Impact of Pump OSNR on Noise

I. INTRODUCTION

W ITH THE recent advances, it is now well-known that the fiber-optical parametric amplifiers (FOPAs) succeed in providing broad-band and high gain amplification. Very few studies are presently focusing on the noise performance. First, Blows and French showed that, in the case of a parametric pump constituted by a continuous source amplified by an erbium-doped fiber amplifier (EDFA), the output noise level of the FOPA is considerably reduced when the pump is filtered so as to remove any broad-band amplified spontaneous emission (ASE) noise present in the amplification band [START_REF] Blows | Low-noise-figure parametric amplifier with a continuous-wave frequency-modulated pump[END_REF]. More recently, it was proved that rather than an optical measurement, an electrical measurement of the noise figure (NF) of the FOPAs was required to take into account other sources of noise than ASE [START_REF] Kylemark | Noise figure characterization of fiber optical parametric amplifiers[END_REF]. Kylemark et al. numerically found that the NF is degraded by the remaining ASE under the pump peak [START_REF] Kylemark | Noise characteristics of fiber optical parametric amplifiers[END_REF]. In this letter, we present an electrical measurement method of the NF. With this method, we experimentally demonstrate the impact of the pump optical signal-to-noise ratio (OSNR) on the FOPA NF. Finally, we show that a pump OSNR as high as 65 dB is required to keep the NF to the quantum limit with a channel input power of 21 dBm.

II. MEASUREMENT METHOD OF THE NOISE FIGURE

The NF of an optical amplifier can be evaluated either optically from the measurements of the input and output OSNRs, or electrically. However, optically, only broad-band noise, like ASE, is detected because of the resolution of optical spectrum analyzers (OSAs). An electrical measurement gives a more accurate estimation of the NF. The electrical NF is determined with a relative-intensity noise (RIN) analyzer and by the RIN subtraction technique [START_REF] Willems | Optical amplifier noise figure determination by signal RIN subtraction[END_REF] (1)

where is the input signal power, and are, respectively, the RIN measured at the amplifier input and output. The measured value of RIN is due to different types of noise: The actual RIN of the signal source but also the beating noise between the signal and any other optical noise

(2) (3)
In these expressions, is the amplifier gain, an input noise, and noise generated in the amplifier responsible of the NF. The notation means that only the polarization component of noise, parallel to the signal one, beats the signal.

In order to validate the electrical method, an experiment was set up to perform simultaneously optical and electrical measurements of noise in the well-known case of an EDFA. Output noise is only constituted by broad-band ASE which can be easily detected either optically or electrically. The output of the amplifier is divided by a coupler. On one port, an OSA measured the gain and the OSNR. On the other port, an optical filter with a 3-dB bandwidth of 0.2 nm selects the signal wavelength and a RIN analyzer measures the output RIN at a particular frequency. The RIN analyzer is calibrated and subtracts the thermal noise and the shot noise from the total noise. To determine the NF, levels of noise are first measured at the input of the amplifier then at the output. A fair agreement was found between both electrical and optical NFs; the discrepancy between both is about 0.35 dB. These measurement techniques can be applied on a parametric amplifier provided we take into account the particularity of the one-pump FOPA, which is to produce polarized ASE contrary to EDFAs as pointed out in [START_REF] Tang | Noise-figure limit of fiber-optical parametric amplifiers and wavelength converters: Experimental investigation[END_REF]. This assertion was experimentally checked out with the following experiment. At the output of the FOPA, a filter selected 0. pump was filtered so that only ASE was detected. A polarization controller placed in front of a polarizer enabled to measure a polarization extinction ratio of 18 dB with an OSA. It corresponds to a degree of polarization of 98% so we can state that the ASE of the FOPA is fully polarized.

At high gain, the standard definition of NF is NF , where is the population inversion factor. For FOPA, the ASE is polarized so and the optical NF is related to the total ASE power as follows:

(4) whereas, in the case of an EDFA, the ASE is not polarized so . Therefore, the optical NF is related to the total ASE power as follows:

(5)

III. PARAMETRIC AMPLIFIER UNDER TEST

The FOPA under test is depicted in Fig. 1. The pump is first phase-modulated by a pseudorandom bit sequence of 3.5 Gb/s to increase the threshold power of the stimulated Brillouin scattering. Then it is successively amplified by two EDFAs with a filtering of the ASE in between. Finally, a circulator and a reflective fiber Bragg grating (Filter 2) reduce the level of ASE from the EDFAs, as proposed in [START_REF] Blows | Low-noise-figure parametric amplifier with a continuous-wave frequency-modulated pump[END_REF]. A variable attenuator, placed in front of the first EDFA, is used to vary the OSNR of the pump, i.e., the level of the residual ASE at the pump wavelength nm. The pump power remains constant to 640 mW. The amplifying fiber used in this FOPA is a highly nonlinear dispersion-shifted fiber ( and nm). On the signal path, a polarization controller allows adjusting the signal polarization to maximize the gain and an isolator protects the input from occasional reflections. The signal is amplified by an EDFA and its power is controlled with a variable attenuator; it is coupled to the pump through the fiber Bragg grating. At the output of the highly nonlinear fiber, a circulator and a fiber Bragg grating (similar to Filter 2 of the pump) separate the amplification spectral band from the pump. Electrical measurements are performed at the frequency 68 MHz, where the pump RIN is below 155 dB/Hz. FOPAs are generally pumped by a low power source, amplified by successive EDFAs. First the pump has to be filtered to suppress ASE produced by the EDFAs in the amplification spectral band of the FOPA. But even with the narrowest filtering centered at the pump wavelength, ASE from the EDFAs remains in the bandwidth of the filter at the pump wavelength. When this noise is injected in the parametric amplifier all together with the pump and the signal, it is reproduced at the signal wavelength through a four-wave mixing (FWM) process between the signal, the pump, and this pump noise. To obtain Fig. 2, we intentionally degrade the FOPA performances by reducing the pump OSNR. In Fig. 3, the transfer of the pump noise to the signal can be observed on the OSA (resolution set to 0.05 nm) as an additional noise, very narrow, located around the signal wavelength. Flat broad-band parametric ASE can be seen as well, at a much lower level. With a better pump OSNR, transfer of pump noise at the signal wavelength can be no longer observed experimentally on the OSA so that it is not taken into account in the measurement of OSNRs.

IV. RESULTS AND DISCUSSIONS

As the noise observed on the OSA is the result of an FWM product between the pump, the signal, and ASE at the pump wavelength, its level is proportional to the signal power and to the level of ASE, i.e., to the inverse of the OSNR of the pump.

These trends have been experimentally verified. The OSNR of the pump and the signal power are controlled with variable attenuators (see Fig. 2). First for a signal wavelength of 1595 nm, the output noise has been measured optically and electrically while the input signal power was increased from 35 to 5 dBm, the pump OSNR remained constant as well as the net gain (19.1 dB). This measurement was carried out for three different pump OSNRs. Then at the same signal wavelength, the output noise was also measured versus the pump OSNR in the case of three input signal powers. The resulting NFs are presented in Figs. 4 and5. In these figures, the electrical and optical NF are represented; for the optical one, the measurements takes into account only parametric ASE. As can be seen, the electrical measurement detects an additional noise, which depends on the input signal power and more particularly on the level of the pump OSNR. This noise gets negligible compared to parametric ASE when the signal power is low and the pump OSNR high. In this case, electrical NF meets optical NF to 4.4 dB. It is the first time to our knowledge that a real noise floor is electrically measured and that this noise floor corresponds to the optical measurement. This 4.4-dB limit is mainly due to relatively high coupling and distributed losses (respectively, about 0.8 and 0.3 dB) and noise produced by the Raman effect [START_REF] Voss | Raman-noise-induced noise-figure limit for (3) parametric amplifiers[END_REF], [START_REF] Tang | Noise-figure limit of fiber-optical parametric amplifiers and wavelength converters: Experimental investigation[END_REF]. Moreover, we numerically simulated the transfer of pump noise toward the signal through FWM by solving the scalar nonlinear Schrödinger equation with a standard split-step Fourier method. The pump and its noise are modeled by a monochromatic continuous wave, and in the spectral domain, we add a square of random noise centered at the pump wavelength, with a bandwidth of 0.2 nm and which average level corresponds to the required OSNR. The signal is also a monochromatic continuous wave. Optical waves are all considered in the same state of polarization. The simulated pump OSNR is higher than the experimental one by 3 dB because pump noise in the orthogonal polarization is not accounted for. At the output of the amplifier, we calculate the level of noise produced around the signal wavelength. The model does not simulate the Raman effect, leading to a minimum NF of 3.8 dB. In experiments, as the pump power remains constant, Raman noise is also constant. A constant noise power was, therefore, added to account for the Raman noise and to achieve the experimental 4.4-dB NF. In Figs. 4 and5, the simulated results are represented by the lines and the experimental ones by the points. A fair agreement is found between simulations and experimental results, so the numerical model can be used to define a pump OSNR specification. In practical systems, amplifiers provide a total output power of 20 dBm, which corresponds, for 128 channels, to a channel input power of 21 dBm. According to our model, a 4.4-dB NF is achieved provided the pump OSNR is higher than 65 dB (0.1 nm), in this case, the pump noise is negligible compared to parametric ASE.

V. C ONCLUSION An electrical measurement method of the NF of an optical amplifier has been implemented. The results of this measurement were first compared to a standard optical measurement and validated in the case of an EDFA. When applied to an FOPA, it allowed us to experimentally quantify, for the first time, the noise limitations caused by the pedestal noise present at the pump wavelength. The FOPA NF was measured versus the pump OSNR and as a function of the signal power. A numerical model was developed to simulate the transfer of pump noise and validated by these measurements. Using this model, we determine, for practical systems, a minimum required pump OSNR (0.1 nm) of 65 dB.

  Fiber-Optical Parametric Amplifiers A. Durécu-Legrand, C. Simonneau, D. Bayart, Member, IEEE, A. Mussot, T. Sylvestre, E. Lantz, and H. Maillotte Abstract-Electrical measurements of the noise figure of a fiberoptical parametric amplifier are presented and compared with optical measurements. The transfer of pump noise by four-wave mixing was clearly demonstrated. A numerical model was developed to simulate the transfer of pump noise and validated by these measurements. Using this model, we determine, for practical systems, a minimum required pump optical signal-to-noise ratio of 65 dB. Index Terms-Noise figure (NF), optical amplifiers, optical fiber communication, optical Kerr effect, parametric amplifiers.

  2 nm in the signal band at the maximum of gain. No signal was injected in the FOPA and the 1041-1135/$20.00 © 2005 IEEE

Fig. 1 .

 1 Fig. 1. Experimental setup for the measurement of the NF of the FOPA.

Fig. 2 .

 2 Fig. 2. Input pump spectrum (res = 0:05 nm).

Fig. 3 .

 3 Fig. 3. Example of noise transferred from the pump pedestal (represented in Fig. 2) to the signal wavelength (res = 0:05 nm).

Fig. 4 .

 4 Fig. 4. FOPA NF as a function of signal power for three different pump OSNRs. (Case of NF . Symbols: Eexperiments. Lines: Simulations).

Fig. 5 .

 5 Fig. 5. FOPA NF as a function of pump OSNR for three different signal powers. (Case of NF . Symbols: Experiments. Lines: Simulations).

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 17, NO.6, JUNE 2005


Giuseppe Iurato
email: giuseppe.iurato@unipa.it

On some historical aspects of Riemann zeta function, 3

In this third note, we deepen, from either an historical and historiographical standpoint, the arguments treated in the previous

both personifying, therefore, those two complementary and opposite typical aspects characterizing the mathematical work. Beyond what had been made by Cauchy, they created the main body of the new complex function theory in the period from about 1850 to 1880 (see (Klein 1979, Chapter VI)). Both received a strong impulse from Jacobi work. The rst elements of the theory of functions according to Weierstrass date back to a period which roughly goes on from 1842-43 to 1854; in the meanwhile, Riemann published, in the early 1850s, his rst works on the foundations of complex analysis, followed by the celebrated works on Abelian functions (which are elliptic functions so named by Jacobi) of the years 1856-57, which dismayed Weierstrass himself, inuencing his next research program. This last point should be taken with a certain consideration. Instead, following (Klein 1979, Chapter VI), in the period from 1830s to the early 1840s, Weierstrass began to self-taughtly study Jacobi's Fundamenta nova theoriae functionum ellipticarum, hence attended Christoph Gudermann (1798-1852) lectures on elliptic functions. He wrote his rst paper in 1841 on modular functions, followed by some other papers wrote between 1842 and 1849 on general function theory and dierential equations. His rst relevant papers were written in the years 1854-56 on hyperelliptic or Abelian functions, which engaged him very much. Afterwards, in the wake of his previous work on analytic, elliptic and Abelian functions, Weierstrass was led to consider the so-called natural boundaries (that is to say, curves or points -later called essential singularities -in which the function is not regularly dened) of an analytic function to which Riemann put little attention. The rst and rigorous treatment of these questions was given by Weierstrass in his masterful 1876 paper entitled Zur Theorie der eindeutigen anatytischen Funktionen, where many new results were achieved, amongst which is the wellknown Casorati-Weierstrass theorem (as we today know it) and the product factorization theorem. Klein (1979, Chapter VI) states that the content of this seminal paper surely dated back to an earlier period, and was chiey motivated by his research interests in elliptic functions. As pointed out in (Hancock 1910, Introduction), nevertheless, it is quite dicult to discern the right contribution to the elliptic function theory due to Weierstrass from other previous mathematicians, because of the objective fact that Weierstrass started to publish his lessons and researches only after the mid-1860s.

Weierstrass' theory of entire functions and their product decompositions, according to Klein, has found its most brilliant application in the (Weierstrass) theory of elliptic functions, to be precise, in the construction of the basic Weierstrassian σ-function σ(u); perhaps -Klein says -Weierstrass' theory of entire functions even originated from his theory of elliptic functions (see also (Bottazzini & Gray 2013, Chapter 6, Section 6.6.3)). Nevertheless, already Gauss and Abel were gone very close to this σ-function and its properties. Again Klein says that he wishes to conclude his discussion of Weierstrass' theory of complex functions, adding only some remarks on the history, referring to R. Fricke distinguished review on elliptic functions for more information (see (Burkhardt et al. 1899-1927, Zweiter Teil, B.3, pp. 177-348)). If we now ask -again Klein says -from where Weierstrass got the impulse to represent his functions by innite products, we nd his principal forerunner in G. Eisenstein (1823Eisenstein ( -1852)), who was also a friend of Riemann with whom often talked about mathematical questions. In this paper (see [START_REF] Eisenstein | Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind[END_REF])), Eisenstein did not attain the fully symmetric normal form, because he still lacked the exponential factors attached to the individual prime factors that will be then introduced by Weierstrass for inducing the product to converge in an absolutely manner. As he himself declared, Weierstrass got this idea from Gauss, who had proceeded in a similar way with his product expansion of the gamma function in 1812 (see the paper on the hypergeometric series in Weierstrass' Mathematische Werke, Band ...). It therefore turns out clear that elliptic and hyperelliptic function theory exerted a notable role in preparing the humus in which grew up the Weierstrass work on factorization theorem, and not only this: in general, it exerted a great inuence on Riemann and Weierstrass work (see (Bottazzini & Gray 2013, Chapter 4, Section 4.5)). Following (Burkhardt et al. 1899-1927, Zweiter Teil, B.3, Nr. 15-17, 25, 45, 55), amongst others, Abel , Euler, Jacobi, Cayley and Gauss (see [START_REF] Bottazzini | Hidden Harmony -Geometric Fantasies. The Rise of Complex Function Theory[END_REF], Chapter 1, Section 1.5.1.1; Chapter 4, Section 4.2.3.1-2)) had already provided product expansions of certain elliptic functions, but it was Eisenstein (see [START_REF] Eisenstein | Genaue Untersuchung der unendlichen Doppelproducte, aus welchen die elliptischen Functionen als Quotienten zusammengesetzt sind[END_REF])), with his innite product expansion ansatz, the closest forerunner of the Weierstrass work on his σ function, in turn based on the previous work made by Jacobi and Gauss. Furthermore, already Cauchy , since 1843, gave some useful formulas involving innite products and innite series which maybe could have played a certain role in the 1859 Riemann paper in deducing some properties of that functional equation related to his ξ function (see later), having seen too the simple fact that Riemann himself known very well Cauchy's work. Therefore, in conclusion, Weierstrass himself acknowledges, in dierent

In (Greenhill 1892, Chapter IX, Section 258), the author states that the well-known expressions for the circular and hyperbolic functions in the form of nite and innite products, have their analogues for the elliptic functions as laid down by Abel in some his researches on elliptic functions published in the celebrated Crelle's Journal, Issues 2 and 3, years 1827 and 1828. Following [START_REF] Hancock | Analysis[END_REF], Chapter V, Article 89), Abel showed, in the 1820s, that elliptic functions, considered as the inverse of the elliptic integrals, could be expressed as the quotient of innite products, then systematically reconsidered in a deeper manner by Jacobi.

In this regards see (Bellacchi 1894, Chapter X), where an interesting discussion of the Jacobi series is made, amongst other things highlighting that already Abel, on the wake of what was done by Johann Bernoulli, had introduced innite product expansions of certain elliptic functions that later Jacobi, in turn, converted into innite series by means of trigonometric arguments, so giving rise to new elliptic functions (see [START_REF] Greenhill | The Applications of Elliptic Functions[END_REF], Chapter IX, Section 258)).

places, his debit both to Gauss and Cauchy, in achieving his celebrated results on entire function factorization theorem.

In 1858, Riemann wrote his unique paper on number theory, which marked a revolution in mathematics. According to Laugwitz (1999, Introduction, Sections 4.1 and 4.2), real and complex analysis has always inuenced Riemann work: algebraic geometry appears, in his works, as a part of complex analysis; he treats number theory with methods of complex function theory; he subsumes physical applications into partial dierential equations; he replaces the usual axiomatic conception of geometry by his novel (Riemannian) geometry, which is a part or real analysis of several variables; and he develops the topology of manifolds as a new discipline derived from analysis. Riemann known the elements of algebraic analysis according to J-L. Lagrange and L. Euler, through the lessons of his teacher, M. A. Stern (1807A. Stern ( -1894)), who was one of the last schoolmasters of the subject. Riemann handled the gamma function in a secure and self-condent way and has dealt with dierential equations and recursions in the Euler's manner. The Stern lessons were of very fundamental importance to achieve many Riemann's results, even if the celebrated 1748 Euler's Introductio in Analysin Innitorum was one of the most inuential textbooks of the time. Nevertheless, Klein (1979, Chapter VI) states that Riemann began already to study elliptic and Abelian functions since the late 1840s, because this subject, in the meantime, has become of a certain vogue in Germany. In the 1855-56 winter term, following the Dirichlet's research lines, Riemann lectured on functions of a complex quantity, in particular elliptic and Abelian functions, while in the 1856-57 winter term he lectured on the same subject, but now with special regard to hypergeometric series and related transcendentals. These lectures, from which he drew publications on Abelian and hypergeometric functions, were partially repeated in the following semesters. Klein (1979, Chapter VI) points out that the years 1857-62 marked the high-point of Riemann's creativity. Moreover, Klein states that before to characterize the specic Riemannian function theory work, he wishes to put forward a remark that may cause some surprise: Riemann did much important work in the theory of functions that does not t into the framework of his typical theory. Klein refers to the notable 1859 paper on the number of primes less than a given magnitude, where it is introduced the Riemann zetafunction ζ(σ + it) given by an analytic expression, namely an innite product. This product is converted into a denite integral, which can then be evaluated by shifting the path of integration. The whole procedure is function theory à la Cauchy. Therefore, according to what Felix Klein states, the mathematical background that was at the basis of the Riemannian analytic treatment of his ζ function, essentially lies on the Cauchy's theory of complex functions. This is also conrmed by [START_REF] Bottazzini | Hidden Harmony -Geometric Fantasies. The Rise of Complex Function Theory[END_REF], Chapter 5, Section 5.1), coherently with what has just been said above in regards to the importance played by Cauchy's work on the Riemann's one.

According to [START_REF] Laugwitz | Turning Points in the Conception of Mathematics[END_REF], Chapter 1), notwithstanding the era of ferment that concerned the 19th century mathematics, an autonomous and systematic account of the foundations of complex analysis is ndable, for the rst time, in the Riemann's works and lecture notes through the winter term 1855-56 to the winter term 1861-62, the latter having been published by the physicist Carl Ernst Abbe in the summer term 1861 (see [START_REF] Ullrich | Review of Riemanns Einführung in die Funktionentheorie. Eine quellenkritische Edition seiner Vorlesungen mit einer Bibliographie zur Wirkungsgeschichte der Riemannschen Funktionentheorie[END_REF]) and references therein) when he was a student of W. Weber and Riemann in Göttingen. The only systematic and congruous historical attempt to organically recognize the various Riemann's lessons has been pursued by E. Neuenschwander in [START_REF] Neuenschwander | Riemanns Einführung in die Funktionentheorie. Eine quellenkritische Edition seiner Vorlesungen mit einer Bibliographie zur Wirkungsgeschichte der Riemannschen Funktionentheorie, Abhandlungen der Akademie der Wissenschaften Zu Göttingen[END_REF]. In any case, Riemann was fully imbedded into the real and complex analysis scenery of the rst middle of the 19th century, which seen involved the outstanding gures of Cauchy, Weierstrass and Riemann himself, whose researches were intertwined amongst them more times. According to [START_REF] Laugwitz | Turning Points in the Conception of Mathematics[END_REF], Chapter 1, Section 1.1.5), just in connection with the drawing up of his paper on the same subject, Riemann was aware of the Weierstrass' papers on Abelian functions wrote between 1853 and 1856-57, for which it is evident that a certain inuence of the latter on the Riemann's oneat least, as concerns such a period -there was, even if Weierstrass will publish these his works only later. Again following [START_REF] Neuenschwander | Riemanns Einführung in die Funktionentheorie. Eine quellenkritische Edition seiner Vorlesungen mit einer Bibliographie zur Wirkungsgeschichte der Riemannschen Funktionentheorie, Abhandlungen der Akademie der Wissenschaften Zu Göttingen[END_REF], one of the key themes of the last 1861 sommersemester Riemann lectures on analytic functions, was the determination of a complex function from its singularities. Thereafter, he claries that this problem regards only single-valued functions dened on C ∪ {0} whose unique singularities are poles (the names pole and essential singularity are respectively due to C.A. Briot and J.C. Bouquet and to Weierstrass). In turn, the resolution of this problem requires the previous knowledge of the zeros of the function which has to be determined. At rst, Riemann considered the case of a function having a nite number of zeros and poles, then he went over the the next question, namely to determine a function with innitely many zeros whose unique point of accumulation if ∞ (which, inter alia, concerns too the Riemann zeta function theory). But, again following [START_REF] Laugwitz | Turning Points in the Conception of Mathematics[END_REF], Chapter 1, Section 1.1.6), in doing so Riemann went over very close to the next Weierstrass' work on the innite product representation of an entire function, using special cases to explain the general procedure. Detlef Laugwitz points out that Riemann has pursued this latter task in such a way that, by following his directions, one could immediately give a proof of the known Weierstrass' product theorem, even if Riemann ultimately failed in reaching the general case; nevertheless, for what follows, this last claim has a certain importance from our historical standpoint.

Indeed, in his renowned paper on the distribution of prime numbers

(s) = - ∑ log(1 -p -s )
remains nite, and since the same holds for the logarithms of the other factors of ξ(t), it follows that the function ξ(t) can only vanish if the imaginary part of t lies between i/2 and -i/2. The number of roots of ξ(t) = 0, whose real parts lie between 0 and T is approximately = (T /2) log(T /2)π -T /2π; because the integral ∫ d log ξ(t), taken in a positive sense around the region consisting of the values of t whose imaginary parts lie between i/2 and -i/2 and whose real parts lie between 0 and T , is (up to a fraction of the order of magnitude of the quantity 1/T ) equal to (T log(T /2)π -T /2π)i; this integral however is equal to the number of roots of ξ(t) = 0 lying within this region, multiplied by 2πi. One now nds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some eeting futile attempts, as it appears unnecessary for the next objective of my investigation. If one denotes by α all the roots of the equation ξ(t) = 0, one can express log ξ(t) as

∑ log ( 1 - tt αα ) + log ξ(0) (2)
for, since the density of the roots of the quantity t grows with t only as log t/2π, it follows that this expression converges and becomes for an innite t only innite as t log ξ(t); thus it diers from log ξ(t) by a function of tt, that for a nite t remains continuous and nite and, when divided by tt, becomes innitely small for innite t. This dierence is consequently a constant, whose value can be determined through setting t = 0. With the assistance of these methods, the number of prime numbers that are smaller than x can now be determined.

latter. To be precise, following [START_REF] Bottazzini | Complex Function Theory[END_REF], just due to this election, Riemann and Dedekind visited Berlin, where they met E.E. Kummer, L. Kronecker and K. Weierstrass.

According to [START_REF] Dedekind | Bernhard Riemann's Lebenslauf[END_REF], very likely, it was just from this meeting that sprung out of the celebrated 1859 Riemann number theory paper that was, then, sent to Weierstrass himself, to be published in the November issue of the Monatsberichte der Berliner Akademie.

So, in his celebrated 1859 paper, Riemann himself had already reached an innite product factorization of ξ(t), namely the (2), which can be equivalently written as follows

log ξ(t) = ∑ α log ( 1 - tt αα ) + log ξ(0) = log ξ(0) ∏ α ( 1 - t 2 α 2 ) (3) from which it follows that ξ(t) = ξ(0) ∏ α ( 1-t 2 /α 2 ).
Thus, questions related to entire function factorizations had already been foreshadowed in this Riemann work. Therefore, we now wish to outline the main points concerning the very early history of entire function factorization theorems, having taken the 1859 Riemann paper as an occasional starting point of this historical question, in which, inter alia, a particular entire function factorization -i.e. the (3) -has been used. In short, this 1859 Riemann paper has been a valuable καιρóς to begin to undertake one of the many study's branch which may depart from this milestone of the history of mathematics, to be precise that branch concerning the entire function theory which runs parallel to certain aspects of the theory of Riemann zeta function, with interesting meeting points with physics. One of the very few references which allude to these Riemann paper aspects is the article by W.F. Osgood in (Burkhardt et al. 1899-1927, Zweiter Teil, B.1.III, pp. 79-80), where, discussing of the gender of an entire function, an innite product expansion of the function sin πs/πs is considered; to be precise, since Johann Bernoulli to Euler, the following form " had already been deduced (see (Bellacchi 1894, Chapter XI))

sin πs πs = ∞ ∏ n=1 ( 1 - s 2 n 2 )
which has gender 0. Following [START_REF] Stopple | A Primer of Analytic Number Theory. From Pythagoras to Riemann[END_REF], Chapter 6, Section 6.1), Euler's idea is to write the function sin πx/πx as a product over its zeros, analogous " As n → ∞ and t ̸ = 0, Weierstrass proved to be

sin πt/πt = ∏ ∞ n=-∞ (1 -t/n)e t/n
-see (Bellacchi 1894, Chapter XI). But, according to (Bellacchi 1894, Chapter XI) and (Hancock 1910, Chapter I, Arts. 13, 14), Cauchy was the rst to have treated (in the Exercises de Mathématiques, IV) the subject of decomposition into prime factors of circular functions and related convergence questions, from a more general standpoint. Although Cauchy did not complete the theory, he however recognized that, if a is a root of an integral (or entire) transcendental function f (s), then it is necessary, in many cases, to join to the product of the innite number of factors such as (1 -s/a), a certain exponential factor e P (s) , where P (s) is a power series in positive powers of s. Weierstrass gave then a complete treatment of this subject. On the other hand, besides what has already been said above, also in [START_REF] Greenhill | The Applications of Elliptic Functions[END_REF], Chapter IX, Section 258)) it is pointed out that, since Abel's work, the innite product expansions of trigonometric functions have been formal models from which to draw inspiration, by analogy, for further generalizations or extensions.

to factoring a polynomial in terms of its roots. For example, if a quadratic polynomial f (x) = ax 2 + bx + c has roots α, β dierent from 0, then we can write f (x) = c(1 -x/α)(1 -x/β). On the other hand, sin πx = 0 when x = 0, ±1, ±2, ... and since sin πx/x = 1 -π 2 x 2 /6 + O(x 4 ), sin πx/πx x→0 → 1 and sin πx/πx = 0 when x = ±1, ±2, ..., Euler guessed that sin πx/πx had a factorization as a product

sin πx πx = ( 1 - x 1 )( 1 + x 1 )( 1 - x 2 )( 1 + x 2 ) ... = = ( 1 - x 2 1 )( 1 - x 2 4 )( 1 - x 2 9 )( 1 - x 2 16 ) ...
which will lead later to a valid proof of this factorization. Then, even in the context of the history of entire function factorization theorems, W.F. Osgood points out that already Riemann, just in his famous 1859 paper, had considered an entire function, the ξ(s), as a function of s 2 with gender 0, but without giving any rigorous prove of this fact, which will be done later by J. Hadamard.

In the next sections, when we will deepen the works of Hadamard and Pólya on the entire function theory related to Riemann zeta function, we also will try to clarify, as far as possible, these latter aspects of the 1859 Riemann paper which mainly constitute one of the central cores of the present work. Following [START_REF] Stopple | A Primer of Analytic Number Theory. From Pythagoras to Riemann[END_REF], Chapter 10, Section 10.1), it was Riemann to realize that a product formula for ξ(s) would have had a great signicance for the study of prime numbers. The rst rigorous proof of this product formula was due to Hadamard but, as himself remember, it took almost three decades before he reached to a satisfactory proof of it. Likewise, also H.M. Edwards (1974, Chapter 1, Sections 1.8-1.19) arms that the parts concerned with (2) are the most dicult portion of the 1859 Riemann's paper (see also [START_REF] Bottazzini | Hidden Harmony -Geometric Fantasies. The Rise of Complex Function Theory[END_REF], Chapter 5, Section 5.10)). Their goal is essentially to prove that ξ(s) can be expressed as an innite product, stating that [...] any polynomial p(t) can be expanded as a nite product p(t) = p(0) ∏ ρ (1 -t/ρ) where ρ ranges over the roots of the equation p(t) = 0 [except that the product formula for p(t) is slightly dierent if p(0) = 0]; hence the product formula (2) states that ξ(t) is like a polynomial of innite degree. Similarly, Euler thought of sin x as a polynomial of innite degree # when he conjectured, and nally proved, the formula sin

x = πx ∏ n∈N ( 1 -(x/n) 2 )
. On other hand, [...] ξ(t) is like a polynomial of innite degree, of which a nite number of its terms gives a very good approximation in any nite part of the # Following (Bottazzini & Gray 2013, Chapter 8, Section 8.5.1), amongst the functions that behave very like a polynomial, there is the Riemann ξ function. In this regards, see also what will be said in the next section 5 about Lee-Yang theorems and, in general, the theory concerning the location of the zeros of polynomials. An historical account of entire factorization theorems from Weierstrass on-ward. To begin, we wish to preliminarily follow the basic textbook on complex analysis of Giulio Vivanti (1859Vivanti ( -1949)), an Italian mathematician whose main research eld was into complex analysis, becoming an expert of the entire function theory. He wrote some notable treatises on entire, modular and polyhedral analytic functions: a rst edition of a prominent treatise on analytic functions appeared in 1901, under the title showing the great historical attention that he always put in drawing up his works. Therefore, he also was a valid historian of mathematics besides to be an able researcher (see (Janovitz & Mercanti 2008, Chapter 1) and references therein), so that his works are precious sources for historical studies, in our case as concerns entire functions. The above mentioned Vivanti's textbook on complex analysis has been one of the most inuential Italian treatises on the subject. It has also had wide international fame thanks to its German edition.

Roughly speaking, the transcendental entire functions may be formally considered as a generalization, in the complex eld, of polynomial functions. Following (Vivanti 1928, Sections 134-135), (Marku²evi£ 1988, Chapter VII) and [START_REF] Pierpont | Function of a complex variable[END_REF], Chapter VIII, Sections 127 and 140), the great analogy subsisting between these two last function classes suggested the search for an equal formal analogy between the corresponding main properties. To be precise, the main properties of polynomials concerned either with the existence of zeros (Gauss' theorem) and the linear factor decomposition of a polynomial, so that it was quite obvious trying to see whether these could be, in a certain way, extended to entire functions. As regards the Gauss' theorem, it was immediately realized that it couldn't subsist because of the simple counterexample given by the fundamental elementary transcendental function e x which does not have any zero in the whole of complex plane. On the other hand, just this last function will provide the basis for building up the most general entire function which is never zero, which has the general form e G(x) , where G(x) is an arbitrary entire function, and is said to be an exponential factor. Then, the next problem consisted in nding those entire functions having zeros and hence how it is possible to build up them from their zero set. In this regards, it is well-known that, if P (z) is an arbitrary non-zero polynomial with zeros z 1 , ..., z n ∈ C \ 0, having z = 0 as a zero with multiplicity λ (supposing λ = 0 if P (0) = 0), then we have the following well-known nite product factorization $

P (z) = Cz λ n ∏ j=1 ( 1 - z z j ) (4)
where C ∈ C \ 0 is a constant, so that a polynomial, except a constant factor, may be determined by its zeros. For transcendental entire functions, this last property is much more articulated respect to the polynomial case: indeed, whilst the indeterminacy for polynomials is given by a constant C, for transcendental entire functions it is larger and related to the presence of an exponential factor which is need to be added to warrant the convergence of innite product development. A great part of history of the approach and resolution of this last problem is the history of entire function factorization. Nevertheless, we also wish to report what says Giacomo Bellacchi (1894, Chapter XI, Section 98) about this last problem. To be precise, he states that Se a 1 , a 2 , a 3 , ..., a n , .... simboleggino le radici semplici di una funzione olomorfa f (z), ed il quoziente f (z) :

∏ (z -a n ) non si annulli per alcuna di esse, la sua derivata logaritmica ψ ′ (z) = f ′ (z)/f (z) - ∑ (1/(z -a n ))
è olomorfa in tutto il piano; moltiplicando i due membri per dz ed integrando, Cauchy giunse $ It is noteworthy the historical fact pointed out by Giuseppe Bagnera (1927, Chapter III, Section 12, Number 73), in agreement to what has been likewise said above, according to which already Cauchy himself had considered rst forms of innite product developments, after the Euler's work. Also Bagnera then, in this his work, quotes Betti's work on elliptic functions and related factorization theorems. Instead, it is quite strange that the Italian mathematician Giacomo Bellacchi (1838Bellacchi ( -1924) ) does not cite Betti, in his notable historical work on elliptic functions [START_REF] Bellacchi | Introduzione storica alla teoria delle funzioni ellittiche[END_REF]) in regards to entire function factorization theorems which are treated in the last chapter of this his work; this is also even more strange because Chapter XI of his book is centered around the 1851 Riemann dissertation on complex function theory, without quoting the already existed Italian translation just due to Betti. Furthermore, Bellacchi studied at the Scuola Normale Superiore of Pisa in the 1860s, for which it is impossible that he had not known Betti (see [START_REF] Maroni | Necrologio di Giacomo Bellacchi[END_REF]). On the other hand, also [START_REF] Loria | Storia delle matematiche dall'alba della civiltà al tramonto del secolo XIX, 2 a edizione riveduta e aggiornata[END_REF], Chapter XLIV, Section 741) refers that Weierstrass found inspiration for his factorization theorem, a result of uncommon importance according to Gino Loria, generalizing a previous Cauchy's formula: indeed, both Cauchy and Gauss are quoted at p. 120 of the 1879 French translation of the original 1876 Weierstrass paper. This, to further conrmation of what has been said above.

alla formula f (z) = Ce ψ(z) ∏ (1 -z/a n ), dove C è una costante [If a 1 ,
a 2 , a 3 , ..., a n , .... represent the simple roots of a holomorphic function f (z), and the ratio f (z) :

∏ (z -a n ) is not zero for each root, then its loga- rithmic derivative ψ ′ (z) = f ′ (z)/f (z) - ∑ (1/(z -a n ))
is holomorphic in the whole of plane; multiplying both sides by dz and integrating, Cauchy reached the formula f (z) = Ce ψ(z) ∏

(1 -z/a n ), where C is a constant], so that it seems, according to Bellacchi, that already Cauchy had descried the utility of exponentials as convergence-producing factors, in a series of his papers published in the Tome XVII of the Comptes Rendus de l'Académie des Sciences (France); this supposition is also conrmed by Hancock (1910, Chapter I, Art. 14). Nevertheless, following (Vivanti 1928, Sections 135-141), the rise of the rst explicit formulation of the entire function factorization theorem was given by Weierstrass in 1876 (see [START_REF] Weierstrass | Zur Theorie der eindeutigen analytischen Funktionen[END_REF]) and was mainly motivated by the purpose to give a solution to the latter formal problem, concerning the convergence of the innite product development of a transcendental entire function f (z) having an innite number of zeros, namely z = 0, with multiplicity λ, and z 1 , ..., z n , ... such that 0 < |z j | ≤ |z j+1 |, z j ̸ = z j+1 j = 1, 2, ..., trying to extend the case related to a nite number of zeros z 1 , ..., z n , in which such a factorization is given by

f (z) = e g(z) z λ n ∏ j=1 ( 1 - z z j ) , (5)
to the case of innite zeros, reasoning, by analogy, as follows. The set of innite zeros z j is a countable set having only one accumulation point, that at innite. Therefore, for every innite increasing natural number sequence {ρ i } i∈N , it will be always possible to arrange the zeros z j according to their modulus in such a manner to have the following non-decreasing sequence

|z 1 | ≤ |z 2 | ≤ ... with lim n→∞ |z n | = ∞.
In such a case, if one wants, by analogy, to extend (5) as follows

f (z) = e g(z) z λ ∞ ∏ j=1 ( 1 - z z j ) , (6)
then it will not be possible to fully avoid divergence's problems inherent to the related innite product. The rst hint towards a possible overcoming of these diculties, was suggested to Weierstrass (see (Weierstrass 1856a)) by looking at the form of the inverse of the Euler integral of the second kind -that is to say, the gamma function -and given by

1 Γ(z) = z ∞ ∏ j=1 ( 1 + z j )( j 1 + j ) z = z ∞ ∏ j=1 ( 1 + z j ) e -z log j+1 j , (7)
from which he descried the possible utility of the exponential factors there involved to, as the saying goes, force the convergence of the innite product of the last equality; these his ideas concretized only in 1876 with the explicit formulation of his celebrated theorem on the entire function factorization.

As we have said above, Weierstrass (1856a) attributes, however, the innite product expansion (7) to Gauss, but some next historical studies attribute to Euler this formula, that he gave in the famous 1748 Introductio in Analysin Innitorum. Indeed, as has been said above, from the 1879 French translation of the original 1876 Weierstrass paper, it turns out that both Cauchy and Gauss are quoted (at page 120), before to introduce the primary factors. Nevertheless, P. Ullrich (1989, Section 3.5) says that the real motivation to these Weierstrass' results about entire function factorization were mainly due to attempts to characterize the factorization of quotients of meromorphic functions on the basis of their zero sets, rather than to solve the above problem related to the factorization of a polynomial in dependence on its zeros. Furthermore, Ullrich (1989, Section 3.5) observes too that other mathematicians dealt with questions concerning entire function factorization methods, amongst whom are just Enrico Betti and Bernard Riemann, the latter, in his important 1861 sommersemester lectures on analytic functions, arguing, as has already been said, upon the construction of particular complex functions with simple zeros, even if, all things considered, he didn't give, according to Ullrich (1989, Section 3.5), nothing more what Euler done about gamma function through 1729 to his celebrated 1748 treatise on innitesimal analysis % . Instead, as we have seen above, D. Laugwitz (1999, Chapter 1, Section 1.1.6) states that Riemann's work on meromorphic functions was ahead of the Weierstrass' one, having been carried out with originality and simplicity. To this point, for our purposes, it would be of a certain importance to deepen the possible relationships between Riemann and Weierstrass, besides to what has been said above: for instance, in this regards, Laugwitz (1999, Chapter 1, Section 1.1.5) says that Riemann was aware of the Weierstrass' works until 1856-57, in connection with the composition of his paper on Abelian functions, in agreement with what has been said in the previous sections. Again according to [START_REF] Laugwitz | Turning Points in the Conception of Mathematics[END_REF], Chapter 1, Section 1.1.6), one of the key themes of Riemann's work on complex function theory was the determination of a function from its singularities which, in turn, implies the approach of another problem, the one concerning the determination of a function from its zeros. In this regards, Riemann limited himself to consider the question to determine a function with innitely many zeros whose only point of accumulation is ∞. What he is after is the product representation later named after Weierstrass. Riemann uses a special case to explain the % Following [START_REF] Lunts | The analytic work of N.I. Lobachevskii[END_REF]), (Marku²eci£ 1988, Chapter VII) and references therein, also Loba£evskij, since 1830s, made some notable studies on gamma function which preempted times.

general procedure. He does it in such a way that by following his direction one could immediately give a proof of the Weierstrass product theorem. Therefore, it would be hoped a deeper study of these 1857-61 Riemann's lectures on complex function theory to historically clarify this last question which is inside the wider historical framework concerning the work of Riemann in complex function theory.

Furthermore, to this point, there seems not irrelevant to further highlight, although in a very sketchily manner, some of the main aspects of the history of gamma function. To this end, we follow the as many notable work of Reinhold [START_REF] Remmert | Classical Topics in Complex Function Theory[END_REF] which, besides to mainly be an important textbook on some advanced complex analysis topics, it is also a very valuable historical source on the subject, which seems to remember the style of the above mentioned Vivanti's textbook whose German edition, on the other hand, has always been a constant reference point in drawing up the Remmert's textbook itself. Following (Remmert 1998, Chapter 2), the early origins of gamma function should be searched into the attempts to extend the function n! to real arguments. Euler was the rst to approach this problem since 1729, giving a rst expression of this function, in a letter to Goldbach (see [START_REF] Whittaker | A Course in Modern Analysis[END_REF], Chapter XII, Section 12.1)), providing a rst innite product expression of this new function, but only for real values. Gauss, who did not know Euler work, also taking into account Newton's work on interpolation (see (Schering 1881, Sections XI and XII)), in 1811 considered as well complex values, denoting such a new function with Π, while it was Legendre, in 1814, to introduce a unied notation both for Euler and Gauss functions, denoting these latter with Γ(z) and speaking, since then, of gamma function. Afterwards, in 1854, Weierstrass began to consider an Euler innite product expansion of the function 1/Γ(z), that he denoted with F c(z) and is given by 1/Γ(z) = ze γz ∏ ∞ j=1 (1 + z/j)e -z/j , where γ is the well-known Euler-Mascheroni constant & , from which he maybe recognized, for the rst time, the importance of the use of exponential factors as innite product convergence-producing elements. However, according to [START_REF] Whittaker | A Course in Modern Analysis[END_REF], Chapter XII, Section 12.1), the formula (7) had already been obtained by F.W. [START_REF] Newman | On Γa especially when a is negative[END_REF] starting from Euler's expression of gamma function given by ( 7).

Following (Vivanti 1928, Section 135-141), [START_REF] Remmert | Classical Topics in Complex Function Theory[END_REF], Chapter 3) and, above all, (Bottazzini & Gray 2013, Section 6.7), Weierstrass extended the product (5) in such a manner to try to avoid divergence problems with the ad hoc introduction, into the product expansion, of certain forcing convergence factors. This attempt was successfully attended, since 1874, as a solution to a particular question -the one which may be roughly summarized as the attempt to build up an entire transcendental function with prescribed zeros -which arose within the general Weierstrass' intent to solve the wider problem to nd a representation for a single-valued function as a quotient of two convergent power series. To be precise, he reached, amongst other things, the following main result Given a countable set of non-zero complex points z 1 , z 2 , ..., such that 0 < |z 1 | ≤ |z 2 | ≤ ... with lim n→∞ |z n | = ∞, then it is possible to nd, in innite manners, a non-decreasing sequence of natural numbers p 1 , p 2 , ... such that the series ∑ ∞ j=1 |z/z j | p j +1 be convergent for every nite value of z, in such a manner that the most general entire function which is zero, with their own multiplicity, in the points z 1 , z 2 , ..., and has a zero of order λ in the origin, is given by '

f (z) = e g(z) z λ ∞ ∏ j=1 ( 1 - z z j ) E j (z) (8)
where

E j (z) = (1 -z)( ∑ j i=1 z i /i) for j ≥ 1 and E 0 (z) = 1 -z, g(z)
being an arbitrary entire function, and the innite product is absolutely convergent for each nite value of |z|.

The factors E j (z) will be later called Weierstrass' factors, whilst the numbers p j will be called convergence exponents. The sequence E j (z) j∈N 0 plays a very fundamental role in the Weierstrass' theorem: from the equation

1 -z = exp(log(1 -z)) = exp(- ∑ i≥1 z i /i), (9)
Weierstrass obtained the formula E j (z) = exp(-∑ i>j z j /j) in proving convergence properties which, on the other hand, would have been easier obtained by means of the following estimates

|E j (z) -1| ≤ |z| j+1 , ∀j ∈ N 0 , ∀z ∈ C, |z| ≤ 1 (10)
that have been proved only later. Amongst the rst ones to have made this, seems there having been L. Fejér (see [START_REF] Hille | Analytic Function Theory[END_REF], Section 8.7)), but the argument appears as early as 1903 in a paper of Luciano [START_REF] Orlando | Sullo sviluppo della funzione (1-z)e (z+z 2 /2+...+z p-1 /(p-1)!) , Giornale di Matematiche di Battaglini per il progresso degli studi nelle università italiane[END_REF] which starts from Weierstrass' theorem as treated by Borel's monograph on entire functions. As has already been said above, Weierstrass was led to develop his ' Historically, in relation to (8), the function f (z) was usually denoted, d'après Weierstrass, by G(z), whilst z λ ∏ ∞ j=1 (1 -z/z j )E j (z) was named canonical (or primitive) function -see [START_REF] Sansone | Lezioni sulla teoria delle funzioni di una variabile complessa[END_REF], Chapter IV, Section 3), where there are too many interesting historical notes.

Luciano Orlando (1887Orlando ( -1915) ) was an Italian mathematician prematurely died in the First World War -see the very brief obituary (Marcolongo 1918) as well as (Rouse Ball 1937, Appendix II, pp. 430-431). His supervisors were G. Bagnera and R. Marcolongo who led him to make researches in algebraic integral equation theory and mathematical physics.

theory by the chief objective to establish the general expression for all analytic functions meromorphic in C except in nitely many points, reaching the scope after a series of previous futile attempts only in 1876, with notable results, spelt out in [START_REF] Weierstrass | Zur Theorie der eindeutigen analytischen Funktionen[END_REF], concerning the class of transcendental entire functions. But what was new and sensational in the Weierstrass' construction was just the introduction and the application of the so-called convergenceproducing factors (or primary factors or Weierstrass' factors) which strangely have no inuence on the behavior and distribution of the zeros. Towards the theory of entire functions. In his necrology to Weierstrass, Poincaré (1899, Section 6) said that Weierstrass' major contribution to the development of function theory was just the discovery of primary factors. Also Hermite was, in a certain sense, astonished and intrigued from the introduction of this new Weierstrass' notion of prime factor, which he considered of capital importance in analysis and making later notable studies in this direction; he also suggested to Èmile Picard to do a French translation of the original 1876 Weierstrass' work, so opening a French research trend on this area. En passant, we also point out the fact that, from the notion of prime factor and from the convergence of the innite product ∏ j∈N E j (z/a j ), representing an entire transcendental function vanishing, in a prescribed way, in each a j , Hilbert drew inspiration to formulate his valuable algebraic notion of prime ideal . Following (Pincherle 1922, Chapter IX, Section 137), (Vivanti 1928, Section 136), (Burckel 1979, Chapter XI), (Remmert 1998, Chapters 3 and 6), (Ullrich 1989, Section 3.5) and (Bottazzini & Gray 2013, Sections 5.11.5 and 6.7), since the late 1850s, Enrico Betti had already reached notable results, about convergence properties of innite products of the type (6), very near to the Weierstrass' ones related to the resolution of a fundamental problem of entire function theory, the so-called Weierstrass' problem (see (Pincherle 1922, Chapter IX, Section 137)). Betti exposed these outcomes in his celebrated 1859-60 Pisa lectures on advanced analysis entitled La teorica delle funzioni ellittiche (see (Betti 1903(Betti -1913, Tomo I, XXII)), published in the Tomes III and Usually, the notion of prime ideal of the commutative algebra, with related operations, would want to be stemmed from the factorization of natural numbers.

Following [START_REF] Forsyth | Theory of Functions of a Complex Variable[END_REF], Chapter V, Section 50) and (Bottazzini & Gray 2013, Section 4.2.3.2), in relation to the innite product expression of an entire transcendental function prior to 1876 Weierstrass' paper, attention should be also paid to a previous 1845 work of A. Cayley on doubly periodic functions. Furthermore, following (Tannery & Molk 1893, Section 85), into some previous 1847 works of G. Eisenstein on elliptic functions, some notable problems having to do with the construction of analytic functions with prescribed zeros as a quotient of entire functions with the involvement of certain transcendental entire functions of exponential type (similar to the Weierstrass problem as historically related to meromorphic functions), had already been considered. See also certain function's quotients stemmed from the developments of certain determinants given in [START_REF] Gordan | Über den grössten gemeinsame Factor[END_REF]. In any case, all these historical considerations conrm, once again, that the prolegomena of entire function factorization theorems should be searched in the general history of elliptic functions.

IV of the Annali di matematica pura ed applicata, Series I, after having published, in the Tome II of these Annali, an Italian translation of the celebrated 1851 Riemann's inaugural dissertation on complex function theory, which can be considered as an introduction to his next lectures on elliptic functions. Indeed, in these latter, Betti before all places an Introduction on the general principles on complex functions, essentially based on these 1851 Riemann lectures. From the point 3. onward of this Introduction, Betti starts to deal with entire functions, their nite and innite zeros (there called roots), as well as on possible quotients between them. In particular, taking into account what is said in [START_REF] Briot | Théorie des functions doublement périodiques et, en particulier, des fonctions elliptiques[END_REF], he considered innite products of the type ρ (1 -z/ρ), where ρ are the zeros of an entire function, with the introduction of a factor of the type e w , where w is an arbitrary entire function, to make convergent this innite product. Furthermore, Betti dealt with this type of innite products starting to consider innite product representations of the following particular function es(z) = z ∏ ∞ m=1 (m/(m + 1)) z (1 + z/m), which satises some functional equations and veries the relation Γ(z) = 1/es(z). Therefore, as Weierstrass too will do later, Betti started from the consideration of the innite product expansion of the inverse of the gamma function for studying the factorization of entire functions. Therefore, Betti guessed the utility of the convergence factors having exponential form, looking at the innite product expansion of Gamma function, similarly to what Weierstrass will do. Afterwards, Betti proved some theorems which can be considered particular cases of the next Weierstrass' results, concluding arming that Da questi teoremi si deduce che le funzioni intere potranno decomporsi in un numero innito di fattori di primo grado ed esponenziali, e qui comparisce una prima divisione delle funzioni intere. Quelle che hanno gl'indici delle radici in linea retta, e quelle che le hanno disposte comunque nel piano; le prime, che sono espresse da un prodotto semplicemente innito, le chiameremo di prima classe, le seconde, che sono espresse da un prodotto doppiamente innito, le diremo di seconda classe. Le funzioni di prima classe si dividono anch'esse in due specie, la prima, che comprende quelle che hanno gl'indici delle radici disposti simmetricamente rispetto a un punto, e che possono esprimersi per un prodotto innito di fattori di primo grado, le altre, che hanno gl'indici delle radici disposti comunque sopra la retta, le quali si decomporranno in fattori di primo grado ed esponenziali. Ogni funzione intera di prima classe della prima specie potrà decomporsi nel prodotto di più funzioni intere della stessa classe di seconda specie, e data una funzione della seconda specie se ne potrà sempre trovare un'altra che moltiplicata per la medesima dia per prodotto una funzione della prima specie. Le funzioni di seconda classe si dividono anch'esse in due specie; la prima comprenderà quelle che hanno gl'indici delle radici disposti egualmente nei quattro angoli di due assi ortogonali, in modo che facendo una rotazione intorno all'origine di un quarto di circolo, gl'indici di tutte le radici vengano a sovrapporsi, le quali funzioni possono esprimersi per un prodotto doppiamente innito di fattori di primo grado; la seconda comprenderà quelle che hanno gl'indici disposti comunque, e si decompongono in un prodotto doppiamente innito di fattori di primo grado e di fattori esponenziali. Data una funzione della seconda specie se ne potrà sempre trovare un'altra che moltiplicata per quella dia una funzione della prima specie.

[From these theorems, we deduce that entire functions might be decomposed into an innite number of rst degree factors and exponential factors, so that here there is a rst classication of entire functions according to that their root's indexes lie along a line or are arbitrarily placed in the plane; the former are said to be of rst class and are expressed by a simply innite product, while the latter are said to be of second class and are expressed by a doubly innite product. The functions of the rst class are, in turn, classied into two kinds: the rst one comprises those functions having the root's indexes symmetrically placed respect to a point and that can be expressed by an innite product of rst degree factors; the second one comprises those functions having root's indexes arbitrarily placed along a line and that can be expressed by an innite product both of rst degree factors and of exponential factors. Each entire function of rst class and of rst kind might be decomposed into the product of other entire functions of the same class and of the second kind; furthermore, given a function of the second kind, it is always possible to nd another function that multiplied by the former, the product gives rise to another function of the rst kind. Likewise, the functions of the second class are divided into two kinds: the rst one comprises those functions having the root's indexes equally placed into the four angles of the two orthogonal cartesian axes in such a manner that all these are overlapped through a π/2 radian rotation around the origin, and are decomposable into a doubly innite product of rst degree factors; the second one includes those functions having the root's indexes arbitrarily placed and that are decomposable into a doubly innite product of rst degree factors and exponential factors. Furthermore, given a function of the second kind, it is always possible to nd another function that multiplied by the former, the product gives rise to a function of rst kind].

Then, Betti carries on treating entire functions in the rst part of his lessons on elliptic functions, followed by a second part devoted to quotients of functions, mentioning either the paper (Weierstrass 1856a) and the paper (Weierstrass 1856b). Therefore, Betti's work on entire function factorization, made in the period 1860-63, was very forerunner of the Weierstrass' one: this is conrmed either by (Rouse Ball 1937, Appendix II, pp. 376-384)) and by (Federigo Enriques 1982, Book III, Chapter I, Section 6), in which it is pointed out that the fundamental Weierstrass' theorem on the factorization of entire transcendental functions from their zeros, had already been discovered by Betti, highlighting however as the Pisa's mathematician, with singular personal disinterestedness, wanted not claim it as due to him. Indeed, following Francesco Cecioni's comments about some works of Ulisse Dini (see [START_REF] Dini | Opere, a cura dell[END_REF], Volume II)), it turns out that Betti's work could easily reach, only with very slight modications, the same generality and abstraction of the Weierstrass' one, as Dini explicitly proved in [START_REF] Dini | Alcuni teoremi sulle funzioni di una variabile complessa[END_REF]; furthermore, Dini proved too that Betti's work could be able to give a particular case, given in the years 1876-77, of the general Gösta Mittag-Leer theorem -see (Mittag-Leer 1884), (Vivanti 1928, Section 145), [START_REF] Loria | Storia delle matematiche dall'alba della civiltà al tramonto del secolo XIX, 2 a edizione riveduta e aggiornata[END_REF], Chapter XLIV, Section 752) and (Bottazzini & Gray 2013, Section 6.7.6) -independently by what Weierstrass himself was doing in the same period, in regards to these latter arguments. Cecioni says that this Dini's work had already been worked out since 1880, whilst the Weierstrass' theorem was published in 1876 -see [START_REF] Weierstrass | Zur Theorie der eindeutigen analytischen Funktionen[END_REF]. Thus, much before, namely in 1860, Betti had proved, as we have already said, a particular but important case of this theorem, albeit he didn't go beyond, because the results achieved by him were enough to his pragmatic scopes concerning Abelian and elliptic functions ! , and, as also Pincherle (1922, Chapter IX, Section 135) has claimed, the Weierstrass' method was essentially the same of the Betti's one with slight modications. In the years 1876-77, also G. Mittag-Leer proved a particular case of a more general theorem that he will give later, to be precise in 1884, after a long series of previous works in which he gradually, through particular cases, reached the general form of this his theorem as nowadays we know it. In the meanwhile, Weierstrass reconsidered Mittag-Leer's works, since the early 1880s, in relation to what himself have done on the same subject. Also F. [START_REF] Casorati | Aggiunte a recenti lavori dei sig. Weierstrass e Mittag-Leer sulle funzioni di una variabile complessa[END_REF] had some interesting ideas similar to the Mittag-Leer's ones, giving further contributions to the subject (see (Loria 1950, Chapter XLIV, Section 750)). Almost at the same time, amongst others, Ernst Schering (1881), Charles Hermite (1881), Émile Picard (1881), Felice Casorati (1882), Ulisse Dini (1881), Paolo Gazzaniga " (1882) and Claude ! In this regards, also Salvatore Pincherle (1899, Chapter IX, Section 175) reports that Betti solved the Weierstrass' problem in a quite general case.

" Some historical sources refer of Paolo Cazzaniga, whereas others refer of Paolo Gazzaniga, but, very likely, they are the same person, that is to say, Paolo Gazzaniga (1853Gazzaniga ( -1930)), an Italian mathematician graduated from Pavia University in 1878 under the supervision of Felice Casorati. In the years 1878-1883, he was interim assistant professor at Pavia, then he spent a period of study in Germany under the Weierstrass and Kronecker supervision. Afterwards, from 1888, he became professor at the high school Tito Livio in Padua, teaching too in the local University. He was also one of the most inuential teachers of Tullio Levi-Civita during his high school studied. Gazzaniga's researches mainly concerned with applied algebra and number theory. Furthermore, Paolo Gazzaniga has to be distinguished from Tito Camillo Cazzaniga (1872-1900), a prematurely died Italian mathematician, graduated from Pavia University in 1896, whose researches concerned with matrix theory and analytic functions according to the research trend of Ernesto Pascal (1865Pascal ( -1940) ) during his teaching in Pavia. Both Tito Cazzaniga (see (Rouse Ball 1937, Appendix II, pp. 412-413)) [START_REF] Guichard | Sur les fonctions entières[END_REF], achieved notable results about the general problem to build up a complex function with prescribed singularities, although related to a generality degree less than that of the Mittag-Leer results. Thus, the history of the Mittag-Leer theorem makes too its awesome appearance within the general history of meromorphic functions, a part of which may be retraced in the same Mittag-Leer 1884 paper in which, amongst other things, also the 1881 work of Ulisse Dini is quoted. However, both Schering (1881, Section XVI) and Casorati (1880-82, p. 269, footnote (***)), in discussing the above mentioned Mittag-Leer results, quote Betti's work on Weierstrass' theorem; in particular, the former speaks of Betti's convergence factors and the latter states that Anche il sig. Dini, nella sua Nota sopra citata, dimostra questo teorema, riducendo lo studio del prodotto innito a quello della serie dei logaritmi dei fattori; riduzione di cui s'era già valso felicemente, per il caso di distribuzione degli zeri a distanze non mai minori di una quantità ssa d, il sig. Betti nella Introduzione della sua Monograa delle funzioni ellittiche (Annali di Matematica, Tomo III, Roma, 1860), dove precede assai più oltre di Gauss nella via che mena al teorema del sig. Weierstrass.

[Also Mr. Dini, in his Note of above, proves this theorem, reducing the study of the innite product to the study of the series of the logarithms of the factors; reduction, this, that had already been used by Mr. Betti in the Introduction to his monograph on elliptic functions (Annali di Matematica, Tome III, Rome, 1860) for the case of a distribution of zeros having reciprocal distances not less than a xed quantity d; in doing so, he much foregoes Gauss in a fashion which leads to the theorem of Mr. Weierstrass.] Therefore, from the Mittag-Leer's works onward, together to all those works made by other mathematicians amongst whom are Dini, Schering, Casorati, Hermite, Picard, Cazzaniga, Guichard and Weierstrass himself, it starts the theory of entire transcendental functions whose early historical lines have been traced in the previous sections. In any case, with Mittag-Leer, we have the most general theorems for the construction, by innite products, of a meromorphic function with prescribed singularities. To be precise, following (Gonchar et al. 1997, Part I, Introduction) and (Vivanti 1901, Section 215), the works by Weierstrass, Mittag-Leer and Picard, dating back to the 1870s, marked the beginning of the systematic studies of the theory of entire and meromorphic functions. The Weierstrass and Mittag-Leer theorems gave a general description of the structure of entire and meromorphic functions, while the representation of entire functions as an innite product à la Weierstrass, served as basis for studying properties of entire and meromorphic functions. The Picard theand Paolo Gazzaniga are quoted in [START_REF] Vivanti | Teoria delle funzioni analitiche, Milano, IT: Ulrico Hoepli Editore-Librajo della Real Casa (with a[END_REF] but not in [START_REF] Vivanti | Teoria delle funzioni analitiche, Milano, IT: Ulrico Hoepli Editore-Librajo della Real Casa (with a[END_REF].

orem opened the theory of value distribution of meromorphic functions, while the J.L.W.V. Jensen works, at the late 1890s, were of a great importance for further developments of the theory of entire and meromorphic functions which started, in the same period, to gradually become a separate and autonomous mathematical discipline after the pioneering works of Laguerre, Hadamard, Poincaré, Lindelö and Borel, up until the Rolf Nevanlinna work of the early 1900s. All that will be in-depth studied in the next section, where we shall deal with the main lines of the history of entire and meromorphic functions whose theory basically starts just from the entire function factorization theorems.

En passant, we also note that the Weierstrass' entire function factorization theorem has had further remarkable applications in many other pure and applied mathematical contexts. In this place, we wish to point out another possible interesting historical connection. To be precise, following (Marku²e-vi£ 1967, Volume II, Chapters 8 and 9), (Burckel 1979, Chapter VII) and (Remmert 1998, Chapter 4), a very similar problem to that considered by Weierstrass was the one considered in (Marku²evi£ 1967, Volume II, Chapter 8, Theorem 8.5) where, roughly, a bounded analytic function with prescribed zeros is constructed by means of certain innite products introduced by Wilhelm [START_REF] Blaschke | Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen[END_REF] in relation to questions related to the Giuseppe Vitali convergence theorem for sequences of holomorphic functions, and dened upon those complex numbers assigned as given zeros of that function that has to be determined. We shall return later on such aspects concerning Blaschke products, which form a special class of Weierstrass' products, when we will go on in deepening the history of entire function theory; due to this, to our historical purposes, we would want to try to analyze whether the previous Weierstrass' work on entire function factorization theorems have played a certain role in the Blaschke's work.

  is nite for all nite values of t, and allows itself to be developed in powers of tt as a very rapidly converging series. Since, for a value of s whose real part is greater than 1, log ζ

	mann stated the following function
	ξ(t)	. =	( 1 2	Γ	( s 2	)	s(s -1)π -s/2 ζ(s)	)	s=1/2+it	(1)
	later called Riemann ξ-function. It is an entire function. Riemann conjectured
	that (ξ(t) = 0) ⇒ (ℑ(t) = 0), that is to say, the famous Riemann hypothesis
	(RH), as it will be called later. Whereupon, he stated that
	This function ξ(t)
										! , Rie-
	! This paper was presented by Riemann, after his nomination as full professor in July

  plane.[...] Hadamard (in 1893) proved necessary and sucient conditions for

	the validity of the product formula ξ(t) = ξ(0) argument by which Riemann went from the one to the other are obscure, to ρ (1 -t/ρ) but the steps of the ∏
	say the very least.
	The last sentence of this Edwards' quotation is historically quite interesting
	and would deserve further attention and investigation. Furthermore, H.M. Ed-
	wards states too that
	Sections 60-64) claimed this last aspect, pointing out, in particular, on the Rie-
	mann's ansätz according to which the entire function ξ(t) is equal, via (3), to
	the Weierstrass' innite product of primary factors without any exponential
	factor. As is well-known, this basic question will be brilliantly solved by J.
	Hadamard in his famous 1892 paper that, inter alia, will mark a crucial mo-

[...] a recurrent theme in Riemann's work is the global characterization of analytic functions by their singularities. See, for example, the Inauguraldissertation, especially Article 20 of

or Part 3 of the introduction to the Riemann article Theorie der Abel'schen Functionen, which is entitled Determination of a function of a complex variable by boundary values and singularities. See also Riemann's introduction to Paper XI of the his collected works, where he writes about [...] our method, which is based on the determination of functions by means of their singularities (Umtetigkeiten und Unendlichwerden)

[...]

. Finally, see the textbook of

[START_REF] Ahlfors | Complex Analysis. An Introduction to the Theory of Analytic Function of One Complex Variable, Third Edition[END_REF]

, namely the section 4.5 of Chapter 8, entitled Riemann's Point of View, according to which Riemann was therefore a strong proponent of the idea that an analytic function can be dened by its singularities and general properties, just as well as or perhaps better than through an explicit expression, in this regards showing, with Riemann, that the solutions of a hypergeometric dierential equation can be characterized by properties of this type. In short, all this strongly suggests us the need for a deeper re-analysis of Riemann ÷uvre concerning these latter arguments, as well as a historical seek for the mathematical background which was at the origins of his celebrated 1859 number theory paper. From what has just been said, it turns out clear that a look at the history of entire function theory, within the general and wider complex function theory framework, is needed to clarify some of the historical aspects of this inuential seminal paper which, as Riemann himself recognized, presented some obscure points. In this regards, also Gabriele

Torelli (1901, Chapter VIII, ment

in the history of entire function theory (see

[START_REF] Maz'ya | Jacques Hadamard[END_REF]

, Chapter 9, Section 9.2) and next sections).

  Teoria delle funzioni analitiche, published by Ulrico Hoepli in Milan, where the rst elements of the theory of analytic functions, worked out in the late 19th-century quarter, are masterfully exposed into three main parts, giving a certain load to the Weierstrass' approach respect to the Cauchy's and Riemann's ones. The importance of this work immediately arose, so that a German edition was carried out, in collaboration with A. Gutzmer, and published in 1906 by B.G. Teubner in Leipzig, under the title Theorie der eindeutigen analytischen funktionen. Umarbeitung unter mitwirkung des verfassers deutsch herausgegeben von A. Gutzmer, which had to be considered as a kind of second enlarged and revised edition of the 1901 rst Italian edition according to what Vivanti himself said in the preface to the 1928 second Italian edition, entitled Elementi della teoria delle funzioni analitiche e delle trascendenti intere, again published by Ulrico Hoepli in Milan, and written following the above German edition in which many new and further arguments and results were added, also as regards entire functions. Almost all the Vivanti's treatises are characterized by the presence of a detailed and complete bibliographical account of the related literature, this

1859, to the Berlin Academy for his consequent election as a corresponding member of this

& See[START_REF] Pepe | Peoples' Friendship University of Russia[END_REF]) for a contextual brief history of the Euler-Mascheroni constant.On some historical aspects of Riemann zeta function, 3