Relations among the Riemann zeta and Hurwitz zeta functions as well as their products

Several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation if the important results of [11]. Also, a relation derived here provides the starting point of a novel approach which in a series of companion papers yields a formal proof of the Lindel\"{o}f hypothesis. Some of the above relations motivate the need for analysing the large $\alpha$ behaviour of the modified Hurwitz zeta function $\zeta_1(s,\alpha)$, $s\in \mathbf{C}$, $\alpha\in \mathbf{R}$, which is also presented here.

The modified Hurwitz zeta function is simply related to the Hurwitz zeta function, ζ(s, α): ζ(s, α) = 1 α s + ζ 1 (s, α), s ∈ C, α > 0. (1.4) In this paper we present certain relations between ζ(s) and ζ 1 (s, α) as well as between products of these functions. In more detail, the following results are presented in sections 2-6: In §2 it is shown that the modified Hurwitz zeta function satisfies the identity where Γ(s), s ∈ C, denotes the Gamma function and (c) denotes the vertical line in the complex z-plane on which Re (z) = c.
It is shown in [3] that (1.5) yields a singular integral equation for |ζ(s)| 2 , 0 < σ < 1, t > 0, and this equation provides the starting point for the proof of the analogue of Lindelöf's hypothesis for a function which differs from the Riemann zeta function only in the occurrence of the log n term. We recall that Lindelöf's hypothesis concerns the growth of ζ(s) as t → ∞ along the critical line σ = 1/2 and states that ζ( 1 2 + it) = O(t ǫ ) for every positive ǫ. The Riemann hypothesis implies Lindelöf's hypothesis and conversely, Lindelöf's hypothesis implies that very few zeros could disobey Riemann's hypothesis [8]. Significant progress has been made by developing ingenuous ways of estimating exponential sums n e 2πif (n) using generalisations of the Vinogradov method [18]. Until recently the best result in this direction had been obtained by Huxley [10], where it is proved that ζ( 1 2 + it) = O t 32/205+ǫ . A short time ago Bourgain announced a further improvement [2] where the exponent 32/205 was reduced to 53/342.
In §3 it is shown that there exists the following asymptotic relation between the Riemann zeta function and the modified Hurwtiz zeta function: (1.6) where B N (α) is defined by B N (α) = 1≤n≤N e 2πinα , α > 0. (1.7) and N = t/2π. A direct consequence of (1.6) is the following theorem: Theorem 1. Let I k (t) denote the 2k-th power mean of the modified Hurwitz zeta function, namely in which the implicit constant is independent of k.
This result immediately implies that Lindelöf's hypothesis is true provided that for each ǫ > 0 I k (t) k,ǫ t ǫ for each k ∈ N.
In connection with (1.9) we recall that an equivalent formulation of the Lindelöf hypothesis involves estimating the 2k-th power mean of the Riemann-zeta function It can be shown [17,Th. 13.2] that the Lindelöf hypothesis holds true if and only if J k (T ) = O (T ǫ ) for each ǫ > 0 and for each k ∈ N.
In §4 the following identities are presented: where Re {u i } 4 i=1 < 1. The above formulae can be generalised in a straightforward way. As a direct application of (1.11) we present in §4 a new derivation of the following exact identity in [11]: (1.14) This immediately yields the estimate It is shown in [5] that (1.12) plays a crucial role for the derivation of an interesting identity between certain double exponential sums. Indeed, it is well known that if 0 ≤ α < 1, then the large t-asymptotics of ζ 1 (s, α) is dominated by the sum S 1 , defined by However, if 1 ≤ α < ∞, the large t-asymptotics of ζ 1 (s, α) is dominated by the sum S 1 defined in (1.16), as well as by a different sum, S ′ 2 (σ, t, α) [4]. Thus, the large t-asymptotics of equation (1.12) provides a relation between two double sums generated from S 1 and S ′ 2 , and the explicit formulae obtained from the large t-asymptotics of the linear and quadratic terms.
Similarly, equation (1.13) yields novel relations between cubic exponential sums. Before using equations (1.12) and (1.13) for the case that Re {u i } 4 i=1 < 1, it is necessary to regularize the terms involving α → ∞; this regularisation is discussed in §4.
Finally, in §5, by considering the Fourier series of the product ζ 1 (u, α)ζ 1 (v, α) with complex numbers u, v satisfying Re u > 0, Re v > 0, by using certain elementary estimates for the resulting coefficients, and by employing theorem 1 together with Parseval's identity, we obtain the following asymptotic result.

An identity involving the Hurwitz function
In order to derive (1.5) we let The definition of the modified Hurwitz zeta function, namely equation (1.3), implies Assuming that | arg(−w)| < π, −Re a < b < 0, (2.4) we observe the Mellin-Barnes type integral identity Thus, Summing over m and n we obtain Substituting (2.6) into (2.2) and then letting u = s, v =s in the resulting expression, we find (1.5).

An asymptotic relation between the Riemann and Hurwitz functions
The approximate functional equation for the Riemann zeta function provides the starting point for the estimation of the ζ(s) along the critical line. In this section we derive a weak analogue of this equation. Throughout we will set N = t/2π and refer to the sum This function is similar to the classical Dirichlet kernel that arises in Fourier analysis. As such, we have the following well-known estimates.
Our first result expresses the approximate functional equation for ζ(s) as an integral equation involving the Hurwitz zeta function ζ 1 (s, α). The proof of Theorem 1 will follow directly from this result. Lemma 2. Let s = σ + it and B N as previously defined. Then we have where χ(s) = 2 s−1 π s sin(πs/2)Γ(1 − s) with Γ(s) denoting the Gamma function and ζ 1 (s, α) denoting the modified Hurwitz zeta function.
Let us first establish (3.1). The identity in (3.2) will be a consequence of this.
Proof. First we recall the approximate functional equations for ζ(s) and ζ 1 (s, α) (see [15] and references therein) and uniformly in 0 < α < 1. The following identity is valid Indeed the left hand side of (3.5) can be rewritten in the form which is the right hand side of (3.5). Using z = s − 1 and employing (3.4) we find We note that Replacing s by 1 − s in (3.4) and taking the complex conjugate of the resulting equation we find Using (3.7) with z = −s and employing (3.8) we find (3.9) Here we have used χ(s) = O t 1/2−σ and a similar estimate as before Using equations (3.6) and (3.7) in (3.3) we arrive at the result in the lemma.
Proof. Using the periodicity of B N (α) on R/Z we find We next estimate the first integral on the right hand side of (3.10), which we denote by I N (s): The integral in the above sum does not have any stationary points. Indeed, candidates for stationary points are the points α * where thus, since 1 ≤ n ≤ N , α * is outside the range of integration. Hence the above integral can be estimated using integration by parts: There error term E n (s) can be evaluated using the second mean value theorem for integrals. For instance, for some ξ ∈ (1, N ) we have and similarly for Im E n (s). It is now straightforward to show We also have the elementary estimate This combined with (3.13) and (3.12) gives the desired result.
The leading order terms in the above expansion are O t −σ/2 log t , thus they can be absorbed into the error term. Indeed, using Lemma 3 it is now straightforward to see that Using similar arguments we also find Combining this observation with the result of Lemma 2 we conclude that for 0 < σ < 1. The proof to Theorem 1 now follows from Lemmas 2-3 with s = 1/2 and the application of Hölder's inequality with exponents p = 2k, q = 2k 2k − 1 .
In particular, using the estimates in Lemma 1 we have This gives rise to the result in Theorem 1.
Using the integral representation of the modified Hurwitz function, namely we find .
Inserting in this equation the identity where and . (4.6) We will next show that Indeed, using the integral representations of Γ(u) and Γ(v), we find Replacing in the RHS of (4.8) ρ 1 and ρ 2 by αx 1 and αx 2 , multiplying the resulting expression by α −(u+v) , and integrating with respect to α from α = 1 to α = ∞, we find which gives (4.7). Finally, we will show that Indeed, using the integral representations of Γ(v) and of ζ 1 (u, α) we find Replacing in the RHS of the above equation ρ by αρ 1 , multiplying the resulting equation by α −v /Γ(v), and then integrating with respect to α from α = 1 to α = ∞, we find (4.9). Inserting in equation (4.1) the expressions for J 0 (v, u), for I 0 (v, u) and for I 0 (u, v) from equations (4.7) and (4.9), we find (4.2).
Eq. (1.12) can be derived following the approach used in Lemmas 4 and 5, and thus it is omitted.

Quadruple formula
Lemma 5. Let ζ 1 (s, α) be defined as in (4.3). Then for Re u i > 1, i = 1, 2, 3, 4, the following identity is valid: where the sums run over permutations of (1, 2, 3, 4) so that the first and third sums contain 4 terms whilst the second sum contains 6 terms.
Proof. Employing the representation (4.3) for each Hurwitz function and integrating over (0, 1) we find that the left hand side of (4.11), which we denote by Q(u 1 , u 2 , u 3 , u 4 ), is given by where the functions {R i } are defined by (4.13) The following identity is valid: (4.14) Using this in (4.12) we find (4. 15) In order to simplify the right hand side of (4.15) we first note the definition of the Gamma function, namely the equation (4.16) implies that Using in the right hand side of this equation the transformations dividing by the product of the four Gamma functions, and multiplying the resulting expression by α −u 1 −u 2 −u 3 −u 4 , we find the identity Integrating this equation over (1, ∞) with respect to α we obtain Employing for ζ 1 (u, α) and Γ(u) equations (4.3) and (4.16) respectively we find Using in the right hand side of this equation the transformations (4.17) but restricted only to i = 2, 3, 4, dividing by Γ(u 2 )Γ(u 3 )Γ(u 4 ), multiplying by α −(u 2 +u 3 +u 4 ) and integrating with respect to α over (1, ∞) we obtain Similarly equations (4.3) and (4.16) imply Using in the right hand side of this equation the transformations (4.17) but only for i = 3, 4, dividing by Γ(u 3 )Γ(u 4 ), multiplying by α −u 3 −u 4 and integrating the resulting expression with respect to α over (1, ∞), we obtain (4.20) A similar procedure yields the identity

A new derivation of the results of [11]
Here we rederive some of the results from [11].
The identity in (1.14) is a consequence of equation (4.2) and of the following exact formula. Lemma 6. Let ζ 1 (u, α), u ∈ C, α > 0, denote the modified Hurwitz function, and let ζ(u), u ∈ C, denote Riemann's zeta function. Then Proof. In order to derive (4.22) we first assume that Re u > 1, so that we can use the sum representation of J 1 (u, α). Furthermore, we assume that Re v < 1, so that the relevant integral converges at α = 0. Then, where we have used the change of variables α = βm in the second equation. Then, the definition of Riemann's zeta function, together with the identity imply equation (4.22). In order to derive equation (4.23) we use integration by parts: Thus, (4.23) follows.
Proof of identity (1.14). Splitting the first integral in the RHS of (4.2) and then using equations (4.22) and (4.23), as well as using the analogous equations where u and v are interchanged, we find equation (1.14).

Remark 1.
In order to derive equation (1.15) we let σ = 1 2 + ε 2 in the LHS of (1.15), and employ the identities as well as the identity Remark 2. By proving a simple estimate for the integrals in the RHS of (1.14), it is shown in [11] that these integrals do not contribute to the leading asymptotics of the LHS of equation (1.14). Actually, it is straightforward to show that if v = σ 1 − it, u = σ 2 + it, σ 1 < 2, σ 1 > 0, (4.28) Indeed, it follows that integrals in the RHS of (4.29) do not possess any stationary points. Then, straightforward integration by parts yields (4.28).

A relation between quadratic products of Hurwitz zeta functions and their Fourier series
Theorem 2 will be proved by examining the Fourier series for the function for Re u, Re v > 0. Following Rane [16] we first construct the Fourier series for ζ 1 (s, α).
Proof. Since ζ 1 (s, α) is a smooth function of α (for fixed s) its Fourier series converges pointwise for α ∈ (0, 1). The Fourier coefficients are defined by a n (s) = Using Euler's integral representation of the Gamma function we arrive at the desired result.
Remark 3. Since a n (s) is expressible in terms of the incomplete Gamma function, we conclude that it has an analytic extension to all complex s = 1.
Note that for σ > 1 we have for σ > 1, and by analytic continuation elsewhere. Now we write where the a n (s) are defined accordingly.
Proof. Since the Fourier coefficients for ζ 1 (u, α) are a n (u) the Fourier coefficients of the product ζ 1 (u, α)ζ 1 (v, α) are given by the convolution Using this in the above we find Since Re u, Re v > 1 the integrands of the first and third terms can be dominated by the integrable functions α −Re u and α −Re v respectively, allowing us to pass the sum inside the integral, q n (u, v) = a n (u + v) We note that both the integrals are absolutely convergent for Re u, Re v > 1.
To establish the main result in this section we must first perform an analytic continuation of the functions q n (u, v) valid for Re u, Re v > 0. To this end, we recall the following result [16]: where s = σ + it and σ > 0. This result can be derived using the Euler-Maclaurin formula. We will need the following lemma to control the final term.
Proof. For α in the stated range we can integrate by parts using We can estimate the sum arising from the first term Computing the derivative and applying integration by parts again, the second term becomes An application of the second mean value theorem for integrals on the real and imaginary parts of this term show it to be so we have established our estimate.
Proof. The proof is essentially the same as that used for Lemma 9. The oscillatory term doesn't have stationary points if |n| > t/2π so integrating by parts yields the desired estimate.
Applying this lemma and using the approximate functional equation we find that for |n| > t/2π the following estimate is valid . .