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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 19. Выпуск 3.
УДК 511.176 DOI 10.22405/2226-8383-2018-19-3-20-34
Суммы Мертенса, требующие меньших значений функции
Мёбиуса
Хаксли Мартин — почетный профессор математики, профессор, доктор наук, Кардиффский университет, Уэльс, Великобритания. e-mail: [email protected]
Уотт Найджел — доктор наук, Данфермлайн, Шотландия. e-mail: [email protected]
Аннотация
Обсудим некоторые тождества с участием ^(п) и М(х) = Y1 п<х Функции Мё-
биуса и Мертенса. Они позволяют вычислить М(Nd) для d = 1,2, 3,... как сумму Оа (Nd(logN)2d-2) членов, каждое произведение вида ^(п\) ■■■ ^(пг) с г < d и п1,...,пг < N. Докажем более общее тождество, в котором М(Nd) заменяется на М(д, К) = Y_1 п<к И1(п)д(п)^№ д(п) - произвольная полностью мультипликативная функция, тогда как каждое щ имеет собственный диапазон суммирования 1,...,Nj. Это не ново, за исключением того, что в N±,..., N^ произвольны, но наше доказательство (вдохновленное тождественным равенством Э. Майсселя, 1854) является новым. Мы главным образом заинтересованы в случае d =2, К = N2, = N2 = N, где тождество имеет вид М(g,N2) = 2М(g,N) — mTAm, при этом А является матрицей N х N элементов атп = T,k<N2/(mn) 9(k), в то время как m = (^(1)д(1),..., p(N)g(N))T. Наши результаты в разделах 2 и 3 данной статьи предполагают, что д(п) равно 1 для всех п. Теорема Фробениуса—Перрона применяется в этом случае: мы находим, что А имеет одно большое положительное собственное значение, приблизительно (n2/6)N2, с собственным вектором приблизительно f = (1,1/2,1/3,..., 1/N)T Т и что при больших значениях N второе наибольшее собственное значение лежит в (-0.58N, —0A9N). Раздел 2 включает оценки для следов А и А2 (хотя для Tr(A2), мы пропустим часть доказательства). В разделе 3 обсуждаются способы аппроксимации mTAm, используя спектральное разложение А или (альтернативно) формулу Перрона: последний подход приводит к контурному интегралу, включающему дзета-функцию Римана. Мы также рассматриваем использование тождества А = N2 ffT — 2uuT + Z, a, Z — матрица N х N элементов zmn = —ф(N2/(mn)), причем ф(х) = х — [жJ — 2- Наши выводы представлены в разделе 4.
Ключевые слова: функция Мёбиуса, функция Мертенса, полностью мультипликативная функция, Майссель, тождество Линника, тождество Вогана, симметричная матрица, teo-рема Фробениуса-Перрона, собственное значение, собственный вектор, формула Перрона, дзета-функция Римана.
Библиография: 15 названий. Для цитирования:
М. Хаксли, Н. Уотт. Суммы Мертенса, требующие меньших значений функции Мёбиуса // Чебышевский сборник, 2018, т. 19, вып. 3, с. 20-34.
CHEBYSHEVSKII SBORNIK Vol. 19. No. 3.
UDC 511.176 DOI 10.22405/2226-8383-2018-19-3-20-34
Mertens Sums requiring Fewer Values of the Möbius Function
Huxley Martin — Emeritus Professor of Mathematics, Professor and Ph.D., Cardiff University, Wales, United Kingdom. e-mail: [email protected]
Watt Nigel — Doctor of Philosophy, Dunfermline, Scotland. e-mail: [email protected]
Abstract
We discuss certain identities involving ^(n) and M(x) = n<x ^(n)-, the functions of Möbius ^d Mertens. These allow calculation of M(Nd), for d = 1,2,3,... , as a sum of Od (Nd(logN)2d-2) terms, each a product of the form ^(n\) ■■■ ^(nr) with r < d and nl,...,nr < N. We prove a more general identity in which M(Nd) is replaced by M(g,K) = Y^n<K ^(n)d(n)-, where g(n) is an arbitrary totally multiplicative function, while each n.,- h^ its own r^ge of summation, 1,..., Nj. This is not new, except perhaps in that Ni,... ,Nd are arbitrary, but our proof (inspired by an identity of E. Meissel, 1854) is new. We are mainly interested in the case d = 2, K = N2, Ni = N2 = N, where the identity-has the form M(g,N2) = 2M(g,N) — mTAm, with A being the N x N matrix of elements amn = Y,k<N2/(mn) d(k), while m = (^(1)g(1),..., ^(N)g(N))T. Our results in Sections 2 and 3 of the paper assume that g(n) equals 1 for all n. The Perron-Frobenius theorem applies in this case: we find that A h^ one large positive eigenvalue, approximately (-k2/6)N2, with eigenvector approximately f = (1,1/2,1/3,..., 1/N)T, and that, for large N, the second-largest eigenvalue lies in (-0.58N, —0.49N). Section 2 includes estimates for the traces of A mid A2 (though, for Tr(A2), we omit part of the proof). In Section 3 we discuss ways to approximate m^Am, using the spectral decomposition of A, or (alternatively) Perron's formula: the latter approach leads to a contour integral involving the Riemann zeta-function. We also discuss using the identity A = N2 ffT — 2> uuT + Z, where u = (1,..., 1)T and Z is the N x N matrix of elements zmn = — ^(N2/(mn)), with ^(x) = x — \_x\ — 5.
Keywords: Möbius function, Mertens function, completely multiplicative function, Meissel, Linnik's identity, Vaughan's identity, symmetric matrix, Perron-Frobenius, eigenvalue, eigenvector, Perron's formula, Riemann zeta-function.
Bibliography: 15 titles. For citation:
M. Huxley, N. Watt, 2018, "Mertens Sums requiring Fewer Values of the Möbius Function" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 20-34.
Dedicated to the memory of Yu. V. Linnik 1. Introduction
The sieve of Eratosthenes will find the prime numbers in N + 1,..., N2 provided that we know all the primes in 2,..., N. In particular the sieve gives a relation for the function w(x) that counts
the number of primes less than or equal to x:
' N 2
n(N2) = tt(N) - 1 + Y^ ^(d)
d<N 2 P (d)<N
(1)
where ß(d) is the Möbius function (which is (—1)u when d has v prime factors, all different, but 0 when d has any prime factor repeated), while P(d) is the greatest non-composite divisor of d, and [x] = max{m e Z : m < x}. The numbers d in (1) are constructed as products of the known primes in 2,..., N, so the values ß(d) can be read off. In general, given a number n, it is very difficult to factorise n and so find ß(n). Thus the Mertens sum
M(x) = £ ß(n) (2)
n<x
is difficult to calculate from the definition. The Dirichlet series ^ ß(n)/ns is 1/C, (s) (the reciprocal of the Riemann zeta function), and, according to folklore, the fastest method of calculating M(x) is by Perron's contour integral formula for the sum of the coefficients of a Dirichlet series.
In this paper we discuss a family of identities which allow M(Nd) to be calculated for each positive integer d as a sum of no more than Od (Nd(log N)2d-2) terms, each a product of the form ß(n\) ■ ■ ■ ß(nr) with r < d and {n\,..., nr} C {1,..., N}. In Theorem 1, below, we state a more complicated form of these identities, in which each of the variables of summation nj (j = 1,... ,r) can have its own independent range of summation: 1,..., Nj (sav).
We actually treat the more general Möbius sum
d
M(g,x) = Y, ß(n)g(n), (3)
n<x
where g(n) can be any totally multiplicative arithmetic function, that is, g(rs) = g(r)g(s) holds for any positive integers r and s. The relevant identity when d = 1 is (of course) the definition (3). The case d = 2 is the next simplest. Let m(g, N) be the column-matrix (/J.(1)g(1),..., y(N)g(N))T, and let A(g, N) be the N x N matrix with elements
amn(g,N)= £ g(k) (m,n <e{1,...,N}). (4)
fc< ^
— mn
Then
M(g, N2) = 2M(g, N) - (m(g, N))T A(g, N)m(g, N) . (5)
In the general case, when d,K,N e N satisfy d > 2 mid K > N > Kl/d — 1, we have: M (g ,K) = dM (g ,N)
d r
— Y,(—i)TdCr £ ••• E E••• E^(ki-kr-i)n^(ni)g(ni), (6)
r=2 m<N nr<N fci fcr_i i=1
nin2...nr kik2...kr-i<K
where dCr = d(d — 1) ■ ■ ■ (d — (r — 1))/(r\).
Note that (5) is just the special case d = 2, K = N2 of (6). Moreover, (6) is itself a special case of another identity (that stated in Theorem 1, below), in which the single range of summation
1,..., N is replaced by d independent ranges of summation. In order to state this more general identity we require some more notation.
Let d be a positive integer greater than 1. Let V = v1v2 ... Vd be a word of length d in the alphabet {0,1}. The support of a word V is the set of indices f for which Vi = 1. The weight w(V) of a word V is the size of the support, so that w(V) = ^ Vi. The combinatorial Mobius function, which we write as to distinguish it from the number-theoretic function is ^*(V) = (—1)w(v\
Let N1,..., Nd be positive integers. For each word V, and each L G N let the notatati on )
signify summation over n1,... ,nd in the ranges ni = 1,..., L when Vi = 0, but ni = 1,..., Ni when Vi = 1. When L = 1 and Vi = 0 the variable of summation ni effectively becomes 'frozen', meaning
{1}
Let K be a positive integer that is less than (1 + N1)(1 + N2)... (1 + Nd). If n1,...,nd are integers satisfying the condition n1n2 ...nd < K, then ni < Ni holds fa at least one index i. It therefore follows by the inclusion-exclusion principle of combinatorics that if f : N ^ C is such that one has |/(n1,..., nd)l > 0 only when n1n2 ■ ■ ■ nd < K, then
K d K
Y,(00... 0)f (m, ...,nd) = Y,(-1)r-1 E E(y )f (ni,...,n*), (?)
1 r=1 v: u(V)=r 1
or, to put it more elegantly, №*(V) ^(V)f (n1,... ,nd) = 0.
Theorem 1. When g(n) is a totally multiplicative arithmetic function, and d, N1,... ,Nd and K are as above, we have:
d
M(g, K) = ^ M(g, min{Ni, K})
i=1
1 d - E (-1)w(y) E(y) E ■■■ E 9(ki ■■■ ku{v )-i) n )g(m). (8)
V: w(V)>2 1 fci ku(V)_i i=1
ki...ku(y )_i<
Proof. We apply (7) with f given by:
f (rn,...,nd)= E ■■^E v(m) ...Knd)g(h ...kd-1U1 ...nd). (9)
ki kd_i ki...kd_i<
For the word V = 11... 1 with w(V) = d, we have
E (11... 1) = E (11...1).
All other words V have Vj = 0 fa at least one index j, so the corresponding summand nj runs over the full range from 1 to K. For these words V we carry out the following 'contraction step'. Take an index j for which Vj = 0. We sum over nj and kd-\ first, observing that by Möbius inversion we have:
K
Y1 Y1 ^(nj)g(njkd-i
nj = 1 , _K_
d-i-ni ...ndk-i...kd-2
_ £ g(rn)Y, Mn)
_K_ nj |m
_ id(1) = ^(1)d(1) ifni... n3-in3+i... ndk\... kd-2 < K, |0 ot h e rwi s e.
We thereby find that the value of the relevant sum over n1,... ,nd and k1,..., kd-1 is unchanged when we omit kd- 1 and freeze n, as the fixed value nj _ 1.
We repeat the contraction step for every index j with v, _ 0, freezing the corresponding variable as nj _ 1, and removing the last variable ki. Exceptionally, when V is 00... 0, we can remove kd-1,kd-2,..., k\, and freeze nd, nd-\,..., n2, but the sum over m remains over the range 1,..., K, giving the term M(g,K) on the left of (8). The summation identity (7), when applied with f given by (9), contracts to give (8). ■
'separate variables' (as, in (6) for example, we separate k\,... ,kr-\ from n\,...,nr by means of the identity g(k\ ■ ■ ■ kr-\n\ ■■■ ,nr) _ g(k\ ■ ■ ■ kr-\)g(n\) ■■■ g(nr)). Indeed, (8) gives a formula for the Mobius function itself, for we can apply (8) to each term in the difference M(g, K) — M(g, K — 1) _ )g(K), and we can then divide through by g(K) to obtain a formula for y(K) that is independent of g. This formula for y(K) may also be deduced from the identity
1 d , Nj , NX d <x /N
d / J f W d ^
n (1 — C(*) £ Cd-1(s) n E ^ (Re(s) > 1), (10)
j=1 ^ n=1 ' j=1 n=1+l
C(s) ^ ns ) 11 ' ns
' j=^ n=1 7 j=1 n=1+Nj
through multiplying out the brackets on the left-hand side, and then computing the coefficient K-
(1 + N1) ■ ■ ■ (1 + Nd) be greater than K. This approach yields a second proof of Theorem 1. We prefer the first proof due to its more obvious connection with Meissel's identity [8 p 303],
rx I j1 if x > 1
^ Yn\ ^(n) \0 if 1 >x> 0,
n—x ^ J 1
which was the initial source of inspiration for our work.
Given any K £ N, any integer d > 2, and any d1,..., 0d > 0 with d1 + ■ ■ ■ + 0d _ 1, it follows from Theorem 1 that (8) will hold when one has also Nj _ [Kei], for j _ 1,..., d. Theorem 1 therefore offers considerably more flexibility of application than (6) does. Although we believe Theorem 1
N1 , . . . , Nd
displayed in (5) and (6) are known results. The result (5) is contained in Vaughan's (slightly more complicated) identity [13 equation (18)] (essentially the special case when u _ Vx, and so S3 _ 0), and one can find in equation (13.38) of [5], for example, a formula for n) that is equivalent to what we have in (6). It is, moreover, clear that even our identity in (8) is akin to formulae of Heath-Brown for sums involving A(n), the von Mangoldt function: compare (10), from which (8) may be deduced, with Lemma 1 of [2]. The earliest formula of this type is due to Linnik himself in [6,7].
We shall refer to the case of (3) (or of (4), (5), (6), or (8)), where the function g(n) takes the constant value 1, as the principal case. The main focus of our work has been on the principal case of the identity (5). Indeed, all subsequent sections of this paper are exclusively devoted to matters connected with this single topic, such as (for example) questions concerning certain properties of the N x ^ matrix A = A(N) that occurs in the principal case of (5) and has, by (4), elements &mn = [N2/(mn)} £ N. In Section 2 we discuss matters related to the spectral decomposition of A = A(N). In the third (and final) section we discuss decompositions (spectral and otherwise) of the quadratic form mTAm, where m = m(N) is the column-matrix (p.(1),..., p.(N))T that occurs in the principal case of (5).
We consider especially the principal case of (5), in the hope that it (modified as necessary) might lead to a new proof of the prime number theorem, or even some new upper bound for the Mertens sum |M(®)|. The following parts of this paper report what we have discovered in the search for such an application of (5).
One of our findings is that the matrix A(N), which (clearly) is real and symmetric, has one exceptionally large positive eigenvalue, approximately N2((2), with eigenvector approximately (1,1/2,1/3,..., 1/N)T. Calculations by the second author show that the second-largest eigenvalue of A(N) lies in an interval of the form [d4N + o(N),c4N + o(N)}, where c4 and d4 are constants that are approximately — 0.496 and -0.572, respectively: for more details, see (18), (25), (31) and (32) below. Hence, for N sufficiently large, the quadratic form on the right-hand side of (5) is neither positive definite nor negative definite in the principal case.
By the principal case of (6), we have a sequence of formulae through which each of M(N2),M(N3),M(N4),... is expressed in terms of ^(1),...,^(N). Although the first of these formulae, the principal case of (5), may be considered analogous to the sieve of Eratosthenes (1), there seems to be no version of (1) for w(N3), because unwanted numbers of the form pq, where p and q are both primes greater than N, survive the sieve process ("Gnoggensplatts" in Greaves's lectures on Sieve Methods).
A connection between Mertens sums and certain symmetric matrices Un (n £ N), that bear some resemblance to our matrices A(N) (N £ N) has previously been established by Cardinal [1]. To define Cardinal's matrix Un, one first takes a1 < a2 < • • • < as to be the elements of the set S = K U {[n/p} : p £ ft.}, where ft = [p £ N : p < /n] (it follows that 0 < 2[/n} — s < 1). Then Un is the s x s matrix with elements Uij = [n/(ai<jj)]. In Propositions 21 and 22 of [1], it is shown that one has TnU-1Tn = Vn, where Tn and Vn are the s x s matrices with elements tij = |[2, s + 1} n [i + j}| and vij = M(uij), respectively.
In the cases where n is a perfect square, so that n = N2 fa some integer then |K| = N, and the N x N principal submatrix of Un consisting of the array of elements from the first N rows and first N columns of Un is our matrix A(N): since 2N — 1 < s < 2N, we can say that A(N) constitutes (exactly, or approximately) the top left-hand quarter of Cardinal's matrix Un. In these same cases, Cardinal's identity TnU-1 Tn = Vn implies that v11, which is M(N2), will be equal to the sum of all s2 of the elements of the inverse of the matrix Un = UN2: we obtain a formula for M(N2) thereby that seems quite different from what we see in the principal case of (5).
As Cardinal observes in Theorem 24 and Remark 25 of fl], information about small eigenvalues of the matrix V,-1 = T,-1UnT,-1 might lead to new upper bounds on M(x). In this respect, the connection that we have found between M(x) and A(N) is quite different from Cardinal's connection between M(x) and Un, fa it is the larger eigenvalues of A(N) and their eigenvectors that matter most in the principal case of (5): see, for example, equation (35), below.
We have scarcely considered non-principal cases of (5), (6), or (8). Certain non-principal cases of (5) may merit further investigation. The first case is when g(n) = x(n)-¡ a non-principal Dirichlet character to some modulus q > 1. The sums J2e<x x(l) that we use to construct the matrix elements amn(x, N) in (4) are periodic step functions of x, whose period is q or some
proper factor of q. In contrast to the principal case, where the set of elements of the matrix A(N) in (5) contains at least N different integers, n amelv [N2/1], [N2/2],..., [N2/N], there is a single finite set, {Eo<£<lx(1) ■ L £ (0,o] n Z}, that contains all the elements of all the matrices A(x, 1),A(%, 2),A(%, 3),... .For % real, A(x,N) will, of course, be real and symmetric just like A(N).
A case of (3) known to be related to the prime number theorem is when g(n) = 1/n (see page 248 of [9], for example). More generally, when g(n) = n-s for some fixed complex number s, then the sum M(g, x) in (3) becomes a partial sum for the Dirichlet series for 1/C, (s). If, for some a0 £ [1/2,1), the only zeros of ((s) with real parts greater than a0 are a pair of simple zeros, p and p (say), and if we put g(n) = n-p (n £ N), then the sum M(g,x) in (3) will grow logarithmically x
Another interesting case of (3) to (5) is when g(n) = X(n), the Liouville function, which is the projection of the Mobius function ^ onto the space of totally multiplicative arithmetic functions.
M( , x) x/ (2)
2. Elementary Estimates for Eigenvalues and an Eigenvector
N A = A( N)
is both real and symmetric, it has eigenvalues Ai < X2 < ... < XN with corresponding eigen-(column-)vectors of unit length ei,..., e n that form an orthonormal basis of Rn. When v £ R n, one has
n
vTAv = ^Ak (ek ■ v)2 (12)
k=1
as a consequence of the spectral decomposition A = ENN=1 AkekeT, and Parseval's identity gives
n
£ (e k ■ v)2 = v ■ v = \\v\\2. (13)
k=1
In order to study the terms appearing in (12) and (13), we estimate: (a) Tr(A) = E Onn (the trace of the matrix A),
2
mm
(b) Tr( A2) = Tr(ATA) = EE®
(c) fTAf, where f = (1, 2, i,..., N)T,
(d) wTAw, where w = u - \\f\\-2(f ■ u)f, with u = (1,1,..., 1)T £ We use the following notation:
0 = £•»-', '= E E {N2/(mn)} and * = * -12
J ^ ^ ^ mn N2 ^ ^ \ mn\
m=1 m<Nn<N m<N n<N v J
where {t} = t — [i] (the fractional part of t). Taking (b) first, we simply observe that
2 / n t2 St. T2 W 2
m n
^^^ £ £ mn ^ ^ \mn Imnj) ^2N + 2) N ' ^
Since Tr(A2) = A1 + ■ ■ ■ + AN, and since 5 > 0 and * < 1, the identity (14) shows already that An < (2N2 + (2(2)-1.
Regarding (c), we are content to note that
T \N2/(mn)] ^ (N2/(mn) - \N2/(mn)}) 2 2
fTAf = V V ±—--^ = V V ^—----i---^ = dN2 - 5 . 15
^^mn^^ mn
m<Nn<N m<Nn<N
We have here ||f ||2 = (2, so bv Ravleigh's Principle it follows from (15) that
(2N2 -~<\n . (16)
2
By (16) and the point noted immediately below (14), we conclude that
(1 + logN)2 ^ , „2 ^ 1
(2 <XN - <2N2 < ^ . (17)
As 0 < S < (f < (0(2 = N(2 < N2(2, the lower bound on XN in (16) is non-negative, and so we may deduce from it that X2N > ((2N2 - 5(-1)2 = C%N4 - 26N2 + 62(-2: this, together with the evaluation of Tr(A2) in (14), is enough to show that
X2 + ■■■ + XN-1 < 2 - S2(-2 < N2 . (18)
From the way we have ordered the eigenvalues, the bound (18) implies:
XI < , • nATff {k = 1,2,...,N - 1)' (19)
y min{&, N - k}
In view of (17) and (19), it is clear that for N large, XN will be exceptionally large, compared with all other eigenvalues of A. Accordingly we consider first the corresponding eigenvector eN, before discussing the estimation (a) of Tr(A). Putting FN = eN ■ f, where f = ||f11—1 f, we find bv (15) and (17), and (12), (19) and (13), that
Xn - (1 + (1 + log N)^ < fTAf < XnFN + N (1 -F2n) .
For N > 1 we have XN > N (this follows by (17) when N > 3), and so, by comparison of the upper f T Af
1>F2 > 1 (2 + (1 + logN)2)
1 >Fn > 1--(Xn - N) .
Choosing the ±-sign so that ±FN = |FNwe therefore find from (17) that
eN
)l = «D -^-g-o ( ^ )
= 0 ^r . (2°)
We now come to the task mentioned in (a) above, which is the estimation of the sum S = Tt(A) = ann- We pick a positive integer K, and we divide the original sum S into two parts: S^, which has the terms with n2 < N2/(K + 1^, and S2, which has the terms with N2 >n2 > N2/(K + 1) (so that ann = [N2/n2] = k for some k G {1,..., K}). We have
s - E = E (N2+o{1))
n2<N2/(K+1) n<N/VK+ïK J
N/vK+i" "" ' ~ {N2)^ ' ~ \VKJ - (2N2 - NVK + N + O(K + -L^ .
-N 2 L -j"_x-2 ix + o(K)) +O( »)
S2
K
S2 = E E * = EE
1<i
K
E
k=1 N <n< ^ 1<i<k<K Vk+i <n< Vk ---
K
' N ' ' N '
.4k. Wk + 1J
1=1
" N ' ' N '
rVl. WK + 1J
K
V- N KN r^T^
> . = + 0(K).
¿= 41 Vk + I v '
Let
Then
Hence
1 1 ^
M=2* - 241-1 -Tl = W + 1 )2 " G N ^ " = ^
K ^ K __, A .
E - = E № - 241-1 - = -a+0{ -jn)-
S2 = 2N Vk -aN - NK + o( -—= + K ) = N VK -aN + o( + K ) ,
\TKTI \rVK J \VK J
and so, putting K = [N2/3], we get:
Tt(A) = S1 + S2 = C2N2 - (a - 1)N + 0 (n2/3^ . By (21) and (17), it follows that
Xi + ■ ■ ■ + Xn-1 = -(a - 1)N + 0 (n2/3^
(21)
(22)
By equations (1.11) to (1.13) of [4] and the case K = 1 of of equation (B.24) of [9] (itself an application of the Euler-Maclaurin summation formula), we find that for a G (0,1) U (1, x>) and K G N,
K
E
1=1
1
K ^ +C(a) + 0(K-a>
1- a
d(K, a) K a
K a
K 1-a -1 ~ . ...
+ —-+ 7 + 22^(a - 1)
=1
1 a
(23)
(24)
where ((s) is Riemann's zeta function, each of 7, 7^ j2, ...is a certain (real valued) absolute constant (the first of these, 7, being Euler's constant) and d(K, a) is a number lying in the (0, 1) a = - (1/2)
a - 1 = -(((1/2) + 1) = 0.4603545 ....
Given that ((2) = n2/6, we find (similarly) that (2 = (n2/6) - N-1 + 0(N-2) in (14) to (18). We also note that ^ =logN + 7 + 0(1/N) (as follows, for example, by letting a ^ 1 in (24)).
We remark that, by combining methods similar to those used to obtain (21) with certain applications of the Euler-Maclaurin summation formula, we have been able to determine that the variable A G [0,1) in (14) and (18) satisfies
A R + 0( 1+logN \
(25)
1
a
where $ = 1 — g — 1 (log(2^) — 1)2 + 1 (1 — 7)2 = 0.32712... . We omit our proof of (25), which shows no features that are truly novel (and would require more than just a few pages). By (25), we
N
Finally we consider the estimation problem (d), stated earlier. Noting firstly that w = u— —( 1/ 2)f
C2
\\w\\2 = N — ^ = N + O ((1 + logN)2) (26)
2
and that
wTAw = uTAu — 2 ((1/(2) uTAf + ((1/C2)2 fTAf . (27)
We have, moreover,
uTAu ^y — =YY — - 2VV
m>N n
m<N n<N
Here
'3(1) - \1lo£"2 îat2
£<N 2
N2
m n
N2
-Di - 2D2 (say). (28)
m n
Di - E r3(£) -f± log2 (N2) + (3-y - 1) log (N2) + N2 + 0 (n^43/48)
2
(29)
where c1 = 372 — 37 + ?j71 + 1; see pages 352-4 of [4] for the second equality in (29). Regarding the sum D2 in (28), we have:
D2- ELL 1-E(E0 E
m>N n k 1<N n\£ 7 N<m<N2/l
(nk)m<N 2
N2 E ^ -N E T2(l)+o(£ T2(l))
1<N l< N \e<N J
1
By partial summation and Huxley's estimate on page 593 of [3] for the remainder term in Dirichlet's divisor problem (namely A(x) = Ye<x T2(l) — (logx + 27 — 1)x), we deduce from the above that
D2 = Q log2 N + (27 — 1) log N + C2^JN2 + 0 (N547/416(log N)3'26) ,
where
f ~ -(x)dx , /(2(a -1) 1 - 1 \ 2
C2 - \ - lim ^--I ----- - 2l—1\-12 - 21 + 2^i + 1
Ji x2 a^2+\ a - 1 (a - 2)2 a - 2 J
7 71
uTAu = (log2N + 27logN + c3) N2 + O (NN547/416(logN)3-26^ , (30)
where c3 = c1 — 2c2 =72 +7 — 71 — 1. Trivial estimates show that one has
uTAf = (1(2N2 + 0((1 + logN )N).
1 2
'T Aw = (log2 N + 27 logN + c3 — (?) N2 + 0 (n547/416 (log N )3-26^
= C4N2 + O (N547/416(logN)3'26) , (31)
w
where C4 = C3 - 72 = 7 -71 - 1 = 0.57721566... - 0.07281584 ... - 1 = -0.495600... (see [10]. Since (26) implies N > ||w||2 > N/10, we find, using (26), (31), and Ravleigh's principle that:
The coefficient of N in this upper bound may well be close to optimal: when N = 10321, for
-0.493678...
an estimate of the value of X1/N in this case. By reasoning similar to that which gives (20), we may deduce from (18), (25) and (32) that, as N ^ ro, we have IX2I/N < (1 + o(1))(R - c4)1/2 ~ ~ 0.2855539 ... and (e1 ■ W)2 > (0.5 + o(1))(1 + (2c24(3-1 - 1)1/2) ~ 0.8540699.... Therefore, for N sufficiently large, the lines {tw : t G R} and {te1 : t G R} will meet at an angle of less than n/8 radians.
We end this section with some speculations driven by certain numerical evidence, gathered with the help of 'GNU Octave'. We omit the detailed evidence, and instead just summarise what it suggests. Let k be any fixed non-zero integer, and let N now be free to vary in the range N > |fc|. Our numerical evidence suggests that X{-k/N}N ~ AkN as N ^ ro, where Ak is a real number that depends only on k, and where each of the two associated sequences, A1, A2, A3,... and -A-1, -A-2, -A-3,... , decreases monotonicallv, and converges to 0. Further numerical evidence suggests that if d G (0,1) is fixed, and if ej,e denotes the l-th component of the normalised eigenvector ej, so that ej = ( ej,1, ej,2,..., ej,N )T for j = 1,...,N, then as N ^ ro we appear to see that
with Ek here being a certain real function independent of l and N that is continuous on (0,1], and with an integer exponent b(N, k) independent of l. The occurrence of the functions E±1, E±2, E±3,... in this might be explained if they were eigenfunctions of a suitable linear operator A :L2[0,1] ^L2[0,1].
3. Various Decompositions of mTAm in the principal case
It is our hope (as vet unrealised) that a study of the quadratic form vTAv (particularly when v is the vector m = (/J.(1),..., p.(N))T), in the principal case of (5), might lead to new results about the Mertens function M(x). In this section we briefly describe (and compare) several different approaches to such an investigation, each involving a different decomposition of the quadratic form.
M( , x) M( , x)
complex number, (rather than a function), to mean M(g, x) for the power function g(n) = n-s.
We consider firstly (12) with v = m. We assume throughout that N is large. As the eigenvalue XN is exceptionally large among all the eigenvalues of A, we handle the term XN(eN ■ m)2 with
-e N e N
+
(32)
e{-k/N} N,1 ^ (-1) b(N,k)N-1/2Ek (l/N) for l = [9 N] + 1, [ON] +2,...,N,
(eN ■ m)2 = ((eN - f) ■ m)2 + 2 ((eN - f) ■ m) (f ■ m) + (f ■ m)2
Here
and, by the Cauchv-Schwarz inequality and (20),
I(eN - f) ■ m| < HeN - f || ■ ||m|| =0
By these results, together with (33) and (17), we have:
\N (eN ■ m)2 = 0 (N log2N) + O (n 3/2(logN)\M (1,N + N2(M (1,N ))2 . (34)
Small eigenvalues make a relatively insignificant contribution here, for (13) and (19) imply that if 1 <K < N/2, then
N-K A7 N A, Ar2
. 2 N v-^ . .2 N .. ..2 N2 m)2 < -= V (e k ■ m)2 = -= ||m||2 <
Z IXkI (e k • m)2 <
k=K VK n=1
vK VK
k=
By this, and by (34) and (12) (for v = m), we find that
= (M(1,N))2 + 01 m\\2/N) £ (Xk/N)(ek • m)2
\<k<N min{k,W-k}<K
-1
+ o(k-1/2 + N-1/2(logN)IM(1,N)I + N-1 log2N^ , (35)
for K = 1, 2,... ,N2. We remark that, if the second of the three terms on the right-hand side of (35) is considered in isolation, then we observe trivially from (19) that the absolute value of this term is 0(VK). Taking account of the context here (the relation (35) and the principal case of (5) and (3)), and noting also that IM(1,N)| < ||m||2/N (a consequence of (11), the trivial bound |[y] — yl < 1, and the fact that [N/1] — ( N/1) =0), it is clear that this term is a bounded function of the pair ( N, K) £ N2. This gives some idea of the gap that must be bridged if (35) is to help in the study of M (x).
To reach (35) we have used the work of Section 2, on Xn and e^. Our next decomposition of mTim avoids such results, but nevertheless has much in common with (35). First we use [x] = x — 2 — ^(x), where ^(x) = {x} — 2- We have
A = N2 ffT — 1 uuT + Z , (36)
where Z is the N x N matrix of elements zmn = —ip(N2/(mn)), whilst f and u are as in Section 2. We have trivially Tr( Z2) < N2/A] with the help of (25), (30), and an estimate for (j, we obtain the sharper result that Tr(Z2) ~ c5N2 as N ^ <x, where c5 = ft + 4 + c3 — i2 = 0.0815206....
K = 1, 2, . . . , N2
mNAm=(mf)2—+mN2m w
= (M (1,N — MNl
+ 01 m||2/N) £ (A,/N) (e* • m)2 + 0 (K-1/2) , (38)
1<k<N min{ k, N +1-k}<K
where A1 < X2 < • • • < An are the eigenvalues of Z, while e1,..., eN form the corresponding orthonormal basis of eigenvectors. We note the presence of the term — 2.N-2(M(N))2 in (38), which is not apparent in (35): in view of our results on Problem (d) of Section 2, one may regard this term as being an approximation to the term (||m||2/N)(X1/N)(e1 • mm)2 = N-2X1(e1 • m)2, which is present in (35) for K > 1.
We remark that (37) permits an alternative, non-spectral, decomposition of mTAm, through substituting the usual truncated Fourier expansion of the function ^ into each element of the matrix Z in (37):
-№= E + 0 (-n \ e „) h = 1h > 1).
) nh + min{|x - ll : l G %}) 1 ' 1
This leads (via estimates from [11]) to the decompositions
T r, H mT Z(h)m (N2(logN)2 logH\ ,r TT A7 .
mTZm = ^-+ 0 i -v 6 HJ--— j (for H = 1, 2,...,N (say)),
h=1 n \ /
where Z(h) is the N x N matrix with elements zmn(h) = sin(27rhN2/(mn)). We have vet to explore making proper use of this truncation idea.
m T Am
equation (A.8) of [4]. We apply Perron's formula as in Lemma 3.12 of [12], adapting the proof to sharpen certain error terms (parts of the improvement come from results of Shiu [11]). We find that if, whenever Re( s) > 1, one has
^(*) = E f' = f^ m7) f^ £) ^ = A(s)B(s)C(s) (say), (39)
1=1 \m<y J \n<z J
where y,z > ^d am, Rn denote complex constants of modulus less than or equal to 1, then, for any fixed e > 0, when x = yz, in the ranges 1 < c < 2 and 3 <T < x1-£, we have
c+i T
_L + Fis)x- ((1 = Eai + 0(+0.(x(ogxT(ogT>) .
s ^ \(c - 1)T
l<x v v '
A
(40)
E°i = E E E Ea™R« = EE Ea™R™ = EE [mdamR
l<x l<x m<y n<z k m<y n<z k m<y n<z
mnk=l mnk<x
1
Setting c = 1 + (logx)-1 in (40), we shift the contour of integration there until it aligns with the line Re(s ) = ¿i in so doing, we pick up a contribution from the residue of ((s) at its pole, s = 1, and also some remainder terms, which are integrals along the line segments joining 1 + iT to c + iT, and 2 - iT to c - iT. By Theorem 7.2 (A) of Titchmarsh [12], we deduce that these remainder term integrals are of size 0(x(\ogx)2^logT/T) for almost all values of T (in a certain sense) lying in any given 'dyadic interval' [To, 2T0] C [3, 2x1-£]. Hence we arrive at the conclusion that, for any given e > 0 when x = yz and 3 <T0 < x1-£, we have
2 ~ 3 N
EE [mk\a^ = A(1)B(1)x + 2- j A(s)B(s)C(s)x°^ +0£(x logx
T
T
for some T G [T0, 2T0]. We specialise this to the case e = 1/2, y = z = N, where N is a positive integer, so that x = N2, and an = Rn = ^(n). We find that when 3 < T0 < N, there exists some T G [T0,2T0] such that
mTAm = M{l,Nyf + MP fT <1»"a1 +;j) (Mii+hp.)2 (
A 0 \ V(l||m|| J
^^ N J-T (n + 2ni t) \ VC1||m||
+ 0 (T0-1 log3N) . (41)
2
If we put E( s) = (1-, 2-s,..., N-S)T fa a fixed complex number s, then the factor M(2 + it,N)/(\/Ci||m||) here may be written as E(2 + it) • mi: the decomposition in (41) may therefore be considered similar in form to that in (35), although (41) involves an integration over the range [—T, T], instead of the summation over a subset of the (discrete) spectrum of A that we had in (35).
4. Conclusions
Using the principal case of (5), and results such as (35), (38), or (41), we are able to approximate M(N2) by ад expression involving only certain limited data: the numbers ц(1),ц(2),..., ц(N) and either the relevant eigenvalues and eigenvectors, or else values of ((1 + it) and gt(n) = n-1 -u.
M( N2)
studying the function M (x). With regard to (35) and (38), it would be helpful to find out more about the relevant eigenvalues and eigenvectors, since that might clarify the possible uses of those results. More generally, it may be worthwhile to study the eigenvalues and eigenvectors of certain
A = A( N) A( , N)
its submatrices.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. J.-P. Cardinal. Symmetric matrices related to the Mertens function // Linear Algebra Appl. 432 (2010), 161-172.
2. D.R. Heath-Brown. Prime Numbers in Short Intervals and a Generalised Vaughan Identity // Can. J. Math. 34, no. 6 (1982), 1365-1377.
3. M.N. Huxley. Exponential Sums and Lattice Points III // Proc. London Math. Soc. (3), 87 (2003), 591-609.
4. A. Ivic. The Riemann Zeta-Function / Dover Publications, Mineola, New York (2003).
5. H. Iwaniec and E. Kowalski. Analytic Number Theory / A.M.S. Colloquium Publications 53, American Mathematical Society, Providence RI, 2004.
6. Yu. V. Linnik. All large numbers are sums of a prime and two squares V // Mat. Sbornik Nov. Ser. 52 (94) (1960), 561-700.
7. Yu. V. Linnik. All large numbers are sums of a prime and two squares II' // Mat. Sbornik Nov. Ser. 53 (95) (1961), 3-38.
8. E. Meissel. Observationes quaedam in theoria numerorum //J. Reine Angew. Math., 48 (1854), 301-316.
9. H.L. Montgomery and R.C. Vaughan. Multiplicative Number Theory I. Classical Theory // Cambridge Studies in Advanced Mathematics 97, Cambridge University Press (2007).
10. On-Line Encyclopedia of Integer Sequences // "Sloane's", http://oeis.org
11. P. Shiu. A Brun-Titchmarsh theorem for multiplicative functions //J. Reine Angew. Math., 313 (1980), 161-170.
12. E.C. Titchmarsh (revised by D.R. Heath-Brown). The Theory of the Riemann Zeta-function / Oxford Univ. Press, 1986
13. R.C. Vaughan. An Elementary Method in Prime Number Theory // Recent Progress in Analytic Number Theory, vol. 1 (Durham, 1979), Academic Press, London - New York, 1981, pp. 341-348.
REFERENCES
1. J.-P. Cardinal, 2010 "Symmetric matrices related to the Mertens function", Linear Algebra Appl. 432, 161-172.
2. D.R. Heath-Brown, 1982, "Prime Numbers in Short Intervals and a Generalised Vaughan Identity", Can. J. Math., 34, no. 6, 1365-1377.
3. M.N. Huxley, 2003, "Exponential Sums and Lattice Points III", Proc. London Math. Soc. (3), 87, 591-609.
4. A. Ivic, 2003, The Riemann Zeta-Function, Dover Publications, Mineola, New York.
5. H. Iwaniec and E. Kowalski, 2004, Analytic Number Theory, A.M.S. Colloquium Publications 53, American Mathematical Society, Providence RI.
6. Yu. V. Linnik, 1960, "All large numbers are sums of a prime and two squares I", Mat. Sbornik Nov. Ser. 52 (94), 561-700.
7. Yu. V. Linnik, 1961, "All large numbers are sums of a prime and two squares II", Mat. Sbornik Nov. Ser. 53 (95), 3-38.
8. E. Meissel, 1854, "Observationes quaedam in theoria numerorum", J. Reine Angew. Math., 48, 301-316.
9. H.L. Montgomery and R.C. Vaughan, 2007, Multiplicative Number Theory I. Classical Theory, Cambridge Studies in Advanced Mathematics 97, Cambridge University Press.
10. On-Line Encyclopedia of Integer Sequences ("Sloane's"), http://oeis.org
11. P. Shiu, 1980, "A Brun-Titchmarsh theorem for multiplicative functions", J. Reine Angew. Math., 313, 161-170.
12. E.C. Titchmarsh (revised by D.R. Heath-Brown), 1986, The Theory of the Riemann Zeta-function, Oxford Univ. Press.
13. R.C. Vaughan, 1981, "An Elementary Method in Prime Number Theory", Recent Progress in Analytic Number Theory, vol. 1 (Durham, 1979), Academic Press, London - New York, pp. 341-348.
Получено 01.06.2018 Принято к печати 10.10.2018
