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On the divisor function and the Riemann zeta-function in short intervals. (English) Zbl 1226.11086

The divisor function \(d(n):=\sum_{\delta|n}1\), i.e., the number of (positive) divisors of \(n\) and \[ \zeta(s):=\sum_{n=1}^{\infty}n^{-s}\quad (\sigma=\text{Re}(s)>1), \] the Riemann zeta-function (see present author’s book of 1985 [The Riemann zeta-function. The theory of the Riemann zeta-function with applications. New York etc.: John Wiley & Sons (1985; Zbl 0556.10026)]) have many features in common. For example, \[ \zeta^2(s)=\sum_{n=1}^{\infty}d(n)n^{-s}\quad (\sigma=\text{Re}(s)>1), \tag{\(*\)} \] as can be easily seen from elementary (or analytic, or other) arguments. In the present paper, the author studies the mean-squares of short-interval remainders linked to \(\zeta(s)\) and to \(d(n)\), i.e., \[ E(T):=\int_{0}^{T}\left| \zeta\left({1\over 2}+it\right)\right|^2 dt-TM_1(\log T), \] where \(M_1\) is a 1-degree polynomial, defined as \(M_1(u):=u-\log(2\pi)+2\gamma-1\), with \(\gamma \approx 0.577\ldots\) the Euler-Mascheroni constant and \[ \Delta(x):=\sum_{n\leq x}d(n) - xP_1(\log x), \] where \(P_1\) is another 1-degree polynomial, very similar to \(M_1\), since \(P_1(u):=u+2\gamma-1\), \(\gamma\) as before.
Actually, since \(r(n):=\sum_{n=a^2+b^2}1\), i.e. the number of ways \(n\) can be written as a sum of squares, is \(r(n)=4\sum_{d|n}\chi(d)\), classically, where \(\chi\) is the unique non-principal Dirichlet character modulo \(4\), we have an analogy with \(d(n)=\sum_{d|n}1\) and, in fact, say \[ P(x):=\sum_{n\leq x}r(n)-\pi x, \] the remainder in the so-called Gauss circle problem, enters the game as a companion of \(\Delta(x)\); this, in turn, because of \((*)\) above, is linked to the so-called Hecke eigenvalues (see the Iwaniec-Kowalski “bible”), say \(a(n)\), through \[ A(x):=\sum_{n\leq x}a(n) \] (the classical example is \(a(n)=\tau(n)\), Ramanujan’s tau-function).
The author gives
1) asymptotic formulae for the “long” mean-squares (both continuous and discrete) of the “short interval” errors \(\Delta(x+U)-\Delta(x)\), in suitable ranges (see Theorem 1 and Corollary 1);
2) asymptotic formulae for the long (continuous) mean-squares of the short interval’s \(P(x+U)-P(x)\) and \(A(x+U)-A(x)\), in suitable ranges (see Theorem 2);
3) \(\Omega-\)results (say, \(f=\Omega(g)\) means \(f=o(g)\) does not hold) for: \(\Delta(x+U)-\Delta(x)\) and \(E(x+U)-E(x)\), in some ranges (Corollary 2) and, also, for: \(P(x+U)-P(x)\) and \(A(x+U)-A(x)\), in some ranges (Corollary 3);
4) two technical results, even of independent interest (but applied here to the other results’ proofs), i.e. about \(|\zeta(1/2+it)|\) large values (Theorem 3) and moments (Corollary 4);
5) finally, upper bounds for the \(4\)th moments of \(E(x+U)-E(x)\) and \(\Delta(x+U)-\Delta(x)\) (see Theorem 4).
We explicitly remark (as the author does) that the results of Theorem 1 and Corollary 1 improve the reviewer’s and S. Salerno’s results of 2004 [Acta Arith. 113, No. 2, 189–201 (2004; Zbl 1122.11062)]; these were proved by elementary arguments, through a Large Sieve estimate; while, here, the author substantially improves them, relying on a 1984 result of M. Jutila [Ann. Univ. Turku., Ser. A I 186, 23–30 (1984; Zbl 0536.10032)] that, in turn, rests on a kind of “truncated Voronoï summation formula”.
In fact, formulae of this kind are available also for \(r(n)\) and \(a(n)\), see the above; hence giving, adapting Jutila mean-square calculations, that Theorem 2 may be proved in a similar fashion as 1) results.
The technical results quoted in point 4) above are part of the well-known experienced skill of the author with the Riemann \(\zeta\)-function (see his 1985 masterpiece about it).
Finally, Theorem 4 results, see 5) above, are the first steps towards proving (since they prove it in a subrange of the \(U-\)variable) a deep Conjecture of Jutila. This, in fact, would (better, will, once reached) prove the “weak sixth moment” for the Riemann zeta-function, i.e., \[ \forall \varepsilon>0 \quad \int_0^T \left|\zeta\left({1\over 2}+it\right)\right|^6 dt \ll_{\varepsilon} T^{1+\varepsilon}. \tag{Weak 6th Moment} \] These few words are not enough to describe exhaustively the rich wealth of results coming from this paper.

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11N37 Asymptotic results on arithmetic functions

References:

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