Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mkma.math.msu.su/Sites/mkma/Uploads/TrigSumSteklov11.docs1.pdf Äàòà èçìåíåíèÿ: Mon Dec 12 10:29:18 2011 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 21:22:34 2016 Êîäèðîâêà: IBM-866

, -
. .
- . ..

. .. , 16 2011


I

1. 2. .. è 3. " " 4. 4n + 1 4n + 3 5. - 6. 7. . 8. 9. 10. 11.


II

12. .. í 13. - 14. 15. 16. L- ,


I
. , 2007 . è , . 190- 14 2011 . è , 14 120 . - . .


II

, .. è . .. : " , è , , ... , è ..."


III
.. è . " " ( .., .V, .7í25), , 1841 . . .. è . : " , , , , , , , , , . , ."


IV

:" , , . , . : , ? !... , , ."


V

, .. è :" x 3 - 2x - 5 = 0. ... 32 ; 48. 5,6,... , , ".


.. è I

.. . (x ) , x 2. . :"è ‡ , (x ) x ". 24 1848 . .. è - " , " (..., .I, .173í190).


.. è II
, lim
x x dx log x

(x ) (x ) 1 lim , x li(x ) li(x )

li(x ) =
2

. , , 1. (x ).

(x )/ li(x ) x , , .. è , li(x ) è 1 -1 = 1 - p -s , (s ) = s n p
n=1

p s > 1. 1748 . Introductio in Analysin Infinitorum.


.. è III
.. è s > 1 (s )


1 (s ) = (s )
0

e

-x

ex - 1

x

s -1

dx ,

1 s = (s ) - s -1 (s )
0

1 1 - e -1 x
x

e

-x s -1

x

dx .

, , n s 1 + 0 .. è , logn p - ps


p

m=2

logn-1 m = ms



(x ) - (x - 1) -
x =2

1 log x

logn x xs

. .. è .


" " I
" " (1850 .) .. è (x ). x0 2 x x0 (0, 92 . . . ) x x (x ) (1, 105 . . . ) . log x log x

, è : (x ) = log p , (x ) = (x ) + ( x ) + ( 3 x ) + . . .
p x

è (x ) (x ) = (m) = log p , 0 m = p r , r N; .
m x

(m),


" " II
log n =
m |n

(m).

. , . (- log (s )) = - (s ) = (s )
p m=1

log p = p ms

n=1

(n) . ns

(s ), (m)
m |n

m=1

(m) ms - (s ) (s )

k =1

1 ks



=
n=1

n


s

=

=

(s ) = - (s ) =
n=1

log n . ns


" " III
, .. è log n =
n x k x

x ( ), k

T (x ) =
n x

log n =
n x d |n

(d ) =
dk x

(d ) =

=
k x d x /k

(d ) =
k x

x ( ). k

, T (x ) = x log x - x + O (log x ) T (x ) - T (x /2) - T (x /3) - T (x /5) + T (x /30), .. è .


" " IV

è , , T (x ) - 2T (x /2) =
k x

(-1)k

-1

(x /k ) = x log 2 + O (log x ).

(x ) , (x ) - (x /2) x log 2 + O (log x ) (x ) - (x /2) + (x /3).


" " V
x /2 < 2r x (x ) - (x /2) x log 2 + O (log x ), (x /2) - (x /4) ... x log 2 + O (log x /2), 2 ... ...

1 r , (x ) 2x log 2 + O (log2 x ). , , x (x ) (x /2)- (x /3)+x log 2+O (log x ) (x /2)+ log 2+O (log x ). 3 , (x )
2x 3

log 2 + O (log2 x ).


" " VI
(x ) - (x /2) .. è , , a > 3 , a, 2a - 2. .. è : a , a, , a + 1, a + 2, . . . , 2a - 2 , , .. . .. , .. , x 4n2 (x , 2x ] n 1 .


4n + 1 4n + 3 I

1853 . .. . , 4n + 1 4n + 3. , (1837 .) p , p l (mod k ) (k , l ) = 1. x + (x ; 4, 1) (x ; k , l ) x , 2 log x (x ; 4, 3) x , 2 log x

p l (mod k ), p x .


4n + 1 4n + 3 II
è : 4n + 3 , 4n + 1 . , ,
c 0+

lim

(-1)(
p >2

p +1)/2 -pc

e

= +.

1918 . . . [4] . [5] , .. è L- 4, .. , (-1)n L(s ) = (2n + 1)s
n=0



s = > 1/2.


- I
.. è , . " , " (1859). - s , s > 1


(s ) =
n=1

1 . ns

. (s ) C s = 1 1 , 1, .. (s ) - s -1 .


- II
s > 1 . , è e -x 1 (s ) = x s -1 dx , (s ) 1 - e -x
0

, , , s . , . , è , .. è , .


- III
, -
-s /2



s (s ) = 2

(1-s )/2



1-s 2

(1 - s ),

(s ) s = < 0 > 1. , - (s /2) - s = -2, -4, -6, . . . , (s ). , 0 1 , = 1/2 .


- IV
, s = > 1


log (s ) = s
1

(x )x

- s -1

dx ,

(x ) = (x )+ ( x )+ ( 3 x )+. . . ,

í , a > 1
a +i

1 (x ) = 2 i
a -i

log (s )

xs ds . s


- V
. 1859 ., , - , . , (x ) -. , 1895 . , (x ) (s ), : (x ) = x -


x (0) 1 - - log (1 - x (0) 2

-2

),

- , 0 < s < 1.


I
" " (1879) .. è , (s ) = (1 - p -s )-1 è ,
p

( ) , : log n =
n x m x



x , m




f (n) log n =
n=2 p

F (p ) log p ,

F (x ) =

f (nx m ).
n=1 m=1


II
f (x ) = x -s , è f (n) = 1, 0 n x ; n > x .

.. : (1) +
H
(y ) =
dmx

²(d )(dm),

H 1 < H N , y , , H , d ( , 1), H , , m .


III
(m) = m-s . 1937 . .. e 2i p ,
p N

( ). 1937 . .. e
p N 2 i (n p n +§§§+1 p )

,

p . , .. , , .


IV
(1934 .) .. . , . .. . , 20 . , , . .. " " (1947 .) , : ,


V

, , , . .. . , . . 70- .. .


VI
1984 . .. . §§§ e 2iF (p1 ,...,pr ) ,
p1 N1 pr Nr

p1 , . . . , pr
n1 nr

F (x1 , . . . , xr ) =
t1 =0

§§§
tr =0

t (t1 , . . . , tr )x11 . . . xrtr , (0, . . . , 0) = 0.

, .. . .


. I
. . .. , . .. : n 2 , 1 , . . . , n . J
1 1 2k 2 i (x n +§§§+1 x )

J = J (P ; k , n) =
0

...
0 x P

e

d n . . . d 1

J = J (P ; k , n) DP = 0, 5n(n + 1)(1 - 1/n) ,
2k -0.5n(n+1)+ ( )

,
n(n+1)

D = D ( ) = (n )6n (2n)4

.


. II

. , . .. 1971 . 1974 . 1975 . .. , , .. . .


. III
, ï P = (P1 , . . . , Pr ), P1 = min (P1 , . . . , Pr ) P1 : ïï J = J (P ; n, k ) = S () ...


|S ()|2k d ,

. S () =
x1 P1 n1 nr

§§§
xr Pr

exp 2 iF (x1 , . . . , xr ),

F (x1 , . . . , xr ) =
t1 =0

§§§
tr =0

t (t1 , . . . , tr )x11 . . . xrtr , (0, . . . , 0) = 0,

(ï) = (t1 , . . . , tr ) F (x1 , . . . , xr ), n1 , . . . nr 1, t ï = (t1 , . . . , tr ) d = t
n1 nr

§§§
t1 =0 tr =0 t1 +§§§+tr 1

d (t1 , . . . , tr ).


I
. .. (1981 .). = max (|n |, . . . , |1 |).
1 2 i (n x n +§§§+1 x )

e
0

dx (1, 32

- 1 /n

).

1976 . . F (x1 , . . . , xr ) (ï), n = max n1 , . . . , nr , = max |(ï)|. t t
ï t 1 1

...
0 0

e

2 iF (x1 ,...,xr )

dx1 . . . dxr min (1, 32

- 1/n

lnr

-1

( + 2)).


II
2k > , .. ,
+ + 1 2k 2 i (n x n +§§§+1 x )

(k ) =
-

...
- 0

e

dx

d n . . . d 1 ,

2k . - 1952 . . 1978 . .. , .. -. , (k ) 2k > 0, 5n(n + 1) + 1 2k 0, 5n(n + 1) + 1.


III
"" f (x ) = n x n + § § § + m x m + r x r , n > § § § > m > r , n + § § § + m + r < 0, 5n(n + 1), .
+ + 1 2k

(k ) =
-

...
- 0

e

2 if (x )

dx

d n . . . d m d r

= n + § § § + m + r . .


I
- q
q

S = S (q , f ) =
x =1

e

2 i

f (x ) q

,

|S | e

nA(n) 1-1/n

q

,

A(n) - A(3) = 6.1, A(4) = 5.5, A(5) = 5, A(6) = 4.7, A(7) = 4.4, A(8) = 4.2, A(9) = 4.05, A(n) = 4 for n 10, f (x ) = an x n + § § § + a1 x , (an , . . . , a1 , q ) = 1, q 1, n 3.


II
. n 2 , n = max (n1 , . . . , nr ), q ,
q q

S (q , F ) =
x1 =1

§§§
xr =1 nr

e

2 i

F (x1 ,...,xr ) q

,

F (x1 , . . . , xr ) =

n1

§§§
t1 =0 tr =0

t a(t1 , . . . , tr )x11 . . . x

tr r

, q . |S (q , F )| e
7nr r (q )

3

( (q ))r

- 1 r - 1/n

q

.


III
1952 . - . a a n 3, f (x ) = q1 x + § § § + qn x n , n 1 (a1 , q1 ) = § § § = (an , qn ) = 1, q = q1 . . . qn .
q

S (q , f ) =
x =1

e

2 i f (x )

( )
+ + qn -1 q1 -1

=
qn =1

§§§
q1 =1 an =0

...
a1 =0

|q

-1

S (q , f )|2k ,

, as qs , s = 1, . . . , n.


IV
- . 2k > 0.5n(n + 1) + 2 2k 0.5n(n + 1) + 2. , , 1 m < r < § § § n , a a n 3, f (x ) = qm x m + § § § + qn x n , m n (am , qm ) = § § § = (an , qn ) = 1, q = qm . . . qn . ""
+ + qn -1 qm -1

=
qn =1

§§§
qm =1 an =0

...
am =0

|q

-1

S (q , f )|2k ,

1981 . "" . 1 m < r < § § § < n, m + r + § § § + n < 0.5n(n + 1) 2k > m + r + § § § + n + 1 2k m + r + § § § + n + 1.


I
í , .. p1 , . . . , pk p1 + § § § + pk = N1 , ......... n n p1 + § § § + pk = Nn , (N1 , . . . , Nn ) , Nk = P k (k + o (1)), k = 0, k = 1, . . . , n P , +. .. -. 1985 .


II
, 1985 .
t1 t t tr p11 . . . pr r1 + § § § + p11 . . . prk = N (t1 , . . . , tr ), k

0 t1 n1 , . . . 0 tr nr , t1 + § § § + tr 1, N (t1 , . . . , tr ) t N (t1 , . . . , tr ) = P11 . . . Prtr ( (t1 , . . . , tr ) + o (1)), (t1 , . . . , tr ) = 0, P1 = min {P1 , . . . , Pr } +. , .. . 2009 . , p n , p , n , .


I
30- .. í , .. n, x 6, . 2002 . .. T (x ) n x , [1 p1 ] + [2 p2 ], p1 , p2 , 1 , 2 , 1 /2 . T (x ) x 2/3+ , > 0 . 1997 . , 1 = 2 = 1 T (x ) x 1- , < 1/10.


.. í I
H 3 = {z = (x1 , x2 , y ) R3 : y > 0}. , u = u (z , z ) = (x1 - x1 )2 + (x2 - x2 )2 + (y - y )2 yy

d (z , z ) z z u (z , z ) = 4 sh2 d (z , z ) . 2

z0 = (x10 , x20 , y0 ) T (x10 , x20 , y0 ch T ) y0 sh T .


.. í II

N (T , w0 , w ) g , d (w0 , gw ) T . .. N (T , w0 , w ) T , í . .. . .


I
k = (k ), k = inf {M }, M < 1,
T -1 1

Ik = Ik (, T ) = T

| ( + it )|2k dt



T ,

k > 0 > 0 . m( ), m( ) = 2f ( ), f ( ) , (k ). , m( ) sup {m}, m > 0 , > 0
T


1

| ( + it )|2m dt



T

1+

.


II
è , , t (1/2 + it )


|t | ,

k > 0 k = 1/2. (s ) , k = 1/2 0 < k 2. k > 2 , k < 1 - 1/k . 1981 . .. - , 8 5/8, m(5/8) 8. , - , .-. (1960 .), 1/2 < < 1 m( )


(1 - )-

3/2

.


III
.. , .. (2003 .) . a, 1 a < 20, t 1 1/2 < < 1 3/2 ( + it ) t a(1-) , k0 = 44 - [22/a]. ) k k 45 k 1 - 1 , (3a(k - k0 ) + (3a(k - k0 ))1/2 )2/3
2701 2880

) 1 =

- 1 . (1 - )3/4

m( ) 1 k0 - 1 + 2 3a(1 - )3

/2

(3a

)1/2


I
(n) n, k -1 (n) = d -1 . x I (x ) =
n x d |n

(n) (n + k )

6 2

-1

(k )x ln2 x ,

.. 1927 . 1931 . . I (x ) = x (A0 ln2 x + A1 ln x + A2 ) + R (x ), R (x ) x 11/12 ln17/3 x , A0 , A1 , A2 . 1979 . .., , R (x ) x 5/6+ , > 0 . , k x 2/3 .. -. 2006 . .. , R (x ) x 3/4 ln4 x .


I
f (n) , p . f (n),
n h x < n x +h

f (n),

, h p , , h p , h , log h 0 as p . log p


I
, p 1, a (mod p ), a = -1, a (mod p ), p 0, a 0 (mod p ), . Sh (x ) =
x < n x +h

n p

.

. . . (1918) 1 x < x + h p |Sh (x )| < p ln p .


II
. . è (1958) : > 0 () > 0 1 , Sh (x ) hp -() as h > p 4 + , .. p 1 [x , x + h] h > p 4 + , . . .è (1952) Sh (x ) . Mp () x , 0 x < p Sh (x ) h1/2 .


1 1 Mp () p 2

e
-

-t 2 /2

dt

p

.


I
p1 , . . . , pk , Q = p1 . . . pk , 1 x < x + h Q , s = ‘1 as 1 s k , T n, n + a1 p1 = 1 , . . . , n + ak pk = k ,

a1 , . . . , ak . . . (2001) . T= h + 2k Q ln Q

|| 1 h > 2k Q ln Q .


II
Sh (x ) = Sh (x ; p1 , . . . , pk ) = n + a1 p1 ... n + ak pk

log h , .. h , log Q 0 Q . NQ {x : . . . } x , , . h (x = (h, Q ) = Sh ) .
y

1 1 N Q {x : < y } Q 2

e
-

- y 2 /2

dy

Q

y .


I
(n) p , 0 x < p , 1 h p , Np {x : . . . } x , , . Gh (x )
x +h

Gh (x ) =
n=x +1

(x )e

2 i

an p

.

h = p , , |Gp (x )| = p . 0 < h < p |Gh (x )| < p log p . . . (2001) .
hx = (h, p ) = G(h ) .

2



1 Np {x : < y } 1 - e p

-y

p

y .


I
p , q . q qq 1 (mod p ), 3 < h < p , 1 x p - 1, Kh (x ) =
q h

e

2 i

xq p

.

. . (2001) .
hx = (x , h) = K(h ) p .

2

1 Np {x : < y } 1 - e p

-y

p

y .


I
m > 1, n, h , p , (n) m, (m) . D=
p h

1, Sh () =
p h

(p ), h () =

Sh () D

2

,

Nm () = # { : h () }. . . (2002) .
h = h () = S(D ) .

2

Nm () 1-e (m)

-

m

.


I
. . (1959) . . (1959). {fn }, f0 = 1, f1 = 1 fn+1 = fn + fn-1 n 1, . m > 1 , > 0 , Nm {n : . . . } n, , . , ,
h -1

Sm (h; a) =
n=0

e

2 i

afn m

, Nm () = Nm {a : 0 a m - 1, |Sm (h; a)| < h}.


II

. . (2001) . m h m h = h(m) h 0.5 log m, = 5+1 . 2 lim Nm () =1-e
-

m

> 0.


L- I

1955 . .. , , L- , . .. , .. , .. , . 2000 . .. . D = p k , p , 2 1 - 4 < < 1, b = 33 , A > 0 1/1024 > > 0 .


L- II2
(1) p e
Ak

, |t | 2D D
b (1- )
3/2

|L(s , )|

ln2/3 D

(1 - )1/4 ln1/6 D + 1

,

(2) D D0 = D0 (A) L(s , ) |t | D , (1) p e 1-
2/3

c ln
2/3

D (ln ln D )1

/3

,c =

1 500b

2/ 3

,

A ln

|t |

, |t | D |t |b
(1- )
3/2

|L(s , )|

ln2/3 |t |

(1 - )1/4 ln1/6 |t | + 1

.

(2) D D0 = D0 (A) L(s , ) c |t | D , 1 - 2/3 . ln |t |(ln ln |t |)1/3


L- III
, .. (1981). " , , . , . . . , . , , . , , . ."


I
[1] è .. , .IíV. .-.: - , 1944í1951. [2] è .. . , 1955, 926. .: -

[3] è .. Lettre de M. le professeur Tch‡ ychev ` M. eb a Fuss, sur un nouveau theor` e relatif aux nombres premiers em contenues dans les formes 4n + 1 et 4n + 3// Bull. de la classe phys.-math. de l'Acad. Imp. Sci. de St.-P‡ ersbourg, 1853, XI, et 208. [4] Hardy G.H., Littlewood D.E. Contribution to the theory of Riemann zeta-function and the theoryof the distribution of primes// Acta Math., 1918, 119-196. è [5] Landau E. Uber einige è re Vermutungen und alte Behauptengen in der Primzahltheorie, I-II// Math. Zeitschr., 1918, 1-24, 213-219.


II
[6] Bombieri E. Problems of Millennium: the Riemann Hypothesis// 2000, pp.11. [7] .. . 2004, 176. .: ,

[8] .. . .: , 1976, 119. [9] Davenport H. Multiplicative Number Theoty. Springer-Verlag, 1980, pp.177. [10] . . New-York:

.-.: , 1948, 543.

[11] .. . .: - - - , 2007, 96.