The structure of multiplicative functions with small partial sums

The Landau-Selberg-Delange method provides an asymptotic formula for the partial sums of a multiplicative function whose average value on primes is a fixed complex number $v$. The shape of this asymptotic implies that $f$ can get very small on average only if $v=0,-1,-2,\dots$. Moreover, if $v<0$, then the Dirichlet series associated to $f$ must have a zero of multiplicity $-v$ at $s=1$. In this paper, we prove a converse result that shows that if $f$ is a multiplicative function that is bounded by a suitable divisor function, and $f$ has very small partial sums, then there must be finitely many real numbers $\gamma_1$, $\dots$, $\gamma_m$ such that $f(p)\approx -p^{i\gamma_1}-\cdots-p^{-i\gamma_m}$ on average. The numbers $\gamma_j$ correspond to ordinates of zeroes of the Dirichlet series associated to $f$, counted with multiplicity. This generalizes a result of the first author, who handled the case when $|f|\le 1$ in previous work.


Introduction
Throughout this paper f will denote a multiplicative function and L(s, f ) = ∞ ∑ n=1 f (n) n s will be its associated Dirichlet series, which is assumed to converge absolutely in Re(s) > 1. We then have for certain coefficients Λ f (n) that are zero unless n is a prime power. Let D denote a fixed positive integer. We shall restrict attention to the class of multiplicative functions f such that for all n. This is a rich class of functions that includes most of the important multiplicative functions that arise in number theory. For example, the Möbius function, the Liouville function, divisor functions, and coefficients of automorphic forms (or if one prefers an axiomatic approach, L-functions in the Selberg class) satisfying a Ramanujan bound are all covered by this framework. When f (p) ≈ v in an appropriately strong form, Selberg [7] built on ideas of Landau [4,5] to prove that where c( f , v) is some a non-zero constant given in terms of an Euler product. Delange [1] strengthened this theorem to a full asymptotic expansion: This problem was studied in the case D = 1 by the first author [3]. If (1.3) holds for some A > 2, then by partial summation one can see that L(s, f ) converges (conditionally) on the line Re(s) = 1. The work in [3] established that on the line Re(s) = 1 the function L(s, f ) can have at most one simple zero. If L(1 + it, f ) = 0 for all t, then lim x→∞ 1 π(x) ∑ p≤x f (p) = 0, DISCRETE ANALYSIS, 2020: 6, 19pp. while if L(1 + iγ, f ) = 0 for some (unique) γ ∈ R then lim x→∞ 1 π(x) ∑ p≤x ( f (p) + p iγ ) = 0.
In this paper we establish a generalization of this result for larger values of D.
Theorem 1. Fix a natural number D and a real number A > D + 2. Let f be a multiplicative function such that |Λ f | ≤ D · Λ, and such that for all x ≥ 2. Then there is a unique multiset Γ of at most D real numbers such that is a constant depending only on f and T , and C 2 is an absolute constant. In particular, The multiset Γ consists of the ordinates of the zeros of L(s, f ) on the line Re(s) = 1, repeated according to their multiplicity. Its rigorous construction is described in Proposition 2.4. The constant C 1 = C 1 ( f , T ) in Theorem 1 can be calculated explicitly in terms of upper bounds for the Dirichlet series L(s, f ) ∏ γ∈Γ,|γ|≤T ζ (s − iγ) and its derivatives, together with a lower bound for this quantity on the line segment Qualitatively Theorem 1 establishes the kind of converse theorem that we seek. There are two deficiencies in the theorem: first, the range A > D + 2 falls short of the optimal result A > D + 1 (which in the case D = 1 was attained in [3], and which we can attain in a special case -see Section 5); and second, one would like an understanding of the uniformity with which the result holds. On the other hand, the proof that we present is very simple, and we postpone the considerably more involved arguments needed for more precise versions of the theorem to another occasion. Definition 2.1. Given a natural number D, we denote by F(D) the class of all multiplicative functions such that |Λ f | ≤ D · Λ.
Lemma 2.2. Let f be an element of F(D). Then its inverse under Dirichlet convolution g is also in F(D), and both | f (n)| and |g(n)| are bounded by τ D (n) for all n.
Proof. Note that so that by comparing coefficients This shows that | f (n)| ≤ τ D (n) for all n. Since the inverse g may be defined by setting Λ g (n) = −Λ f (n), it follows that g is in F(D) and that |g(n)| ≤ τ D (n) as well.
For later use, let us record that if f ∈ F(D), then for σ > 1 we have We now introduce the class F(D; A), which is the subclass of multiplicative functions in F(D) with small partial sums.
The class F(D; A) consists of all functions lying in F(D; A, K) for some constant K.
The following proposition about the class F(D; A) is an important stepping stone in the proof of Theorem 1. In particular, it gives a description of the multiset Γ appearing in Theorem 1.    We next establish the following lemma which contains part (a) of Proposition 2.4 and more. The remaining parts of the proposition will be established in Section 4. In particular, the series L ( j) (s, f ) converges uniformly in compact subsets of the region Re(s) ≥ 1. Furthermore, it satisfies the pointwise bound for s = σ + it with σ ≥ 1 and t ∈ R.
Proof. Since | f | ≤ τ D by Lemma 2.2, all claims follow in the region Re(s) ≥ 2 from the bound ∑ n>N τ D (n)/n 2 D (log N) D−1 /N. Let us now assume we are in the region 1 ≤ Re(s) ≤ 2. Using partial summation, we have We estimate both terms on the right-hand side of (2.5) using our assumption on the partial sums of f , thus obtaining that This establishes (2.3). In particular, L ( j) (s, f ) converges uniformly in compact subsets of the region 1 ≤ Re(s) ≤ 2 by Cauchy's criterion.
To obtain (2.4), we let M → ∞ in (2.3) to find that 2, the first term on the right side of (2.6) is bounded in size by

Two lemmas
Here we collect together a couple of disparate lemmas that will be used in the future. Both of them are of a standard nature, and proofs are provided for completeness. We begin with an asymptotic formula for partial sums of generalized divisor functions.
. . , α m } be a multiset, consisting of k distinct elements, and arranged so that α 1 , . . . , α k denote these k distinct values. Suppose that these distinct values α j appear in A with multiplicity m j . Let τ A (n) denote the multiplicative function Then for large x we have where P j,A denotes a polynomial of degree m j − 1 with coefficients depending on A, and δ = δ (A) is some positive real number.
Proof. Note that in the region Re(s) > 1 Now the lemma follows by a standard application of Perron's formula to write (with c > 1) and then shifting contours to the left of the 1-line and evaluating the residues of the poles of order m j at 1 + iα j .
Our second and final lemma gives a variant of the Brun-Titchmarsh theorem for primes in short intervals. Define Λ j (n) by means of Thus Λ 0 (n) = 1 if n = 1 and 0 for n > 1, while Λ 1 (n) = Λ(n) is the usual von Mangoldt function. Using the identity one can check easily that Λ j (n) ≥ 0 for all j and n. In addition, Λ j (n) is supported on integers composed of at most j distinct prime factors, and is bounded by C j (log n) j on such integers for a suitable constant C j .
Proof. We argue by induction on j. The base case j = 1 is a direct corollary of the classical Brun-Titchmarsh inequality (for example, see [6, Theorem 3.9]). Now suppose that j ≥ 2 and that the lemma holds for Λ 1 , . . . , Λ j−1 .
The number of integers in (x, x + y] all of whose prime factors are ≥ √ y may be bounded by y/ log y ε y/ log x (see [6,Theorem 3.3]). Therefore, with P − (n) denoting the least prime factor of the integer n, we have that To establish the lemma, it remains to show that Let p be a prime and suppose n = p a m with a ≥ 1 and p m. Note that If a = 1, then we deduce that On the other hand, if a > 1, then we use the bound Λ j− (m) j (log m) j− to conclude that We now return to the task of estimating (3.3), using the above two estimates. Let p denote the smallest prime factor of n, so that p ≤ √ y. The terms with p n contribute, using the induction hypothesis and (3.4), Lastly, using (3.5), we find that the terms with p 2 |n contribute Combining (3.6) and (3.7) yields (3.3), completing the proof of the lemma.

Proof of Proposition 2.4
Recall that part (a) of Proposition 2.4 was already established in Lemma 2.5. We now turn to the remaining three parts of the proposition, with the next lemma settling part (b).
Proof. As σ → 1 + , Taylor's theorem 1 shows that But since Re( f (p)p −iγ ) ≥ −D for all p, relation (2.2) implies that Therefore L ( j) (1 + iγ, f ) = 0 for some 0 ≤ j ≤ D, and the notion of multiplicity is well defined. If m ≤ D denotes the multiplicity, then a new application of Taylor's theorem gives Writing σ = 1 + 1/ log x and taking logarithms, we find that as desired.
We now turn to the task of proving part (c) of Proposition 2.4. Suppose 1 + iγ 1 , . . . , 1 + iγ k are distinct zeros of L(s, f ), and let m j denote the multiplicity of the zero 1 + iγ j . We wish to show that m 1 + · · · + m k ≤ D, so that part (c) would follow. A key role will be played by the auxiliary function where N is an integer that will be chosen large enough. By expanding the (2N)-th power, it is easy to see that A N (x) is non-zero only for those real x that may be written as j 1 γ 1 +· · ·+ j k γ k with | j 1 |+· · ·+| j k | ≤ N. Note that there may be linear relations among the γ j , so that A N (x) could have a complicated structure. The following lemma summarizes the key properties of A N (x) for our purposes. Proof. (a) Let γ 0 = 0. Then, the multinomial theorem implies that By restricting our attention on the 'diagonal' terms with j i = j i for all i, we infer that When | j i − N/k| ≤ √ N for all i, Stirling's formula implies that the corresponding binomial coefficient has size k (k + 1) N N −k . Since there are k N k/2 tuples ( j 0 , j 1 , . . . , j k ) ∈ Z k+1 ≥0 with j 0 + j 1 + · · · + j k = N, the claimed lower bound on A N (0) follows readily.
provided that N is large enough. Hence, if T is sufficiently large,
Proof of Proposition 2.4(c). Let N be a large integer to be chosen later, and consider the behavior of as x → ∞. We will estimate this quantity in two distinct ways, one of which will produce a lower bound and another one which will produce an upper bound. Comparing these bounds will then show that m 1 + · · · + m k ≤ D.
For the lower bound on λ N (x), we note that our assumption that | f (p)| ≤ D for all primes p implies that with the second relation following from the Prime Number Theorem.
The number of choices of j 1 , . . ., j 2N that lead to γ = γ is exactly A N (γ ). If γ is not γ for some 1 ≤ ≤ k, then by Lemma 4.1 we see that the sum over p is bounded above by a constant. Indeed if γ is not an ordinate of a zero of L(s, f ) on the 1-line, then the sum over p is simply O(1); a priori, there could be other zeros of L(s, f ) besides 1 + iγ 1 , . . ., 1 + iγ k and γ could be one of these zeros, but nevertheless the sum over p is bounded above by O(1). In conclusion, Comparing the above inequality with (4.2), we infer that To complete the proof, we apply Lemma 4.2(b) with ε = 1/(m 1 + · · · + m k + 1) to find that the left-hand side of (4.3) is > A N (0)(m 1 + · · · + m k − 1), as long as N is large enough. Since A N (0) > 0, we conclude that m 1 + · · · + m k < D + 1, as desired.
It remains lastly to prove part (d) of Proposition 2.4. Suppose that the multiset Γ consists of k distinct values, and has been arranged so that γ 1 , . . . , γ k are these distinct values, and each such γ j occurs in Γ with multiplicity m j . As in Lemma 3.1, put τ Γ (n) = ∑ d 1 ···d m =n d iγ 1 1 · · · d iγ m m and define f Γ to be the Dirichlet convolution f * τ Γ .
for some constant C( f ), and with m denoting the maximum of the multiplicities m 1 , . . . , m k .
Proof. As in the hyperbola method we may write, for some parameter 2 ≤ z ≤ √ x to be chosen shortly, Using our hypothesis on the partial sums of f , and since √ x ≤ x/z ≤ x/b, we see that the second term above is since |τ Γ (b)| may be bounded by the D-th divisor function. On the other hand, Lemma 3.1 implies that there is some δ = δ ( Γ) > 0 such that the first term equals where P j, Γ denotes a polynomial of degree m j − 1 with coefficients depending on f and Γ. Since | f (a)| is bounded by the D-th divisor function, the error term in (4.5) is easily bounded by x(log x) D /z δ . Now consider the main term in (4.5). Applying (2.3) (with N = x/z and M → ∞ there), for any 0 ≤ ≤ m j − 1 we have and we conclude that the quantity in (4.5) is Combine this with (4.4), and choose z = exp((log log x) 2 ) to obtain the lemma.
Combining Lemma 4.3 with the argument of Lemma 2.5, part (d) of Proposition 2.4 follows.

Proof of Theorem 1 in a special case
In this section we establish Theorem 1 in the special case when L(s, f ) has a zero of multiplicity D, say at 1 + iγ. By Proposition 2.4 there can be no other zeros of L(s, f ) on the 1-line. In this special case, we can in fact prove a stronger result, obtaining non-trivial information in the optimal range A > D + 1. In the next section, we shall consider (by a very different method) the remaining cases when the multiplicity of any zero is at most D − 1.
Write g(n) = f (n)n −iγ , and consider G = τ D * g. We begin by establishing some estimates for ∑ n≤x G(n) and ∑ n≤x |G(n)|/n. Note that G(n) = n −iγ f Γ for the multiset Γ composed of D copies of γ. Hence, Lemma 4.3 and partial summation imply that Since |D + g(p)| 2 = D 2 + 2DRe(g(p)) + |g(p)| 2 ≤ 2D(D + Re(g(p))), an application of Cauchy-Schwarz gives After these preliminaries, we may now begin the proof of Theorem 1 in this situation. We shall consider the function G * G = τ 2D * g * g. Note that Λ G * G (n) = 2DΛ(n) + Λ g (n) + Λ g (n) is always real and non-negative. Thus G * G is also a real and non-negative function, and we have We bound the right side above using the hyperbola method. Thus, using (5.1) and (5.3), and using |D + g(p)| 2 ≤ 2D(D + Re(g(p))) and Cauchy-Schwarz we conclude that Once the estimate (5.4) has been established, it may be input into the above argument and the bound (5.4) may be tidied up. Partial summation starting from (5.4) leads to the bound ∑ p≤x |D + g(p)|/p f 1 in place of (5.2). In turn this replaces (5.3) by the bound ∑ n≤x |G(n)|/n f 1. Using this in our hyperbola method argument produces now the cleaner bound As mentioned earlier, the estimate (5.5) obtains non-trivial information in the optimal range A > D + 1. If we suppose that A > D + 2, then the right side of (5.5) is f x/ √ log x, and Theorem 1 follows in this special case if |γ| ≤ T . If |γ| > T then note that so that the theorem holds as stated in this case also.
Our goal now is to bound the left-hand side of (6.1), and to do this we split the integral into several ranges. There is a range of small values |t| ≤ T , and the range of larger values |t| > T , which we further subdivide into dyadic ranges 2 r T < |t| ≤ 2 r+1 T with r ≥ 0.

Small values of |t|
We deduce that for a suitable constant C 1 ( f , T ).

Large values of |t|
Now we turn to the larger values of |t|, namely when 2 r T < |t| ≤ 2 r+1 T for some r ≥ 0. Writing the desired integral splits naturally into two parts. Now for |t| ≥ T we have It remains now to estimate To help estimate this quantity, we state the following lemma whose proof we postpone to the next section.
Lemma 6.1. Let X ≥ 2 and σ > 1 be real numbers. Let f ∈ F(D) and suppose j ≥ 1 is a natural number.
Returning to (6.4), in the notation of Lemma 6.1, we have Using this identity, the integral in (6.4) splits into two parts, and using Lemma 6.1 we may bound the second integral (with X = 2 r T ) by x X(X/T 0 ) 10 X<|t|≤2X |G 1 (c + it)| 2 dt x X(X/T 0 ) 10 X(log X) 2 + log x x log x 2 r T 0 . (6.5) Finally, we must bound the integral arising from G 2 (s). To this end, we define x s s e s/T 0 − 1 s/T 0 10 ds , (6.6) so that we require a bound for I(2; 2 r T, 0). We shall bound I( j; X, α) in terms of I( j + 1; X, α + β ) for suitable β > 0, and iterating this will eventually lead to a good bound for I(2; X, 0). Lemma 6.2. Let X ≥ T , and α ≥ 0 be real numbers. For j ≥ 2 and all k ≥ 1 we have I( j; X, α) k Using this in the definition of I, we obtain that I( j; X, α) 1 0 I( j + 1; X, α + β )dβ + x X(X/T 0 ) 10 X + 1 0 2X X |G 1 (c + α + β + it)G j (c + α + β + it)|dtdβ .
Thus we conclude that I( j; X, α) .
We now return to the task of bounding I(2; 2 r T, 0). Applying Lemma 6.2, we see that for any k ≥ 1 we have I(2; 2 r T, 0) k 1 0 I(2 + k; 2 r T, β )β k−1 dβ + x log x 2 r T 0 . (6.9) We choose k to be the largest integer strictly smaller than A − 3. Since A > D + 2, we have Applying Lemma 2.5, we find that L (2+k) (c + β + it, f ) (1 + |t|). Furthermore, since f ∈ F(D), we have 1 Thus, with this choice of k, it follows that Since k ≥ D − 1, we infer that 1 0 I(2 + k; 2 r T, β )β k−1 dβ x log x 2 r T 0 , by a similar argument to the one leading to (6.8). In conclusion, I(2; 2 r T, 0) x log x 2 r T 0 .
Combining this with (6.5), and summing over all r ≥ 0, we obtain Combining (6.11) with (6.1) and (6.2), it follows that Partial summation now finishes the proof of Theorem 1.
7 Proof of Lemma 6.1 Write g j (n) for the Dirichlet series coefficients of G j (n). We claim that |g j (n)| ≤ D j Λ j (n) for all n. For j = 1 this is just the definition of the class F(D). To see the claim in general, we use induction on j, noting that G j+1 (s) = −G j (s) + G 1 (s)G j (s), (7.1)