
Problem 1
20 points Let A be Liouvilles function: if n = p31 mp, then A n = 1)el+"+e*.
1 k
_. (i) Prove that Mn) = Z p(-:2), where p, is the Mbius function.
d2|n
(ii) Express the Dirichlet series F(3) = 2 LEE-)- in terms of the Riemann zeta function.
n=1