NOTES ON PRIME NUMBER THEOREM - II
K. Ramachandra, A. Sankaranarayanan and K. Srinivas
1. Introduction
In a series of papers the Soviet mathematician I. M. Vinogradov developed a very
important method of dealing with estimation of trigonometric sums. (See Chapter
VI of [ECT] and [ECT, DRHB]). An epoch making result due to I. M. Vinogradov
is his mean value theorem which we state first.
Theorem. Let
f (n) = αk nk + · · · + α1 n + α0
be a polynomial of degree k ≥ 2 with real coefficients and let a and q be integers,
X
exp(2πf (n)),
S(q) =
a<n≤a+q
Z
1
Z
...
J(q, l) =
0
1
|S(q)|2l dα1 . . . dαk
0
where l > 0 is any integer. Let r be any positive integer and l ≥
Then
1
J(q, l) ≤ K r (log q)r q 2l− 2 k(k+1)+δr
1
4
k 2 + 14 k + kr.
where
1
1
1
k(k + 1)(1 − )r , K = 482l (l!)2 lk k 2 k(k−1) .
2
k
From his theorem I. M. Vinogradov deduced the following
δr =
Corollary. We have
3/2
|ζ(σ + it)| < C1 tC2 (1−σ)
(1)
where
99
100
(log t)2/3
≤ σ ≤ 1 and t ≥ 3, (C1 > 0, C2 > 0 are absolute constants).
Remark 1. Following Vinogradov’s method, D. R. Heath-Brown has arrived at
C2 = 18.4. (See notes at the end of chapter VI in [ECT, DRHB]). Further work on
getting economical constants C1 and C2 has been done by Kevin Ford (see some of
his recent work published in Illinois J. of Math. and Proc. London Math. Soc.).
Remark 2. For many important purposes the constant 32 in (1) is not very important. Any exponent A ≥ 23 in place of 23 gives, as a corollary, the zero-free
region
(2) σ ≥ 1 −
C3
, (C3 > 0 is a constant and t ≥ 30),
(log t)2/3 (log log t)1/3
for ζ(s) (see [KR]1 and also [KR]2 )
Remark 3. The result (1) is essentially due to I. M. Vinogradov (in the sense that
he arrived at some A > 32 in place of 2/3). He knew
(3)
2
ζ(1 + it) = O((log t) 3 ),
1
t≥3
2
and many more things. In spite of the importance of results like (1) and (3),
improvement even of (3) to say
2
ζ(1 + it) = o((log t) 3 ), t ≥ 3
(4)
has remained as a great challenge even today. The results (1) and (3) are more
than 80 years old.
2. Some Queries
Let F (n) = log n if n is prime and 0 otherwise. Put F (1) = 0. Then an easy
deduction from (2) is
X
1
3
(5)
(F (n) − 1) = O(X) with X = x (exp((log x ) 5 (log log x )− 5 )−C4
n≤x
where C4 > 0 is a constant. There are many other important (easy) deductions
from (2). For example
X
(6)
µ(n1 ) . . . µ(nk ) = O(X)
n1 ... nk ≤x
for any fixed integer k > 0.
If instead of (2) we assume that the region
σ ≥1−
(7)
C5
(log t)α
(α a constant, C5 > 0 is any fixed constant t ≥ 3, 0 < α < 1), is free from zeros of
ζ(s), many easy deductions follow. For example (with any fixed k ≥ 1)
X
(8)
F (n1 ) . . . F (nk ) = O(Y )
n1 ... nk ≤x
X
(9)
µ(n1 ) . . . µ(nk ) = O(Y )
n1 ...nk ≤x
1
where Y = x(exp(log x) α+1 )−C6 where C6 > 0 is a constant. P. Turan [P.T] was
the first to prove that the converse is true and J. P. Serre [J.P] has a more precise
result in this direction. But these proofs are somewhat complicated and depend on
Turan’s method of power sums. In ([KR, AS, KS] pages 24-26) the authors have
given a simple proof of the converse result for k = 1 in (9). The object of this note
is to show that their proofs go through even for general k ≥ 1 without any serious
changes. But Turan has proved that (8) implies (7) (though by a complicated
method).
3. The equation (9) with any fixed k ≥ 1 implies (7)
The notation in this section is different from what preceeded. Let k ≥ 1 be a
fixed integer and let
X
µ(n1 ) µ(n2 ) . . . µ(nk ) = O(x exp(−c(log x)θ )
n1 ...nk ≤x
1
(for all x ≥ 1), where θ = 1+α
(0 < α < 1) and c > 0 are constants. Let ρ =
99
β + iγ(γ > 0, 100 ≤ 1) be a zero of ζ(s). We prove that
(1 − β)−1 << (log γ1 )α , (with γ1 = γ + 100).
3
Put
X
MX (s) =
µ(n1 ) . . . (µ(nk )(n1 . . . nk )−s , (X ≥ 1),
n1 ...nk ≤X
F (s) = (ζ(s))k MX (s), (X ≥ 1)
We have
F (s) = 1 +
X
an n−s (f or Re s > 1),
n>X
where |an | ≤ d2k (n) (and an = 0 for 1 < n ≤ x), d2k (n) being defined by
(ζ(s))2k =
∞
X
d2k (n) n−s .
n=1
Note that
∞
X
µ(n1 ) . . . µ(nk )(n1 . . . nk )−s = (ζ(s))−k in σ > 1,
n1 ,n2 ,...,nk =1
and that
!
X
µ(n1 ) . . . µ(nk )(n1 . . . nk )−s
(ζ(s))k =
where
a0n n−s , (σ > 1),
n=1
n1 ...nk >X
a0n
∞
X
= 0 if 1 ≤ n ≤ X. Subtracting we obtain


X

µ(n1 ) . . . µ(nk )(n1 . . . nk )−s  (ζ(s))k
(σ > 1)
n1 ...nk ≤X
∞
X
= (ζ(s))−k (ζ(s))k −
a0n n−s
(σ > 1)
n=1
X
=1+
a00l l−s with |al00 | ≤ d2k (l), (σ > 1).
l>X
We first prove a lemma.
Lemma. We have (under the assumptions made on MX (s)) the estimate
1
3
1
|MX ( + it)| << (|t| + 10)X 4 exp(− c(log X)θ ).
4
2
Proof. We have with s = 43 + it, and Mu = Mu (0),
Z X+0
MX (s) =
u−s d(Mu )
1−0
X+0
Z X
Mu
Mu
+s
= s
du
u 1−0
us+1
1
Z
1
= O X 4 exp(−c(log X)θ ) + O((|t| + 10)
X
1
u 4 exp(−c(log u)θ )
1
1
4
θ
= O(X (|t| + 10)(exp(−c(log X) ) log(X + 1))
since
1
1
max(u 4 exp(−c(log u)θ ) = O(X 4 exp(−c(log X)θ )
Thus the lemma is completely proved.
du
)
u
4
We next write (with X ≥ 10, w = u + iv, Y = (logXX)2 ) the identity
Z 2+i∞
X
n
1
1
F (ρ + w)Y w Γ(w)du = exp (− ) +
an n−ρ e− Y
2πi 2−i∞
Y
n>X
where clearly the RHS is 1 + o(1) uniformly in ρ. In the integral on the LHS we
move the line of integration to u given by β + u = 43 . Since our choice of X will be
subject to
X ≤ exp((log(|γ| + 10))1+α D)
where D > 0 is a constant and |Γ(1 − ρ)| ≤ e−|γ| , the contribution from the pole
(at w = 1 − ρ) of ζ(ρ + w) is o(1). The integral has no pole at w = 0 since ζ(ρ) = 0.
The integral on the line u = 34 − β is (by the lemma above)
43 −β
Z ∞
1
X
1
(|γ + v| + 10)2
X 4 exp( c(log X)θ exp(−|v|)dv
2
(log
X)
2
−∞
1
γ12 X 1−β exp(− c(log X)θ )
4
1
2
δ(log γ1 )−α
γ1 X
exp(− c(log X)θ ) (if 1 − β ≤ δ(log γ1 )−α ).
4
We put X = exp((log γ1 )1+α D) and obtain finally
1
1
)
1 + o(1) γ12 (exp(δD log γ1 )) exp(− c(log X)θ ), (θ =
4
1+α
1
1
γ12+δD−λ , (λ = cD 1+α )
4
1
= o(1) if δ =
and D is large.
D
This contradiction proves that 1 − β cannot be ≤ δ(log γ1 )−α i.e. ζ(s) 6= 0 if
σ ≥ 1 − dk (log t)−α (t ≥ 10) under the assumption on
X
µ(n1 ) . . . µ(nk ).
n1 ...nk ≤x
This proves the theorem completely.
Acknowledgement
The authors are thankful to Professor Roger Heath-Brown for informing them
of the work of J. Pintz and that in this paper J. Pintz uses Turan’s method of
Power sums. Finally the authors are thankful to Smt.J. N. Sandhya for technical
assistance.
References
[J.P ] J. Pintz, On the remainder term of the prime number formula and the zeros
of the Riemann zeta-function. Number Theory, Noordwizkerhout 1983 (Noordwizkerhout 1983) 186-197, Lecture notes in math.1068, Springer 1984.
[K.R. ]1 K. Ramachandra, Notes on prime number Theorem-I, Number Theory
(ed. Professors R. P. Bambah (Mrs) R.J.Hans-Gill, and V.C.Dumir), INSA
and HBA publications (2000), 351-370.
[K.R. ]2 K. Ramachandra, Riemann zeta-function, Ramanujan Institute, Chennai
(1979).
5
[K.R, A.S, K.S ] K. Ramachandra, A. Sankaranarayanan and K. Srinivas, Ramanujan’s
lattice point problem, Prime Number Theory and other Remarks, HardyRamanujan J., Vol. 19 (1996), 2-56 (see pages 24-26).
[E.C.T ] E. C. Titchmarsh, The Theory of the Riemann zeta-function, Clarendon
Press, OXFORD (1951), Second edition [E.C.T, D.R.H-B] (Revised and
edited by D. R. Heath-Brown (1986)).
[P.T ] P. Turan, On the Remainder-term of the Prime-number formula, Akademiai
Kiado (Budapest VI, Stalin-ut 31) (1950).
Remark. In [J.P] it is proved that if


Z t
 X

du
1−
R(x) = max |
|, 2 ≤ t ≤ x


2 log u
p≤t
then log(x(R(x))−1 ) is asymptotic to
min {(1 − β) log x + log |γ|}
as x → ∞. Here β + iγ runs through all the non-trivial zeros of ζ(s).
K. Ramachandra, Hon. Vis. Professor, Nat. Inst. of Adv. Studies, I. I. Sc. Campus,
Bangalore-560012, India. And also Retd. Professor, TIFR Centre, P.O.Box 1234, I. I.
Sc. Campus, Bangalore-560012, India.
E-mail address: [email protected]
A. Sankaranarayanan, T I F R, Homi Bhabha Road, Colaba, Mumbai - 400 005. India.
E-mail address:
[email protected]
K. Srinivas, Institute of Mathematical Sciences, CIT Campus, Tharamani, Chennai
600 113, India
E-mail address: [email protected]