Euler Product Asymptotics on Dirichlet L-Functions 3

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Before stating our results, let us look back to a few traditional results. Let χ (mod q) be a primitive character, then L(s,χ) satisﬁes the functional equation (see Chapter 9 of [Dav00])

q s/2 s + ν Λ(s,χ) := Γ L(s,χ)= ǫ(χ)Λ(1 s, χ), π 2 − where ν = (1 χ( 1))/2 and ǫ(χ) = i−ν q−1/2τ(χ) with τ(χ) = χ(a)e a being the Gauss sum. − − a (mod q) q The completed L-function Λ(s,χ) has a meromorphic continuation to C and is entire if χ = 1. Throughout P this paper φ(q) denotes the number of Dirichlet characters (mod q). Let π(x; q,a) be the number6 of primes p up to x belonging to the arithmetic progression a (mod q); particularly we set π(x) = π(x;1, 1). Such an abbreviation also applies to other functions below. Then the prime number theorem for arithmetic progressions shows that in any residue class a (mod q), the primes are eventually equidistributed amongst the plausible arithmetic progressions (mod q), ie π(x) π(x; q,a) ∼ φ(q) as x whenever (a, q) = 1 and q > 1. An important question here is as to uniformity in q, which is solidly→ related ∞ to distribution of zeros of L(s,χ). Further, the Siegel-Walfisz theorem [Sie35, Wal36] says that 1 (1.2) π(x; q,a)= Li(x)+ O(x exp( c log x)) φ(q) − p for any q 6 (log x)A where c and the implied constant depend on A alone (not effectively computable if A > 2). Yet, it is indispensable to weaken the restriction on q for numerous applications. Indeed, the assumption of the GRH yields (1.2) in a much wider range q 6 √x/(log x)2+ǫ. Since we assume the GRH throughout this paper, such restriction on q always adheres to our argument. It is conjectured that the following tighter estimates are available: 1 π(x; q,a)= Li(x)+ O(x1/2+ǫ) φ(q) uniformly for q 6 x1/2−2ǫ, or perhaps even the Montgomery hypothesis [Mon76] 1 (1.3) π(x; q,a)= Li(x)+ O(q−1/2x1/2+ǫ) φ(q) for q 6 x1−2ǫ. As is customary, we denote ϑ(x; q,a)= log p and ψ(x; a, q)= Λ(n), p6x n6x p≡r X(mod q) n≡r X(mod q) where Λ(n) is the usual von Mangoldt function, and whence we define the remainder term x E(x; q,a)= ϑ(x; q,a). φ(q) − It is well-known that the estimate E(x; q,a) √x(log x)2 is tantamount to the GRH. With this notation, the Bombieri-Vinogradov theorem| ([Bom65,| Vin65, ≪ Vin66]) states that Theorem 1.1. Fix A> 0. For all x > 2 and all √x(log x)−A 6 Q 6 √x, we have

5 (1.4) max max E(y; q,a) A √xQ(log x) . y6x (a,q)=1 | | ≪ 6 qXQ This shows that the remainder terms max(a,q)=1 E(x; q,a) behave as if they satisfy the GRH in an average sense. See also [FI10] and references therein for many applications. Although the GRH is not capable of showing (1.2) in a range q > x1/2 it was conjectured by Elliott and Halberstam [EH70] that the left hand side of (1.4) with Q = x1−ǫ is bounded by x(log x)−A. It is well known that [FG89] their conjecture fails at the endpoint, that is to say Q = x. EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS 3

In order to capture the asymptotic behaviour of (1.1) as x tends to inﬁnity, we shall follow an exquisite idea of Ramanujan [Ram97] and adapt its technique (such as appealing to a precise version of the explicit formula for ψ(x)) to the present problem. An approach via this explicit formula is a peculiarity of his method. That was originally created in the process of scrutinising the maximal order of the divisor function

−s σ−s(n)= d Xd|n by introducing generalised superior highly composite numbers. Note that he unemphatically assumed the Riemann Hypothesis (RH), whence certain sums over nontrivial zeros of the Riemann zeta function ζ(s) was estimated nicely. Ramanujan’s result we now utilise is contained in the rest of Ramanujan’s paper [Ram15] in 1915 entitled ‘Highly Composite Numbers’, and the rest was published later in 1997 ([Ram97]). There are two intriguing manuscripts by him on sums involving primes (but these are handwritten by G. N. Watson). These can be found on pages 228–232 in [Ram88]. The original manuscripts by Ramanujan are in the library at Trinity College, Cambridge. Several claims in the ﬁrst original manuscript are fallacious, indicating that it emanates from an earlier portion of Ramanujan’s career sometime before he departed for England in March 1914. Several valuable comments on his argument are written in [AB13, Rob91]. In order to state the result of Ramanujan, let

xρ−s S (x)= s , s − ρ(ρ s) X − where the sum runs over the nontrivial zeros of the Riemann zeta function ζ(s). He ascertained the following: Theorem 1.2 (Ramanujan [Ram97]). If the RH for ζ(s) is assumed, we then have Case I. 0

Case II. s =1/2: 1 −1 1 1+ S (x) 1 1 = √2 ζ exp Li( ϑ(x)) + 1/2 + O ; − √p − 2 log x (log x)2 p6x Y p Case III. s> 1/2: 2sx1/2−s S (x) x1/2−s (1 p−s)−1 = ζ(s) exp Li(ϑ(x)1−s)+ + s + O . − | | (2s 1) log x log x (log x)2 6 pYx − This is a rather close approximation and a cruder one can be derived by materials flourished in analytic number theory now available. The √2 phenomenon says that the extra √2 factor arises only at s = 1/2. One of striking points in Ramanujan’s method is that it suddenly emerges from the formula (1.5) lim (Li(1 + h) log h )= γ. h→0 − | | c Generically, as for Hasse-Weil type zeta functions, √2 is to be replaced by √2 with c being the order of the pole of the second moment Euler product L(2)(s)= L(s, Sym2)/L(s, 2) at s = 1. Conrad figured out that ∧ the appearance of this factor is governed by ‘second moments’. The role of √2 was borne out in Section 4 of his paper [Con05] in detail; thereby it turned out that the √2 arises from some series rearrangement. There have been some conjectural and related topics beyond the scope of the GRH. In fact, Theorem 1.2 is related to the Deep Riemann Hypothesis (DRH)—a refined version of the GRH, and an outstanding open 4 IKUYA KANEKO problem that may play an important role in development of the GRH. In general, one can see that the DRH manifests the GRH. The DRH states, in short, overconvergence or divergence behaviour of partial Euler products in the critical strip 1/2 6 s < 1 (in the present situation the DRH is concerning convergence of (1.1)). ℜ Throughout this paper we assume that χ is primitive. The aim of this paper is to generalise Theorem 1.2 and deliberate upon further applications. Theorem 1.3. Let χ be a primitive Dirichlet character (mod q). Denote s = σ + it with s> 0, ℜ 1 if χm = 1 (1.6) ηm(q,a)= (0 otherwise and s xρ−s S (x, χ)= χ(a) ψ(a) , s −φ(q) ρ(ρ s) a (modX q) ψ (modX q) L(ρ,ψX)=0 − where the sum over ψ runs over all Dirichlet characters (mod q) and the innermost sum is over nontrivial zeros ρ of L(s,χ) for each χ. If the GRH for a Dirichlet L-function attached with χ is assumed, for q 6 √x/(log x)2+ǫ we then have Case I. 0 < s< 1/2: ℜ 1 1 log (1 χ(p)p−s)−1 = χ(a)Li((φ(q)ϑ(x; q,a))1−s) Li(x1−φ(q)s) − φ(q) − φ(q) − · · · 6 pYx a (modX q) 1 (2s 1+ η (q,a))x1/2−s S (x, χ) x1/2−σ Li(x1−ns)+ − 2 s + O − n (1 2s) log x − log x (log x)2 − with n the largest multiple of φ(q) not exceeding [1 + 1/2σ];

Case II. s =1/2: ℜ (1 χ(p)p−s)−1 − 6 pYx 1/2−s 1 1−s x + Ss(x, χ) 1 = L(s,χ)exp χ(a)Li((φ(q)ϑ(x; q,a)) )+ + O 2 φ(q) log x (log x) ! a (modX q) √2 s =1/2 and χ2 = 1 η (q,a)x1−2s(2xs−1/2 1) ; × exp 2 − otherwise  2(2s 1) log x − Case III. s> 1/2:  ℜ 1 (1 χ(p)p−s)−1 = L(s,χ)exp χ(a)Li((φ(q)ϑ(x; q,a))1−s) − φ(q) 6 pYx a (modX q) (2s 1+ η (q,a))x1/2−s S (x, χ) x1/2−σ + − 2 + s + O . (2s 1) log x log x (log x)2 − ! The proof of Theorem 1.3 occupies the bulk of the paper and it is carried out in Sections 2–4. Overcon- vergence of (1.1) is discussed in Section 5 in connection with reducing E(x; q,a). Notice that if we do not assume the GRH, the precision of resulting asymptotic formula gets worse embarrassingly. Putting the exponential multipliers in Theorem 1.3 aside, it is expected that for χ = 1 6 (1.7) lim (1 χ(p)p−s)−1 = L(s,χ) on the half-plane s> 1/2. x→∞ − ℜ 6 pYx EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS 5

It is confirmed by Conrad [Con05] that the GRH for L(s,χ) is equivalent to (1.7). As a strengthening of (1.7), the DRH asserts the following for the case of Dirichlet L-functions. Conjecture 1.4 (DRH for Dirichlet L-functions). If χ = 1 and a complex number s is on the critical line s =1/2 with m the order of vanishing of L(s,χ), we have6 ℜ L(m)(s,χ) √2 if s =1/2 and χ2 = 1 (1.8) lim (log x)m (1 χ(p)p−s)−1 = . x→∞  −  emγm! × otherwise 6 (1 pYx This is crisper than the GRH and seems to be far-reaching; for several L-functions, extensive numerical experiments ([KKK14, KK18]) have confirmed the DRH. The second moment of L(s,χ) over Q is L(s,χ2) for non-principal character χ, so in this case, the second moment hypothesis is satisfied, with a simple pole at s = 1 when χ is quadratic and with neither zeros nor poles on s = 1 otherwise. The theorem of Conrad [Con05] tells us that the following two statements are equivaℜlent: the limit in the left hand side of (1.8) exists for some s on s = 1/2 and it exists for every s on s = 1/2. This kind of fact seems to be not applicable only for s on ℜs = 1/2 but also in the critical strip.ℜ Moreover the conjecture (1.8) is known to be equivalent to ℜ (1.9) E(x; q,a)= o(√x log x) as is stated in [Con05, Theorem 6.2] (in which the modulus q seems to be fixed). This is indeed much stronger than what we know under the GRH. Case II of Theorem 1.3 shows that the DRH holds when the remainder term is bounded as (1.9). We point out that that figures illustrating (1.8) can be found in Section 3 of [KKK14]. Generally, Conrad [Con05] analysed partial Euler products for a typical L-function along its critical line, and demystified the √2 factor in the general context. He found the equivalence between the Euler product k k asymptotics and the estimate ψ (x) := k (α + + α ) log Np = o(√x log x) which is stronger L Np 6x p,1 ··· p,d than the GRH. Given an elliptic curve E/Q with N the conductor, Kuo and Murty [KM05] established the P equivalence between the Birch and Swinnerton-Dyer conjecture (see, eg [BSD63, BSD65, Bir65, Wil06]) and the growth condition n6x cñ = o(x). Herec ñ indicates that k k P αp +βp k + k n = p , p is a prime, k Z , p ∤ N cñ = ∈ (0 otherwise 1 with αp and βp the Frobenius eigenvalues at p. Akatsuka [Aka17] has very recently studied the DRH for ζ(s)—capturing the divergent behaviour of the partial Euler product by means of a modified logarithmic integral. Taking account of a simple pole of ζ(s), the DRH is equivalent to the estimate ϑ(x) x = o(√x log x). The function field analogues are known to be true and this is powerful evidence in favor− of the DRH ([KKK14, KS14]). Nowadays the proof of the DRH over function fields becomes the epitome of examining the behaviour of partial Euler products for various L-function. On the ground of the analogue of the RH for Selberg zeta functions (characteristic zero), the Euler product asymptotics was brought out in the recent paper [KK18]. In the forthcoming paper [KKK19], we prove the DRH for the Artin L-function attached with a nontrivial irreducible representation of Gal(K/K¯ ) as well as for the standard L-function of an n-dimensional nontrivial cuspidal automorphic representation π = 6 π of GL (A ). ⊗p ∞ p n K 2. Prelude to the asymptotics As explained in the introduction, our approach is based upon the aforementioned method provided by Ra- manujan. Unsurprisingly, for Dirichlet L-functions, the discussion becomes somewhat complicated than that for ζ(s). Before executing our Euler product asymptotics, we note that in his method, there is ambiguity on derivation of a so-called ‘constant term’. Conrad’s theorem [Con05, Theorem 3.1] supplements the ambiguity so that this constant can be specified.

1Unfortunately, as of January in 2019, we should note that a serious misunderstanding lies in the review of Akatsuka [Aka17] in the MathSciNet, where the reviewer states that Akatsuka studies one of equivalence conditions to the RH. 6 IKUYA KANEKO

Following Ramanujan, it is appropriate that we begin with considering the partial summation that if Φ′(x) is continuous between q1 and x, x Φ(q ) log q + Φ(q ) log q + + Φ(q ) log q = Φ(x)ϑ(x; q,a) Φ′(t)ϑ(t; q,a)dt 1 1 2 2 ··· n n − Zq1 with q < q < < q being an ascending sequence of consecutive primes of the form qm + a and q 1 2 ··· n 1 (resp. qn) standing for the smallest (resp. largest) prime below x of such form. Integrating by parts gives x 1 x x Φ(x)ϑ(x; q,a) Φ′(t)ϑ(t; q,a)dt = const+ Φ(t)dt Φ(x)E(x; q,a)+ Φ′(t)E(t; q,a)dt − φ(q) − Zq1 Zq1 Zq1 where ‘const’ depends solely on Φ, q and a. In what follows let us assume the GRH for Dirichlet L-functions, which permits us to work with q 6 √x/(log x)2+ǫ. Taylor’s theorem then yields that φ(q)ϑ(x;q,a) x 1 Φ(t)dt = Φ(t)dt Φ(x)φ(q)E(x; q,a)+ (φ(q)E(x; q,a))2Φ′(x + O(√x(log x)2)). − 2 Zq1 Zq1 Gathering together these equations we have (2.1) Φ(q ) log q + Φ(q ) log q + + Φ(q ) log q 1 1 2 2 ··· n n 1 φ(q)ϑ(x;q,a) x = const + Φ(t)dt + Φ′(t)E(t; q,a)dt φ(q) Zq1 Zq1 1 φ(q)E(x; q,a)2Φ′(x + O(√x(log x)2)), − 2 Let us call this underlying equation (2.1) ‘Ramanujan sum formula’. One sees plenty of asymptotic formulae and estimates each time as long as a suitable test function Φ is chosen.

3. Partial Euler products for Dirichlet L-functions Let χ be a primitive character (mod q). We now exploit the Ramanujan sum formula (2.1) φ(q) times and incorporate them each other. For our purpose, let us suppose that Φ(x) = χ(a)/(xs χ(a)) for each a with (a, q) = 1. We assume, for the sake of simplicity, that s> 0 throughout this paper.− Hence we derive ℜ χ(2) log 2 χ(3) log 3 χ(5) log 5 χ(p) log p (3.1) + + + + 2s χ(2) 3s χ(3) 5s χ(5) ··· ps χ(p) − − − − 1 φ(q)ϑ(x;q,a) χ(a)dt = const+ s φ(q) q1 t χ(a) a (modX q) Z − x E(t; q,a)dt −s 4 s χ(a) 1−s s 2 + O(x (log x) ), − q1 t (t χ(a)) a (modX q) Z − with the sums above being over a with (a, q) = 1, and p being the largest prime below x. Furthermore, by invoking the explicit formula (eg cf. [IK04]) for ψ(t; q,a) subject to (a, q) = 1, we obtain (3.2)

φ(q)E(t; q,a)= t φ(q)ϑ(t; q,a) − tρ tρ/2 = δ (q,a)√t + δ (q,a)t1/3 + ψ(a) ψ(a) + O(t1/5), 2 3 ρ − ρ ψ (modX q) L(ρ,ψX)=0 ψ (modX q) L(ρ,ψX)=0 where δ (q,a) = # x (mod q): xm a (mod q) , m { ≡ } the outer sums over ψ range over all Dirichlet characters (mod q) and the inner sums are over nontrivial zeros of L(s, ψ). Notice that the number of primitive Dirichlet characters (mod q) is given by dr=q µ(d)φ(r), which follows from Lemma 1 of [Sou07]. By the Chinese remainder theorem, δ (q,a) is multiplicative with 2 P EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS 7

n2 n3 n5 np1 2 respect to q if q = 2 3 5 p1 . To be more precise, by counting carefully the solutions to x a ··· a ≡ (mod q), we infer that for a with q =1 4 if n2 > 3

np ω(q)−1 2 if n2 =2 δ2(q,a)= δ(p ,a)=2  , × 1 if n2 =1 p  Y 2 otherwise  where ω(q) is the number of diﬀerent prime factors of q and satisﬁes ω(q) log q/ log log q, and ω(q) 6 (1/ log 2 ǫ) log log q for almost all q. Then it turns out that the contribution≪ from the fourth term of the right hand− side of (3.2) is estimated as

x s+ρ/2−1 ρ/2−s 1 x dx x 1/4−s ψ(a) s 2 x ρ q1 (x χ(a)) ≪ ρ(ρ/2 s) ≪ ψ (mod q) L(ρ,ψ)=0 Z − ψ (mod q) L(ρ,ψ)=0 − X X X X s −2 −2s −3s for every 1 6 a 6 q. Inasmuch as (x χ(a)) = x + O(x ), the contribution from the third term of the right hand side of (3.2) becomes −

1 x xs+ρ−1dx χ(a) ψ(a) s 2 ρ q1 (x χ(a)) a (modX q) ψ (modX q) L(ρ,ψX)=0 Z − xρ−s = const+ χ(a) χ(a) + O(x1/2−2s). ρ(ρ s) χ ρ a (modX q) X X − Hence recalling that s xρ−s S (x, χ)= χ(a) ψ(a) , s −φ(q) ρ(ρ s) a (modX q) ψ (modX q) L(ρ,ψX)=0 − we arrive at the following asymptotic formula:

χ(2) log 2 χ(3) log 3 χ(p) log p (3.3) + + + 2s χ(2) 3s χ(3) ··· ps χ(p) − − − φ(q)ϑ(x;q,a) x 1/3 1 χ(a)dt δ2(q,a)√t + δ3(q,a)t = const+ s s χ(a) 1−s s 2 dt φ(q) q1 t χ(a) − q1 t (t χ(a)) ! a (modX q) Z − Z − 1/2−2s 1/4−s + Ss(x, χ)+ O(x + x ).

4. Logarithmical approach to Euler products For further computation we produce the following idiosyncratic orthogonality: Lemma 4.1. Let χ be a Dirichlet character to the modulus q > 1 and let m > 1 be a positive integer. We then have φ(q) if χm = 1 χ(a)δm(q,a)= . (0 otherwise a (modX q) Proof. The proof relies on the identity

m χ(a)δm(q,a)= χ(a) a (modX q) a (modX q) to which we then apply the usual orthogonality. 8 IKUYA KANEKO

We now recall the quantity (1.6). One appeals to Lemma 4.1 after expanding the integrands of the two integrals in the right hand side of (3.3) respectively. By truncating the unnecessary order, the sum of the contributions from these integrals can be calculated as 1 χ(a) χ(a)2 const + (φ(q)ϑ(x; q,a))1−s + (φ(q)ϑ(x; q,a))1−2s φ(q) 1 s 1 2s a (modX q) − − x1−φ(q)s x1−ns 2η (q,a)sx1/2−s 3η (q,a)sx1/3−s 4η (q,a)sx1/2−2s + + + 2 3 2 + O(x1/2−2s), 1 φ(q)s ··· 1 ns − 1 2s − 1 3s − 1 4s − − − − − where n is the largest multiple of φ(q) not exceeding [2 + 1/(2s)]. From this, there follows that χ(2) log 2 χ(3) log 3 χ(p) log p (4.1) + + + 2s χ(2) 3s χ(3) ··· ps χ(p) − − − L′(s,χ) 1 χ(a) χ(a)2 x1−φ(q)s = + (φ(q)ϑ(x; q,a))1−s + (φ(q)ϑ(x; q,a))1−2s + + − L(s,χ) φ(q) 1 s 1 2s 1 φ(q)s ··· a (modX q) − − − x1−ns 2η (q,a)sx1/2−s 3η (q,a)sx1/3−s 4η (q,a)sx1/2−2s + 2 3 2 + S (x, χ)+ O(x1/2−2σ + x1/4−σ). 1 ns − 1 2s − 1 3s − 1 4s s − − − − Note that const = L′(s,χ)/L(s,χ) is justified by the result of Conrad [Con05]. If we consider the Euler products at s =1, 1−/2, 1/3or 1/4, we must take the limit of the right hand side of (4.1) when s gets close to 1, 1/2, 1/3or 1/4. In order to end our asymptotics, we have to replace s = σ + it with u + it and integrate the asymptotic formula (4.1) in u once from to σ. We deduce that for n defined above, ∞ 1 (4.2) log (1 χ(p)p−s)= log L(s,χ)+ χ(a)Li((φ(q)ϑ(x; q,a))1−s) − − φ(q) 6 pYx a (modX q) η (q,a) η (q,a) 1 2 Li(x1−2s)+ 2 Li(x1/2−s) Li(x1−φ(q)s) − 2 2 − φ(q) − · · · 1 x1/2−s + S (x, χ) x1/2−σ Li(x1−ns) s + O , − n − log x (log x)2 where we have used that σ xa+b(u+it) 1 σ S (x, χ) x1/2−σ du = Li(xa+bs) and S (x, χ)du = s + O . a + b(u + it) b u+it − log x (log x)2 Z∞ Z∞ Exponentiating (4.2) and classifying it into three parts 0 < s < 1/2, s = 1/2 and s > 1/2 finishes the proof of Theorem 1.3. ℜ ℜ ℜ

5. Observing the size of E(x; q,a) We can describe the special case where the asymptotic formula is valid for all s lying on s =1/2. ℜ Theorem 5.1. If the GRH is assumed, for any χ and s C such that χ = 1, s =1/2, then we have ∈ 6 ℜ (log x)m (1 χ(p)p−s)−1 − 6 pYx L(m)(s,χ) 1 x1/2−s + S (x, χ) = exp χ(a)Li((φ(q)ϑ(x; q,a))1−s)+ s + O((log x)−2) emγ m! φ(q) log x  a (modX q)   √2 if s =1/2 and χ2 = 1 × (1 otherwise for q 6 √x/(log x)2+ǫ, where γ is the Euler constant. EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS 9

Proof. We omit the proof, remarking that the proof rests on exploiting the special case of the result due to Conrad ([Con05, Theorem 5.11]), that is B e−mγ (1 χ(p)p−s)−1 t − ∼ (log x)m 6 pYx where Bt is the leading Taylor coeﬃcient of L(s,χ) at s =1/2+ it. To ascertain that the partial Euler products (1.1) for every non-principal character converge on the critical line s =1/2, we require the estimate ℜ (5.1) χ(a)Li((φ(q)ϑ(x; q,a))1−s) 0 → a (modX q) on the line. But one fails because the bound φ(q)ϑ(x; q,a)= x + O(√x(log x)2) asserts that Li((φ(q)ϑ(x; q,a))1−s) = Li(x1−s)+ O(log x). As expected, the deeper estimate φ(q)ϑ(x; q,a)= x + o(√x log x) yields the limit (5.1) on s =1/2. Conse- quently we infer, with attention to Conjecture 1.4, that our present knowledge of the sumℜ in (5.1) puts the bound (1.9) out of reach.

For notational convenience, we now introduce the twisted counting function 1 ψ(x, χ)= χ(n)Λ(n) so that ψ(x; q,a)= χ(a)ψ(x, χ). φ(q) 6 nXx χ (modX q) Let us shift the topic to improving the bound on ψ(x, χ). We should point out that, putting the hypothesis (1.3) aside, Montgomery [Mon80] provided a reason to believe that the true order of magnitude of ψ(x) x is at most √x(log log log x)2. Specifically, He conjectured that − ψ(x) x 1 (5.2) lim − = √x(log log log x)2 ±2π in the process of considering the hypothesis that the nontrivial zeros of ζ(s) are all simple and (under the RH for ζ(s)) the imaginary parts of the nontrivial zeros are linearly independent over Q. These simplicity and linear independence hypothesis can fail for general L-functions, eg the L-function of an elliptic curve might have a multiple zero at the real point on its critical line. When the L-function is primitive (this terminology was first proposed in [RS96]), it still seems reasonable to believe that ψ(x, χ) √x(log log log x)2 (more 2 ≪ generally ψL(x) √x(log log log x) ), which would imply ψ(x, χ) = o(√x log x). It would be fascinating if these estimates≪ on ψ(x, χ) is related to statements about the vertical distribution of zeros on the line s =1/2. Incidentally, Gallagher and Mueller [GM78] succeeded in showing that the RH and Montgomery’s ℜpair correlation conjecture (in the general form) yields ψ(x) = x + o(√x(log x)2) by tailoring their analysis to the Weyl-van der Corput technique for evaluating exponential sums. In another paper, Gallagher made a magnificent achievement for estimating ψ(x) x; under the RH, he used an explicit formula for ψ(x) to refine the estimate ψ(x) x √x(log x)2 coming− from the RH to ψ(x) x √x(log log x)2 off a closed subset of [2, ] with finite− logarithmic≪ measure. His argument is trivially− applicable≪ to ψ(x, χ) and other psi-functions.∞

6. Appendix We devote this section to explain several prospects on the DRH (in particular for the Riemann zeta function). In view of the discussion in this paper, there seems to exist a connection between the DRH and Mertens’ conjecture on the distribution of the summatory function of the M¨obius function, viz. 1 if n =1, M(x)= µ(n) with µ(n)= 0 if n is not squarefree, n6x k ( 1) if n is squarefree and n = p1 ...pk. X −  10 IKUYA KANEKO

This is closely related to the reciprocal of the Riemann zeta function, since it follows that 1 ∞ ∞ = µ(n)n−s = s M(x)x−s−1dx ζ(s) n=1 1 X Z for s> 1, and ℜ 1 c+i∞ xs M(x)= ds 2πi sζ(s) Zc−i∞ valid for c> 1 and x / Z. The best known unconditional bound on M(x) is as follows [Ivi85]: ∈ M(x) x exp( c (log x)3/5(log log x)−1/5) ≪ − 1 for some c1 > 0. In passing, the RH is equivalent to the estimate c log x M(x) √x exp 2 ≪ log log x for some c2 > 0 (see [Tit87]). It is very interesting to compare Gonek’s unpublished work on bounding M(x) with Montgomery’s conjecture (5.2) on (ψ(x) x)/√x. − Conjecture 6.1 (Gonek’s conjecture). There exists a number B > 0 such that M(x) lim = B. √x(log log log x)5/4 ± It was noticed in [Ng04] that, in the early 1990’s, Gonek had annunciated this conjecture at several conferences. The exponent of the iterated triple logarithm is 2 in Montgomery’s conjecture, whereas it is 5/4 in Conjecture 6.1. This difference actually stems from the different discrete moments of 1 1 (log T )2 and (log T )5/4, ρ ≍ ρζ′(ρ) ≍ 6 6 γXT | | γXT | | where the second asymptotics is currently conjectural. Notice that the constant 5/4 is precisely due to the following Gonek-Hejhal conjecture: Conjecture 6.2 (Gonek-Hejal [Gon89, Hej89]). For k R, let ∈ 1 J (T )= . −k ζ′(ρ) 2k 6 0<γXT | | We then have 2 J (T ) T (log T )(k−1) . −k ≍ Based on our experience for the estimation of ψ(x) x, the equivalence between some strong bound on M(x) and the DRH should be derived upon supposing− something like J (T ) T . Also, Conjecture 6.1, in −1 ≪ which M(x) is replaced by M(x, χ)= n6x χ(n)µ(n), may be connected with a matter of understanding the size of ψ(x, χ). More generally, we have to treat instead the moment 1/ L′(f,ρ) 2k, where L(f,s) is P 0<γ6T | | an ‘analytic’ L-function associated to some arithmetic object f as handled in [Con05]. To this topic, we may P revisit elsewhere. We remark the relation between M(x) and the summatory function of the Liouville function λ(n), where λ(n) = ( 1)Ω(n) with Ω(n) indicating the total number of prime factors of n. The Liouville function is a completely− multiplicative function and it is easy to see that ∞ ζ(2s) λ(n)n−s = ζ(s) n=1 X for s> 1. If we define ℜ L(x)= λ(n), 6 nXx EULER PRODUCT ASYMPTOTICS ON DIRICHLET L-FUNCTIONS 11 then from partial summation we can derive

∞ ∞ λ(n)n−s = s L(t)t−s−1dt. n=1 1 X Z From numerical data, P´olya conjectured that the inequality L(x) 6 0 always persisted. It should be pointed out that P´olya’s conjecture implies the RH. However, Haselgrove [Has58] proved that this conjecture cannot be true. In this way, the DRH would generally be related to properties of

k k αp + + α b(n)λ(n) with b(n)= ,1 ··· p,d k 6 k nXx NXp =n for some αp,j’s such that αp,j 6 1 (the notation is similar as in [Con05]). Finally, in the paper of| Gonek-Keating| [GK10], it has been noted that one should have

1 2T 1 2k 1 2T 1 2k ζ + it dt Px + it dt T T 2 ∼ T T 2 Z Z for suﬃciently large values of x (in the process of pondering the Splitting Conjecture in [GHK07]). Here Λ(n) P (s) = exp x ns log n 6 6 2 Xn x with remarking that ∞ 1 Λ(n) lim = log √2. x→∞ kpk/2 − √n log n 6 6 6 pXx Xk=1 2 Xn x Hence the famous moment conjecture [KS00] would then imply that

2k 2T 2 1 −1/2−it −1 k (6.1) (1 p ) dt akgkT (log T ) , T T − ∼ Z p6x Y where we have denoted as G2(k + 1) g = k G(2k + 1) with G(s) the Barnes G-function, and

2 ∞ 1 k Γ(m + k) 2 a = 1 p−m . k − p m!Γ(k) p m=0 ! Y X 2 γ k Gonek-Hughes-Keating [GHK07] showed that the left hand side of (6.1) is asymptotically ak(e log x) for 2 6 x (log T )2−ǫ. If we can improve on such an upper bound on x, it turns out that the extended version of the≪ their consequence does not coincide with the conjecture (6.1). The aforementioned reasoning should be extended to other families of L-functions, such as ‘analytic L-functions’.

Acknowledgement This paper is an outgrowth of the article of Kaneko-Koyama [KK18], where we have investigated the partial Euler products of Selberg zeta functions in the critical strip, and the connection between the Prime Geodesic Theorem and the region of overconvergence of the products. The author would like to appreciate his advisors Nobushige Kurokawa and Shin-ya Koyama for helpful information and advice. 12 IKUYA KANEKO

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Tsukuba Kaisei High School, 3315-10 Kashiwada, Ushiku, 300-1211 Japan E-mail address: [email protected]