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In 2017, Aistleitner, Munsch and the second author [2] used the resonance method to prove that there is a constant C such that γ max ζ 1 it ≥ e log2 T log3 T C . (2) t √T ,T ∈[ ] Note that this result essentially matchesS ( + )S (1), however,( + the size+ of) the interval here is much larger. Unfortunately, over shorter intervals T, T H , very little seems to be known regard- ing large values of ζ 1 it (see [5], [6] for further details). [ + ] In this paper, we( generalize+ ) (2) to a large class of L-functions, namely G, which conjec- turally contains the Selberg class S. We establish (2) for elements in G with non-negative Dirichlet coefficients. The key difference between G and S is that elements in G satisfy a polynomial Euler-product which is a more restrictive condition than that in S. However, the functional equation in S is replaced by a weaker “growth condition” in G. This is a signifi- cant generalization because most Euler products, which have an analytic continuation exhibit a growth condition, but perhaps not a functional equation. As applications, we prove the analogue of (2) for Dedekind zeta-functions ζK s and Rankin-Selberg L-functions given by L s,f f . We also prove a generalized Merten’s theorem for G as a precursor to the proof of our main theorem. ( ) ( × ) The resonance method with a similar resonator was used by Aistleitner, Munsch, Peyrot and the second author [3] to establish large values of Dirichlet L-functions L s,χ with a given conductor q at s = 1. Perhaps, a similar method can also be used to establish large values over more general orthogonal families of L-functions in G. ( )
1.1. The class G. In 1989, Selberg [20] introduced a class of L-functions S, which is expected to encapsulate all naturally occurring L-functions arising from arithmetic and geometry. Definition 1.1 (The Selberg class). The Selberg class S consists of meromorphic functions F s satisfying the following properties. (i) Dirichlet series - It can be expressed as a Dirichlet series ( ) ∞ aF n F s = Q s , n=1 n ( ) ( ) which is absolutely convergent in the region R s > 1. We also normalize the leading coefficient as aF 1 = 1. (ii) Analytic continuation - There exists a non-negative( ) integer k, such that s 1 kF s is an entire function( ) of finite order. ( − ) ( ) LARGE VALUES OF L-FUNCTIONSON1-LINE 3
(iii) Functional equation - There exist real numbers Q > 0 and αi ≥ 0, complex numbers βi and w ∈ C, with R βi ≥ 0 and w = 1, such that s ( Φ) s = Q SMS Γ αis βi F s (3) i satisfies the functional equation( ) ∶ ( + ) ( )
Φ s = wΦ 1 s . (iv) Euler product - There is an Euler( ) product( − of) the form F s = M Fp s , (4) p prime where ( ) ( ) b k F s ∞ p log p = Q ks k=1 p kθ ( ) with bpk = O p for some θ < 1 2. (v) Ramanujan hypothesis - For any ǫ > 0, ( ) ~ ǫ aF n = Oǫ n . (5)
The Euler product implies that theS coefficients( )S ( aF) n are multiplicative, i.e., aF mn = aF m aF n when m,n = 1. Moreover, each Euler factor also has a Dirichlet series repre- sentation ( ) ( ) ( ) ( ) ( ) a pk F s ∞ F , p = Q ks k=0 p ( ) which is absolutely convergent on R s (>)0 and non-vanishing on R s > θ, where θ is as in iv . ( ) ( ) ( For) the purpose of this paper, we need a stronger Euler-product to ensure that the Euler factors factorize completely and further require a zero free region near 1-line, similar to what we notice in the proof of prime number theorem. However, we can replace the functional equation with a weaker condition on the growth of L-functions on vertical lines. This leads to the definition of the class G. Definition 1.2 (The class G). The class G consists of meromorphic functions F s satisfying (i), (ii) as in the above definition and further satisfies (a) Complete Euler product decomposition - The Euler product in (4)( ) factorizes completely, i.e., 1 k α p − F s j = M M 1 s (6) p = p j 1 ( ) ( ) ∶ − with αj ≤ 1 and R s > 1. (b) Zero-free region - There exists a positive constant cF , depending on F , such that F s ShasS no zeros in( ) the region
cF ( ) R s 1 , ≥ log I s 2 ( ) − except the possible Siegel-zero of F s . (S ( )S + ) ( ) 4 ANUPB.DIXITANDKAMALASKHYAMAHATAB
(c) Growth condition - For s = σ it, define λ µF∗ σ = inf +λ > 0 F s ≪ t 2 . Then, ( ) ∶ { ∶ S ( )S (S S+ ) } µF∗ σ 1 2σ ( ) is bounded for σ < 0. − One expects S to satisfy conditions a and b . In fact, the Riemann zeta-function, the Dirichlet L-functions, the Dedekind zeta-functions and the Rankin-Selberg L-functions can be all shown to satisfy conditions a (and) b .( Furthermore,) for elements in S the growth condition c is a consequence of the functional equation (3). However, it is possible to have L-functions not obeying a functional( ) equation( ) to satisfy the growth condition. One can consider linear( ) combination of elements in S to see this. A family of L-functions based on growth condition was introduced by V. K. Murty in [17] and the reader may refer to [11] for more details on this family. Also the Igusa zeta-function, and the zeta function of groups have Euler products but may not have functional equation, which is discussed in [19].
1.2. The Main Theorem. In this paper, we produce a lower bound for large values of L- functions in G on the 1-line. For a meromorphic function F s having a pole of order m at s = 1, define m ( ) c m F = lim s 1 F s . (7) − s→1
Theorem 1.3. Let F ∈ G have non-negative( ) Dirichlet( − ) coefficients( ) aF n and a pole of order m at s = 1. Then, there exists a constant CF > 0 depending on F s such that ( ) γF m max F 1 it ≥ e log2 T log3 T ( C)F , t∈ √T ,T [ ] S ( + )S (( + ) − ) where γF = mγ log c m F and γ is the Euler-Mascheroni constant. − In the above+ theorem,( since) aF n ≥ 0, we clearly have m ≥ 1. This is important because if F has no pole at s = 1, it is possible for F s to grow very slowly on the 1-line. ( ) As an immediate corollary, we get the following( ) result for Dedekind zeta-functions ζK s . Let K Q be a number field. The Dedekind zeta-function ζK s is defined on R s > 1 as ( ) 1 ~ 1 1( ) − ( ) ζ s 1 , K = Q Na s = M Np s 0≠a⊆OK p ( ) ∶ − where a runs over all non-zero integral ideals( ) and p runs over( ) all non-zero prime ideals of K. The function ζK s has an analytic continuation to the complex plane except for a simple pole at s = 1. Furthermore, ζK satisfies properties (a), (b), (c) and therefore ζK ∈ G. Thus, by Theorem 1.3, we( have)
Corollary 1. For a number field K, there exists a constant CK > 0 depending on K such that
γK max ζK 1 it ≥ e log2 T log3 T CK , t∈ √T ,T [ ] S ( + )S ( + − ) where γK = γ log ρK, with ρK being the residue of ζK s at s = 1. + ( ) LARGE VALUES OF L-FUNCTIONSON1-LINE 5
The L-function associated to the Rankin-Selberg convolution of any two holomorphic new- forms f and g, denoted by L s,f g , is in the Selberg class. Moreover, it can also be shown that L s,f g ∈ G. Here f and g are normalized Hecke eigenforms of weight k. It is known that if L s,f g has a pole( at s× =)1, then f = g. Hence, from Theorem 1.3, we have the following.( × ) ( × ) Corollary 2. For a normalized Hecke eigenform f, there exists a constant Cf > 0 such that γf max L 1 it,f f ≥ e log2 T log3 T Cf , t∈ √T ,T [ ] where γf = γ log ρf , with ρfS being( + the× residue)S of (L s,f +f at s−= 1. ) The result+ obtained in Theorem 1.3 is a refined( version× ) of the bound established by Aistleitner-Pa´nkowski [4], which states that if F s is in the Selberg class and satisfies the prime number theorem, namely, x ( ) x Q aF p = κ O 2 , p≤x log x log x then for large T , S ( )S + κ max F 1 it = Ω log log T . (8) t∈ T,2T [ ] Furthermore, since we are assumingS ( the+ zero-free)S (( region) in)G, using [13, Theorem 1], we have κ = m. Hence, we get a slightly more refined result than (8), but on a larger interval T , T . √ [ The] poles of any element F in the Selberg class S are expected to arise from the Riemann m zeta-function. More precisely, if F s has a pole of order m at s = 1, then F s ζ s is ex- pected to be entire and in S. Thus, it is not surprising to expect the lower bound in Theorem 1.3 to be of the order log log T m.( ) ( )~ ( )
It is possible to generalize( Theorem) 1.3 to the Beurling zeta-function [8] by constructing a suitable resonator over Beurling numbers instead of integers. However, this will carry us far afield from our current focus. Hence, we relegate it to future research.
2. Mertens’ theorem for the class G In 1874, Mertens [15] proved the following estimate for truncated Euler-product of ζ s , which is also known as Mertens’ third theorem given by 1 ( ) 1 − γ M 1 = e log x O 1 , p≤x p where γ denotes the Euler-Mascheroni − constant. The analogue+ ( ) of Mertens’ theorem for number fields was proved by Rosen [18], who showed that 1 1 − γ M 1 = ρK e log x O 1 , NP Theorem 2.1. Let F s ∈ G. Suppose that F s has a pole of order m at s = 1 and c m F − be as in (7). Then, for a constant 0 < CF ≤ 1, ( ) 1 ( ) ( ) k α p − j γm m CF √log x M M 1 = c m F e log x 1 O e− . p≤x = p − j 1 ( ) Proof. We closely follow − the method of Yashiro( ) [23].( Denote) + by k 1 αj p − F 1; x = M M 1 . p≤x = p j 1 ( ) Let ( ) ∶ − b n F s ∞ F . log = Q s n=1 n ( ) r θ By the complete Euler product (6), we have( ) bF n = 0 if n ≠ p and bF n ≪ n for some θ < 1 2. Since k( ) ( ) r 1 r ~ bF p = Q αj p , r j=1 r we have bF p ≤ k. Write ( ) ( ) b pr S ( )S F x ∞ F log 1; = Q Q r p≤x r=1 p ( ) ( ) b n b pr b pr F F F . = Q Q Q r Q Q r (9) n≤x n < ≤ pr>x p ≤ pr>x p ( ) √x p x ( ) p √x ( ) It is easy to estimate the second and third+ term above as follo+ ws. r bF p ∞ 1 1 1 Q Q r ≪ Q Q r ≪ Q 2 ≪ . < ≤ pr>x p < ≤ r=2 p < ≤ p x √x p x ( ) √x p x √x p x Also, √ r bF p 1 1 Q Q r ≪ Q ≪ . ≤ pr>x p ≤ x x p √x ( ) p √x √ From (9), we get bF n 1 log F 1; x = Q O . n≤x n x ( ) √log x Setting 1 log x = u and e =(T and) using Perron’s+ formula,√ we get u iT s bF n 1 + x cF √log x ( ~ ) Q = S log F 1 s ds O e− . n≤x n 2πi u iT s ( ) − ( + ) + Let u′ = CF log T = CF log x. Choosing x sufficiently large, we can ensure that there are no Siegel zeros for F 1 s √in the region u′, u . Hence from the condition b , F 1 s has no zeros in the~ region u′~≤ R s ≤ u and I s ≤ T and has a pole of order m at s = 0. ( + ) [− ] ( ) ( + ) Consider the contour−C joining( ) u iT, uS ′( iT,)S u′ iT and u iT . By the residue theorem, we have s s x − − − 1− + x + Ress=0 log F 1 s = log F 1 s ds. (10) s 2πi SC s ( + ) ( + ) LARGE VALUES OF L-FUNCTIONSON1-LINE 7 We now estimate the above integral. Suppose s = σ it. By the growth condition c , we have F s t µF σ , ≪ +( ) ( ) where µ σ 1 2σ . Thus, for our choice of u and u , we get for σ u , u ≪ S ( )S S S ′ ∈ ′ log F 1 σ iT log T 2. ( ) ( − ) ≪ [− ] Hence, we have ( + + ) ( ) u′ iT s 2 u′ 1 − + x log T − σ log F 1 s ds ≪ x dσ 2πi Su iT s T Su + ( ) W ( + ) W W √log x W ≪ log x e− c′ √log x ≪ (e− F ) , (11) for some 0 < cF′ < 1. Similarly, we also get u iT s ′ 1 + x c √log x log F 1 s ds ≪ e− F . (12) 2πi S u′ iT s − + We use the following resultV due to Landau( (see+ ) [16,V p. 170, Lemma 6.3]) to esimate the other terms in (10). Lemma 2.2. Let f z be an analytic function in the region containing the disc z ≤ 1, supposing f z ≤ M for z ≤ 1 and f 0 ≠ 0. Fix r and R such that 0 < r < R < 1. Then, for z ≤ r we have ( ) S S S ( )S S S f ′ ( ) 1 M S S z = Q O log , f ρ ≤R z ρ f 0 S S where ρ is a zero of f s . ( ) + − S ( )S m Let f z = z 1 2 ( it) F 1 z 1 2 it , R = 5 6 and r = 2 3 in the above Lemma 2.2. Using the zero-free region b , we get ( ) ( + ~ + ) ( + + ( ~ + )) ~ ~ m log t 4 , t ≥ 7 8 and σ ≥ u′, log s F( 1) s ≪ 1 t ≤ 7 8 and σ ≥ u′. (S S+ ) S S ~ − We now have the estimateS ( + )S ′ S S ~ − u′ iT s T u − + x x− m m log F 1 s ds ≪ log s log s F 1 s dt S u′ s S0 s − c′′ √log x W ( + ) W e F (S, S + S ( + )S) ≪ − S S (13) for some 0 < cF′′ < 1. Similarly, we also have ′ u s ′′ − x c √log x log F 1 s ds ≪ e− F . (14) S u′ iT s − − Using the estimates (11),W (12), (13) and( (14)+ ) inW the Equation (10) and choosing CF = min cF , cF′ , cF′′ , we get u iT s s 1 + x x CF √log x ( ) log F 1 s ds = Ress=0 log F 1 s O e− 2πi Su iT s s − Let C denote the circle of radius( +u′ centered) at 0. Then, ( + ) + 1 xs xs log F 1 s ds = Ress=0 log F 1 s . 2πi SC s s ( + ) ( + ) 8 ANUPB.DIXITANDKAMALASKHYAMAHATAB Hence, it suffices to estimate the above integral. Since F s has a pole of order m at s = 1, m c m F = lim s 1 F s (≠)0. − s→1 m m Writing F s 1 = s− s F s 1( ), we get( − ) ( ) s s ( + ) 1( )(x ( + )) m x log F 1 s ds = log s ds log c m F . (15) 2πi SC s 2πi SC s − The integral on the right hand side( + is) − + ( ) ′ iθ s π u e x x iθ iθ log s ds log u′e iu′e dθ (16) SC = S iθ s π u′e − π π u′eiθ log x u′eiθ log x = i log u′ ( e )( dθ ) θe dθ. (17) S π S π − − By the series expansion of exponential( function,) we have− π π r π u′eiθ log x ∞ u′ log x irθ S e dθ = S dθ Q S e dθ π π r=1 r! π − − ( ) − = 2π + Similarly, π π r π u′eiθ log x ∞ u′ log x irθ θe dθ = θdθ Q θe dθ S π S π = r! S π − − r 1 − (r )r ∞ u′ log+ x 1 2π = Q r=1 r! ir ( r ) u′ log(−x ) 2π ∞ 1 r 1 = Q S w − dw i r=1 r! 0 (−′ ) u log x w 2π e− 1 = dw. i S0 w − But the Euler-Mascheroni constant γ satisfies the identity 1 w w 1 e− ∞ e− γ = dw dw. S0 w S1 w − Thus, we have − u′ log x w u′ log x w e− 1 dw ∞ e− dw = γ dw S0 w S1 w Su′ log x w − C √log x = γ + log log x log−u′ O e− F . (18) Combining the estimates above (16)-(18),+ we get + + CF √log x log F 1; x = log c m F mγ m log log x O e− . − y Taking exponential on( both) sides and( using)+ the+ fact that e += 1 O y for y < 1, we are done. + ( ) S S LARGE VALUES OF L-FUNCTIONSON1-LINE 9 3. Proof of the main theorem For F ∈ G, define k 1 αj p − F s; Y = M M 1 s . ≤ = p p Y j 1 ( ) We use the following approximation( lemma.) ∶ − Lemma 3.1. For large T , 1 F 1 it F 1 it; Y 1 O , = log T 10 10 ( +1 10) ( + ) + for Y = exp log T and T ~ ≤ t ≤ T . ( ) Proof. From(( the Euler) ) product of FS Ss , we have for R s > 1, k k l ( ) αj p ( ) αj p log F s = Q Q log 1 s = Q Q Q ls . p j=1 p p j=1 lp ( ) l ( ) ( ) − − Let t0 > 0 and let α > 0 be any sufficiently large constant. Define 1 10 1 1 T ~ σ0 , σ1 and T0 . = α log T = log T 20 = 2 ∶ ∶ ∶ Applying Perron’s summation formula as in( [21,) Theorem II.2.2], we get σ1 iT0 s k + Y αj p 1 log F 1 it0 s ds log 1 O . S = Q Q 1 it0 10 σ1 iT0 s ≤ j=1 p + log T − p Y ( ) ( + + ) − − + Now, we shift the path of integration to the left. By the zero-free region( of F) ∈ G, the only pole of the above integrand in R s ≥ σ0 and I s ≤ T0 is at s = 0. Therefore, we have k s ( α) j p ( ) 1 Y log F 1 it0 log 1 O log F 1 it0 s ds , (19) = Q Q 1 it0 10 SC ≤ = p + log T s p Y j 1 ( ) C( + ) − − + + ( + + ) where is the contour joining σ0 iT0,σ1 iT0,σ1( iT0 and) σ0 iT0. Since, log F σ it ≪ log t on C, we get − − + + S ( + )S σ0 iT0 Y s log T σ1 iT0 Y s log T − − F it s ds , + F it s ds , log 1 0 ≪ 1 10 log 1 0 ≪ 1 10 (20) Sσ1 iT0 s T S σ0 iT0 s T − ~ − + ~ and ( + + ) ( + + ) σ0 iT0 s 2 − + Y log T log F 1 it0 s ds ≪ 1 , (21) S σ0 iT0 s exp log T 9 − − ( α ) ( + + ) where all implied constants are absolute. Substituting the bounds ( from) (20) and (21) in (19), 1 10 for T ~ ≤ t0 ≤ T , we obtain k αj p 1 log F 1 it log 1 O . 0 = Q Q 1 it0 10 ≤ = p + log T p Y j 1 ( ) ( + ) − − + Similarly we may argue when t0 is negative. ( ) 10 ANUPB.DIXITANDKAMALASKHYAMAHATAB By Lemma 3.1, it suffices to show Theorem 1.3 for F 1 it; Y . We closely follow the argument in [2]. Set 1 ( + ) X log T log T = 6 2 and for primes p ≤ X set ( )( ) p q 1 . p = X Also set q1 = 1 and qp = 0 for p > X. Extend − the definition completely multiplicatively to define qn for all integers n ≥ 1. Now define it 1 R t = M 1 qpp − . p≤X Then we have ( ) ( − ) log R t ≤ Q log X log p p≤X (S ( )S) = π X( log X− ϑ X) , where π X is the prime counting function( and) ϑ X −is( the) first Chebyshev function. By partial summation, we know that ( ) X π t ( ) X π X log X ϑ X = dt = 1 o 1 . S2 t log X ( ) By our choice of X, we( get) − ( ) ( + ( )) 2 1 3 o 1 R t ≤ T ~ + ( ). (22) From the Euler product, R t has the following series representation S ( )S ∞ it ( ) R t = Q qnn , n=1 and hence we get ( ) it 2 ∞ it ∞ it ∞ m R t = Q qnn Q qnn− = Q qmqn . n=1 n=1 m,n=1 n We have S ( )S ( )( ) k it 1 αj p p− − F 1 it; Y = M M 1 ≤ = p p Y j 1 ( ) Since αj p ≤ 1, we get ( + ) − k 10k S ( )S F 1 it; Y ≪ log Y ≪ log T . t2 Set Φ t = e− and recall thatS ( its+ Fourier)S transform( ) is( positive.) Using (22), we have ( ) ∶ 2 log T F 1 it; Y R t Φ t dt ≪ 1, S t ≥T T S S and V ( + )S ( )S V 2 log T 5 6 o 1 F 1 it; Y R t Φ t dt ≪ T ~ + ( ). S t ≤√T T S S V ( + )S ( )S V Using the fact that q1 = 1 and the positivity of the Fourier coefficients of Φ, we also have the following lower bound LARGE VALUES OF L-FUNCTIONS ON 1-LINE 11 T 2 log T 1 o 1 R t Φ t dt T + ( ). S√T T By a similar argument, again usingS ( the)S positivity of the≫ Fourier coefficients, we have log T log T ∞ F 1 it; Y R t 2Φ t dt ∞ F 1 it; X R t 2Φ t dt. S T S T −∞ −∞ So, we restrict( ourselves+ )S to( primes)S p X in the≥ truncated( + Euler-product.)S ( )S This is to ensure both R t and F 1 it; X have the terms with same q’s. ≤ Write( F) 1 it;(X+ as ) ∞ it ( + ) F 1 it; X = Q akk− , n=1 ( + ) ∶ where ak ≥ 0. This is because the Dirichlet coefficients of F s are non-negative. Now define log T I ∞ F 1 it; X R t 2Φ t( dt) 1 = S T −∞ it ∶ ∞ (∞ + ∞ )S it( )S m log T = Q ak Q k− qmqn Φ t dt. = S n T k=1 m,n 1 −∞ We also define log T I ∞ R t 2Φ t dt. 2 = S T −∞ Notice that since we are working∶ withS truncated( )S Euler-produ cts, everything is absolutely convergent. Now, using the fact that the Fourier coefficients of Φ are positive and that qn are completely multiplicative, we get the inner sum of I1 as it it ∞ ∞ it m log T ∞ ∞ it m log T Q S k− qmqn Φ t dt ≥ Q Q S k− qmqn Φ t dt m,n=1 n T n=1 k m n T −∞ S −∞ it ∞ ∞ ∞ r log T = qk Q Q qrqn Φ t dt. = = S n T n 1 r 1 −∞ Thus, we have k 1 I1 ∞ αj p − ≥ Q akqk = M M 1 qp I2 = ≤ j=1 p k 1 p X ( ) k − 1 k αj p − p αj p = M M 1 M M (23) ≤ j=1 p ≤ j=1 p αj p qp ⎛p X ( ) ⎞ ⎛p X − ( ) ⎞ − Using the generalized Merten’s⎝ Theorem 2.1, we⎠ have⎝ − ( ) ⎠ 1 k α p − j γF m M M 1 = e log X O 1 ≤ j=1 p p X ( ) γF m − = e (log2 T) log+ 3(T ) O 1 (24) ( + ) + ( ) 12 ANUPB.DIXITANDKAMALASKHYAMAHATAB The second product in (23) can be bounded as follows. k k p αj p p αj p log M M = Q Q log ≤ j=1 p αj p qp ≤ j=1 p αj p qp ⎛p X − ( ) ⎞ ⎛p X − ( ) ⎞ − − 1 ⎝ − ( ) ⎠ ≪ Q⎝ − ( ) ⎠ (25) p≤X X 1 . (26) ≪ log X From (23),(24) and (25), we get I1 γF m ≥ e log2 T log3 T O 1 . I2 In other words, we have ( + ) + ( ) T 2 log T √ F 1 it; X R t Φ t dt ∫ T T γF m ≥ e log2 T log3 T O 1 . T R t 2 log T t dt U ∫(√T+ )SΦ( )ST U ( + ) + ( ) Hence, we conclude S ( )S γF m max F 1 it ≥ e log2 T log3 T CF . t∈ √T ,T [ ] S ( + )S (( + ) − ) References [1] C. Aistleitner. Lower bounds for the maximum of the Riemann zeta function along vertical lines. Math. Ann., 365, no. 1-2, 473–496, 2016. [2] C. Aistleitner, K. Mahatab, M. Munsch. Extreme values of the Riemann zeta function on the 1-line. Int. Math. Res. Not. IMRN, no. 22, 6924–6932, 2019. [3] C. Aistleitner, K. Mahatab, M. Munsch, A. Peyrot. On large values of L(σ, χ). Q. J. Math., 70, no. 3, 831–848, 2019. [4] C. Aistleitner,L. Pa´nkowski. Large values of L-functions from the Selberg class. J. Math. Anal. Appl., 446, no. 1, 345–364, 2017. [5] L. Arguin and D. Belius, P. Bourgade, M. Radziwi ll, K. Soundararajan. Maximum of the Riemann zeta function on a short interval of the critical line. Comm. Pure and Applied Math, 72, no. 3, 500–535, 2019. [6] Louis-Pierre Arguin, Fr´ed´eric Ouimet, Maksym Radziwi ll. Moments of the Riemann zeta function on short intervals of the critical line. arXiv:1901.04061 [math.NT]. [7] R. Balasubramanian, K. Ramachandra. On the frequency of Titchmarsh’s phenomenon for ζ(s)-III. Proc. Indian Acad. Sci., 86, 341–351, 1977. [8] A. Beurling. Analyse de la loi asymptotique de la distribution des nombres premiers g´en´eralis´es. I. Acta Math., 68, no. 1, 255–291, 1937. [9] A. Bondarenko, K. Seip. Large greatest common divisor sums and extreme values of the Riemann zeta function. Duke Math. J., 166, 1685–1701, 2017. [10] R. de la Bret`eche, G. Tenenbaum. Sommes de G´al et applications. Proc. London Math. Soc. (3), 119, no. 1, 104–134, 2019. [11] A. B. Dixit. The Lindel¨of class of L-functions. PhD Thesis, University of Toronto, 2018. [12] A. Granville, K. Soundarajan. Extreme vaues of Sζ(1+it)S. The Riemann zeta function and related themes: papers in honour of Professor K. Ramachandra, Ramanujan Mathematical society Lecture Notes Series, 2, 65–80, 2006. [13] J. Kaczorowski, A. Perelli, On the prime number theorem for the Selberg class, Arch. Math., 80, 255–263, 2003. [14] N. Levinson. Ω-theorems for the Riemann zeta-function. Acta Arithmetica, 20, 317–330, 1972. [15] F. Mertens. Ein Beitrag zur analytischen Zahlentheorie. J. Reine Angew. Math., 78, 46-62, 1874. [16] H. Montgomery, R. C. Vaughan. Multiplicative number theory. I. Classical theory. Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 2007 LARGE VALUES OF L-FUNCTIONS ON 1-LINE 13 [17] V. K. Murty. The Lindel¨of Class of L-functions. eds. L. Weng and M. Kaneko, The conference of L- functions, World Scientific publication, 165–174, 2007. [18] M. Rosen. A generalization of Mertens’ theorem. J. Ramanujan Math. Soc., 14, 1999. [19] M. du Sautoy. Natural boundaries for Euler products of Igusa zeta functions of elliptic curves. Int. J. Number Theory, 14, no. 8, 2317–2331, 2018. [20] A. Selberg. Old and new conjectures and results about a class of Dirichlet series. Collected Papers, 2, Springer-Verlag, Berlin Heidelberg New York, 47–63, 1991. [21] G. Tenenbaum. Introduction to analytic and probabilistic number theory (English summary). Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics 46, Cambridge University Press, 1995. [22] E. C. Titchmarsh. The theory of Riemann-Zeta function. Second edition, revised by D.R. Heath-Brown, Clarendon Press-Oxford, 1986. [23] Y. Yashiro. Mertens’ theorem and prime number theorem for Selberg class. arXiv:1311.0754v4 . Department of Mathematics and Statistics, Queen’s University, Jeffrey Hall, 48 University Ave, Kingston, Canada, ON, K7L 3N8 E-mail address: [email protected] Kamalakshya Mahatab, Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, FIN 00014 Helsinki, Finland E-mail address: [email protected], [email protected]_